Lecture 1 | Modern Physics: Special Relativity (Stanford)



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Lecture 1 of Leonard Susskind’s Modern Physics course concentrating on Special Relativity. Recorded April 14, 2008 at Stanford University.

This Stanford Continuing Studies course is the third of a six-quarter sequence of classes exploring the essential theoretical foundations of modern physics. The topics covered in this course focus on classical mechanics. Leonard Susskind is the Felix Bloch Professor of Physics at Stanford University.

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this program is brought to you by Stanford University please visit us at stanford.edu this quarter we're going to learn about field theory classical field theory fields such as the electromagnetic field gravitational field other fields in nature which I won't name right now propagate which means they change according to rules which give them a wave-like character moving through space and one of the fundamental principles of field theory in fact more broadly nature in general is the principle of relativity the principle the special printless the the principle of special relativity in this particular case the principle of special relativity well let's just call it the principle of relativity goes way back there was not an invention of Einstein's I'm not absolutely sure when it was first announced or articulated in the form which I'll spell it out I don't know whether it was Galileo or Newton or those who came after them but those early pioneers certainly had the right idea it begins with the idea of an inertial reference frame now inertia reference frame this is something a bit tautological about an inertial reference frame Newton's equations F equals MA are satisfied in an inertial reference frame what is an inertial reference frame it's a frame of reference in which Newton's equations are satisfied I'm not going to explain any further what an inertial reference frame is except to say that the idea of an inertial reference frame is by no means unique a reference frame first of all was a reference frame in tale of a reference frame first of all entails a set of coordinate axes in ordinary space X Y & Z and you know how to think about those but it also entails the idea that the coordinate system may be moving or not moving relative to whom relative to whomever we sitting here or you sitting here in this classroom here define a frame of reference we can pick the vertical direction to be the z axis the horizontal direction along my arms here to be the x axis X plus that way X my X is minus in that direction and which one have I left out I've left out the y axis which points toward you from me so there are some coordinate axes for space XY and Z and I didn't this in addition to specify a frame of reference one also imagines that this entire coordinate system is moving in some way relative to you sitting there presumably with a uniform velocity in a definite direction if your frame of reference is an inertial frame of reference in other words if when you throw balls around or juggle or do whatever is supposed to do in an inertial frame of reference if you find yourself in an inertial frame of reference then every other frame of reference that's moving with uniform velocity relative to you now remember what uniform velocity means it doesn't just mean with uniform speed it means with uniform speed in an unchanging direction such a frame of reference is also inertial if it's accelerated or if it starts standing still and then suddenly picks up some speed then it's not an inertial frame of reference all inertial frames of reference according to Newton and also I think also Galileo Galileo was often credited with the idea but I never read enough of Galileo to know whether he actually had it or not neither did I read enough of Newtons they both wrote in languages that I don't understand what was I saying oh yes right according to both Newton and anybody else who thought about it very hard the laws of physics are the same in all inertial reference frames laws of physics meaning F equals MA the forces between objects all the things that we would normally call laws of nature or laws of physics don't distinguish between one frame of reference of and another if you want a kind of pictorial example that I like to use a lot when I'm explaining this to the children or to grownups I like to think about the laws of juggling there are very definite procedures that you train your body to do uh in order to be able to juggle balls correctly now you can imagine yourself being in a railroad car moving with perfectly uniform velocity down the x axis and trying to juggle do you have to compensate for the fact that the train is moving and for particular when you throw a ball up into the air that you have to reach over to the right to compensate for the fact that the train is moving to the left my left your right the answer is no you don't the laws of juggling are the same in every reference frame and every inertial reference frame whatever you do in one reference frame you do exactly the same thing and you'll succeed or fail depending on whether you're a good juggler or not but it will not depend on whether you're moving with uniform velocity so the laws of juggling are the same in every inertial reference frame the laws of mechanics are the same in every inertial reference frame the laws Newtonian laws of gravity are the same in every inertial frame according to Newton what about the laws of electrical phenomena well there there was a clash the clash had to do with Maxwell's equations Maxwell's equations were the field equations the field theory that governed the electromagnetic field and the way that it propagated and sent waves electromagnetic waves that we ordinarily call light or radio waves or so forth and the fundamental dilemma as you all know I'm sure you all know the fundamental dilemma was both according to well here was the dilemma Maxwell's equations said light moves with a certain velocity if you take the various constants that appear in Maxwell's equations and put them together in the right way you get the velocity of waves moving down an axis and that velocity comes out to be a certain number out of Maxwell's equations you have two choices one is to believe that Maxwell's equations are true laws of nature as good as any other laws of nature in which case the principle of relativity says they should be the same in every reference frame but if it follows from Maxwell's equations that the speed of light is three times ten to the eighth meters per second which is about what it is if it follows from Maxwell's equations that light moves that fast and if Maxwell's equations are laws of physics fundamental laws of physics and if the laws of physics are the same in every reference frame then the speed of light must be the same in every reference frame but that was very hard to swallow because if a light beam is going down that axis and you chase it and run along with it that lets say three-quarters of the speed of light then you want to see that light ray moving much more slowly than three times ten to the eighth meters per second relative to you on the other hand the light ray going in the other direction since you're sort of running into it you should see going even faster so all these possibilities could not simultaneously be correct that the laws of nature are the same in every reference frame and that Maxwell's equations are laws of physics in the same sense that Newton's laws of physics namely the same in every reference frame something had to give well the point was of course that they were good laws of nature and that they were the same in every reference frame the thing that had to give is our concepts of velocity space and time and how we measure velocity especially velocities were up which are up near the speed of light now I'm not going to spend the full amount of time that I did previously on the special theory of relativity that can be found on lectures from how long ago and there on the Internet I believe relativity and electromagnetism I think that was maybe about three quarters ago I've lost track yeah they're up there they're on the net and they're the lectures on relativity special relativity and electromagnetic theory we're just going to cut through it real fast we're going to cut through the basic ideas of relativity a little more mathematically than I would do if I were teaching it for the first time I teach it the first time I tend to teach it the way Einstein first conceived of it how do you measure distances how do you measure velocities how do how does the propagation of light influence these things instead I'm going to take a more mathematical view of it and think about the properties of various kinds of coordinate transformations coordinates now consists not only of XY and Z but also time T so imagine every event in the world is characterized by just like every particle would be characterized by a position x y&z every event taking place in space-time is characterized by four coordinates X Y Z and T let's suppress for the moment y&z let's just forget I forget them for the moment and concentrate on X and T that would be appropriate if we were mainly interested in motion along one axis let's focus on that motion along the x axis let's suppose there is no motion along y&z then we can forget y&z for the moment momentarily we'll come back to them and think of motion along X and T and the various reference frames that might be moving along the x axis alright here's here's time vertically is space horizontally physicists always draw space horizontally and time vertically I found out that mathematicians are at least certain computer scientists always draw time going horizontally I didn't know that and I got into an enormous argument with a quantum computer scientist which was ultimately resolved by the fact that he had time going horizontally and I had it going vertically these are traditions I guess traditions grow up around subjects but time is north and X is east I guess or at least time is upward yeah yeah yeah that's what that that that's the point that is the point yes they're thinking of time is the independent variable and everybody knows that it's a law of nature that the independent variable should be horizontal ok all right now let's in let's imagine a moving observer moving down the x axis with a velocity V let's take his origin of spatial coordinates his origin of spatial coordinates at time T equals zero is just the same let's assume that my I'll be the moving observer I move down the x-axis I am my own origin there's nobody who was your origin that seat is vacant over there so that absent a human over there is the center of the x-coordinates in your frame I'm the X prime coordinates and of course I being very egocentric will take my x-acto is origin to be where I am there I do I move down the x-axis we pass each other our origins pass each other at t equals 0 so that means at T equals 0 your axis and my axes are the same or your origin in my origin is the same but then as I move down the x axis my core my coordinate center moves to the right most of the right that's supposed to be a straight line that's as good as I can do under the circumstances that's a straight line and it's moving with velocity V which means it's X prime equals SR it means x equals VT but it's also that's the way you describe it in terms of your coordinates my centre you described by saying x equals VT how do I describe it I just say X prime my coordinate X prime is 0 X prime equals 0 is the same as x equals VT all right what's the relationship between X Prime and X and T well it's easy to work out if you believe this picture the X prime coordinate is the distance from my origin the x coordinate is the distance from your origin so one of these is X the other is X prime the upper one here is X prime the low and here is X and the relationship between them is that they differ by an amount VT in particular X is equal to X prime minus VT or X prime is equal to X plus VT will have it wrong yes I do X prime is X minus BT and X is X prime plus VT yeah I think I have that's correct now all right what about time itself well according to Newton and according to Galileo and according to everybody who came afterward up until Einstein time is just time is just time is just time there was no notion that time might be different in different reference frames Newton had the idea of a universal time sort of God's time God upon his cloud ticking off with his with his super accurate watch and that time was universal for everybody no matter how they were moving and so everybody would agree on what on the time of any given event in this map of space and time here and so the other equation that went with this is that T prime is equal to T let's forget the top equation here let's just forget it one might say that this was the Newtonian or the Galilean transformation properties between X and T your coordinates and the coordinates that I ascribe to a point in space-time now let's examine a light ray moving down the plus x axis if it starts at the origin here then it moves along a trajectory which is x equals CT C being the speed of light now shortly I'm going to set C equal to 1 we're going to work in units in which C is equal to 1 but not quite yet incidentally once you understand a bit of relativity working in coordinates in which C is not equal to 1 is about as stupid as using different units for x and y are if we used yards for x and feet for y then we will have all kinds of funny factors in our equations which would be conversion factors from X which is measured in feet to Y which is measured in our yards the cycle has its uses log scale has its uses no long skilling long scale well let common interest yep I'm not sure we good but okay I'm just saying it is quite often in practical circumstances that one uses different scales yeah you sometimes you might there might be a good reason I mean um it wouldn't be totally unreasonable for a sailor to use different units for horizontal direction and vertical direction hmm I mean he's used to moving around horizontally he might use what miles miles versus fathoms or something nautical miles versus paddles yeah Persian is relative but um when you talk about a frame of reference you need to specify a period of time because obviously goes that 15 billion years there is no yeah we're ignoring now the fact that the universe began at some time and we're imagining now as Newton did and as the early Einstein did that the universe has just been here forever and ever and ever unchanging totally static and space and time have properties which don't change with time now of course that's incorrect in the real world and at some point we will take up the subject of cosmology and find that's not right but as long as we're interested in time intervals which are not I suspect this is what you're getting at as long as we're interested in time intervals which are not too long in particular time intervals over which the universe doesn't expand very much and so forth we can mainly say the properties of space don't change over a period of time and so everything just stays the same as always was is that what you're asking it seems that that this assumption if it is made it needs to what you're describing so well so the question is without imagining to some point as it doesn't lead it doesn't lead to what I'm describing where is this this room for different formulas here this is a formula which is based on an assumption the assumption being that time is universal that's what Einstein found was wrong basically what he found is that when you're in a moving frame of reference to different the observers will not agree about what time a particular event takes place this is the culprit here this one and some modifications to this one but in any case to see what's wrong let's go to Maxwell's equations Maxwell's equations say that light always moves with this velocity C being some numbers in meters per second okay 3 times 10 to the 8th meters per second we will later as I said say C equals 1 let's imagine a light beam moving down the x axis let's describe how X prime sees it in other words you see the light move this way to the right how do I see the light well let's see what I see let's just work it out X prime will be X which is CT for that light ray minus VT which is the same as C minus VT all this says is that I see the light moving with a diminished velocity a velocity C minus V why is that because I'm moving along with the light so naturally I see it move slowly the slow compared to what you see it what about the light going in the other direction supposing it was a light beam going in the other direction then how would you describe it you would describe it as x equals minus CT and if I do exactly the same thing I will find that X prime is equal to X that's minus CT – VT which is the same as minus C plus V times T so what this says is that I will see the light moving also in the negative direction that's the minus sign but I'll see it moving with an enhanced velocity C plus V if this were the right story and if these were the right transformation laws for space and time then it could not be the case that Maxwell's equations are laws of physics or laws of nature in the sense that they were true in every reference frame they would have to be corrected in moving frames just like the juggler who had to reach to the right who didn't actually but who thought he had to reach to the right to collect the ball when train is moving the physicist interested in light beams would have to correct things for the motion of his reference frame now it's an experimental fact that this is not the case that you don't have to correct for motion was the famous Michelson Morley experiment Einstein he just rejected he just felt this can't be right Maxwell's equations were much too beautiful to be relegated to the approximate or to the contingent on which reference frame and so he said about to find a framework in which the speed of light would be the same in every reference frame and he basically focused on these equations and after various very very beautiful Gedanken experiments thought experiments about light and about measuring and so forth he came to a set of formulas called the Lorentz transformations I'm going to explain them the Lorentz transformations in a more mathematical way not fancy mathematics but just get we want to get right to the heart of it and not spend the three weeks doing it the best way is to a mathematical problem but before I do let me set up a different mathematical problem which is for most of you you've seen me do this before but nonetheless let's go through it again the problem of rotation of coordinates we're going to do this quickly let's just take spatial coordinates now for the moment two dimensional spatial coordinates let's forget X and T and just concentrate on X&Y two coordinates in space instead of events in space-time concentrate on a point in space a point in space has coordinates and we can determine those coordinates the x and y coordinates just by dropping perpendicular to the x axis in the y axis and we would describe this point as the point at position let's just call it X Y now there's nothing sacred about horizontal and vertical so somebody else may come along some crazy mathematician a really nutty one who wants to use coordinates which are at an angle relative to the vertical maybe a couple of beers and you don't know the difference between vertical and worth worth worth we should give this direction a name oblique yeah all right the oblique observer the blue observer can blue be seen everybody can see blue okay good ah the blue observer also characterizes points by coordinates which he calls X Prime and Y Prime the X Prime and the Y prime coordinates are found by dropping perpendicular to the X Prime and the Y prime axis so here's X prime is y prime and given a point X Y there's a role it must be a role if you know the value of x and y you should be able to deduce the value of X I'm in y-prime if you know the angle between the two coordinates between the x coordinate and the X prime coordinate and the formulas simple we've used it least in these classes many times I'll just remind you what it is that's X prime is equal to x times cosine of the angle between the two frames between the two coordinate systems minus y times sine of the angle and Y prime is equal to minus plus I think X sine of theta plus y cosine theta I just want to remind you about a little bit of trigonometry all of trigonometry is encoded in two very simple formulas I've used them this signs on these signs of are on the right let's Ella and X prime is bigger than X for small theta since ours here are all so it's Auto Expo Rhine is bigger than it is is it yeah let's see if you rotate it to the next so that y is y prime is zero it's further out X prime rook will have it backward yeah what's your gift I'm not gonna fit nobody so let's say just make sure the links take survive is the little perpendicular there no my life primary so that's y prime y prime is this is why I'm here right right that's why I'm in X prime is bigger than X so there has to be a plus sign on the second you know its prime is bigger than X let's see um yeah X prime is bigger than X yeah X prime is bigger than X looks like that's probably right probably sign but then this one must be man negative yeah okay there's an easy way to correct for it another way to correct for it just call this angle minus theta that would also do the trick because cosine of minus theta is the same as cosine of theta and sine changes sign when you change theta 2 minus theta so if instead of calling this angle theta I called it minus theta then my previous formulas would be right it's true true but the it's an excuse all right what do we know about sine and cosine it's important to understand sine and cosine everything you ever learned about trigonometry can be codified in two very simple formulas if you know about complex numbers the two very simple formulas are that cosine of theta is e to the I theta plus e to the minus I theta over 2 and sine of theta is e to the I theta minus e to the minus I theta over 2i those two formulas contain everything about trigonometry you don't have to know any other formulas other than these for example I will assign you the homework problem of using these two formulas to find cosine of the sum of two angles but the way you would do it is just write the sum of two angles in here and then reexpress the Exponential's in terms of cosine and sine that's easy to do e to the I theta is equal to cosine of theta plus I sine theta and e to the minus I theta is cosine of theta minus I sine theta so work through these formulas get familiar with them they're extremely useful formulas once you know them you will never have to remember any trigonometric formulas again the other thing to know is that e to the I theta times e to the minus I theta is 1 all right e to the anything times e to the minus the same thing is one those things characterize all trigonometric formulas in particular as was explained to me by Michael a number of times if we multiply e to the I theta times e to the minus I theta we will get one on this side but on this side we will get cosine squared of theta plus sine squared of theta naught minus sine squared but plus sine squared cosine squared and then ice minus I squared sine squared that gives us cosine squared plus sine squared cosine squared theta plus sine squared theta so that's equivalent to the fact that e to the I theta times e to the minus I theta is 1 all right now the most important fact that again follows from the simple trigonometry is that when you make the change of coordinates from XY to X prime Y prime something is left unchanged namely the distance from the origin to the point XY that's something which is you know you count the number of the molecules along the blackboard from here to here and that doesn't change when I change coordinates so the distance from the origin to the point XY has to be the same independent of which coordinate axes we use well let's take the square of that distance the square of that distance we know what it is let's call it s squared I'm not sure why I use s but s for distance s s for distance s for space I think it must be for space that I'm using it for the spaces for the spatial distance from the origin to the point XY we know what that is it's Pythagoras theorem x squared plus y squared but as I said there's nothing special about the XY axes we also ought to be able to calculate it as X prime squared plus y prime squared well it's not too hard to work out that X prime squared plus y prime squared is x squared plus y squared it's easy to use do X prime squared plus y prime squared will have x squared cosine squared theta it will also have x squared sine squared theta when you add them you'll get x squared plus y squared you know you know the rigmarole so it follows from cosine squared plus sine squared equals 1 that X prime squared plus y prime squared equals also equal is equal to x squared plus y squared work that out make sure that you have this on the control that you understand why from the trigonometry not from the the basic physics of it or the basic geometry of it is clear make sure that you understand that you can see that from the trigonometry okay one last thing about sines and cosines if I plot on the blackboard for every angle if I plot sine or cosine along the horizontal axis supposing I plot cosine of theta along the horizontal axis and sine of theta along the vertical axis then if I plot all possible angles they will correspond to a bunch of points that lie on a unit circle Y on a unit circle because sine squared plus cosine squared equals 1 so one might call the properties of sine and cosine the properties of circular functions circular in that they're convenient for rotating they're convenient for describing unit circles points on unit circles are described in terms of coordinates which are cosines and sines of angles and so forth it's natural to call them circular functions these are these are not the functions that come in to the transformation the new transformation properties first of all these are wrong and I don't want to use X what's X ya ya now just wrong Newton had it wrong Newton or Galileo however it was postulated who postulated it Einstein modified it now we're going to have to make sure that Einstein's modification doesn't change things in situations where Newton knew where Newton's equations were good approximations the situations where I'm Stan's modifications are important is when we're talking about frames of reference moving very rapidly up near the speed of light before the 20th century nobody or nothing had ever moved faster than a hundred miles an hour probably well of course some things did light did but for all practical purposes light didn't travel at all it's just when you turned on the switch the light just went on so light didn't travel nothing and anybody's experienced direct experience traveled faster than 100 or 200 miles an hour and well I should say nothing travels faster than 100 miles an hour and then live to tell about it so all of experience was about very slow velocities on the scale of the speed of light on the scale of such velocities newton's formulas must be correct they work they're they're very useful they work Nutan got away with it so there must be good approximations it better be that whatever einstein did to the equations in particular to these two equations here had been a reduced to newton's equations in the appropriate limit okay let's come back now to light light according to the Newton formulas doesn't always move with the speed of light but let's let's try to figure out what it would mean of a better formula of a replacement for this but light always moves with the speed of light first of all let's set the speed of light equal to one that's a choice of units in particular it's a choice of the relation between space units and time units if we work in our light years for spent for a distance and years for time then light moves one light year per year the speed of light is one if we use seconds and light seconds it's also one whatever whatever scale we use for space if we use for time the time that it takes light to go that distance one unit of space if we use that for time units then the speed of light is equal to one now from the ordinary point of view of very slowly moving things those are odd units but if we were electrons with neutrinos and whizzing around like photons they would be the natural units for us speed of light equals one so let's set the speed of light equal to one as I said it's just the choice of units and then a light ray moving to the right just moves along a trajectory x equals T C is just equal to one a light ray moving to the left is x equals minus T how can we take both of these equations and put them together sorry x equals minus T can I write a single equation which if it's satisfied is a light ray either moving to the left or to the right yes here's an equation x squared equals T squared it has two solutions x equals T and X equals minus T the two square roots or x squared equals T squared is equivalent to either x equals T or x equals minus T in other words this equation here has the necessary and sufficient condition for describing the motion of a light ray either to the right or to the left supposing we found a replacement for this equation which had the following interesting property that whenever let's let's write it this way X square minus T squared equals 0 this is even better for our purposes x squared minus T squared equals 0 that's the necessary and sufficient condition to describe the motion of a light ray supposing we found a new set of rules a new set of transformation properties which which um had the property that if x squared minus T squared is equal to 0 then we will find that X prime squared minus T prime squared is equal to 0 in other words supposing this implied this and vice-versa then it would follow that what the unprimed observer you and your seats see is a light ray the primed observer me moving along also see as a light ray both of us agreeing that light rays move with unit velocity now this doesn't work for Newton's formula here it just doesn't work if X is equal to T it does not follow that X prime is equal to the T prime in fact it says something quite different okay so the form of these equations must be wrong let's look for some better equations now at this point let's in fact let's even be a little bit more ambitious it turns out being a little bit more ambitious actually simplifies things let's not only say that when X square minus T squared is equal to zero then X prime squared minus T prime squared is equal to zero let's say something even bolder let's say the relation between XT and X prime T prime is such that x squared minus T squared is equal to X prime squared minus T prime squared in other words pick any X and any T and calculate X square minus T squared then take the same point except reckoned in the primed coordinates in other words we take a certain event a light bulb goes off someplace you say that corresponds to X and T I say it corresponds to X Prime and T Prime but let's require just to try it out see if we can do it let's look for transformations so that X square minus T squared will always be equal to X prime squared minus T's prime squared that would be enough to ensure that everybody will agree about the speed of light why if x squared minus T squared equals X prime minus T prime squared for all X and T and so forth then when X square minus T squared equals zero X prime minus T prime squared will be zero and then if this is a light ray so is this a light ready everybody get the logic ok good so let's assume now that let's ask can we find transformations which have this particular property now it's not so different from looking for transformations which preserve x squared plus y squared equals x prime squared plus y prime squared it's just a little minus sign other than a minus sign here X square minus T squared look of these two is very similar and the mathematics is quite similar here are the transformations which preserve x squared plus y squared what are the transformations which preserve x squared minus T squared well they are the Lorentz transformations they are the fundamental transformations of the special theory of relativity they're not this but they're closely related or perhaps one should say closely analogous to these equations here but we have to substitute for circular trigonometry hyperbolic trigonometry so let's go back and remember a little bit about hyperbolic functions instead of circular functions well I didn't want to erase that all right these are the basic rules governing circular functions cosine theta this sine theta is equal to this and the e to the I theta in terms of cosine and sine all right let's see if we have a yeah we do have a blank blackboard here let me write whoops what did I do here I erased something I didn't mean to erase incidentally does everybody see how I got this side from the side you just add and subtract the equations appropriately and you isolate it to the I theta e to the minus R theta that's elementary exercise alright hyperbolic functions what are hyperbolic functions alright those are functions of the form hyperbolic cosine cosh hyperbolic cosine first of all the angle theta is replaced by a variable called Omega which I will call Omega Omega is called a hyperbolic angle it doesn't go from zero to two pi and then wind around on a circle it goes from minus infinity to infinity goes from minus infinity to infinity so it's a variable that just extends over the entire real axis but it's defined in a manner fairly similar to cosine and sine cosh Omega is by definition you're not allowed to ask why this is definition e to the Omega plus e to the minus Omega over 2 all we do is substitute for theta or for Omega theta I theta substitute Omega and that gives you hyperbolic functions likewise or similarly there's the hyperbolic sine and that's given by e to the Omega minus e to the minus Omega over 2 essentially you throw away all eyes out of that formula out of the top formulas just throw away all Sun all eyes the equations on the right-hand side become e to the Omega equals hyperbolic cosh Omega plus sin Chi Omega and e to the minus Omega equals cosh so mega- cinch Omega I think that's right is it right gosh – cinch it is yeah it is right okay now what about the analog of cosine squared plus sine squared equals one that simply came by multiplying this one by this one so let's do the same operation multiplying e to the Omega by each by e to the minus Omega gives one and now that gives cosh squared minus cinch squared you see we're getting a minus what we want we want that minus the minus is important we want the well somewhere is under here was a formula with a minus sign yeah we want to get that – into play here that's cos Omega squared knockouts Prakash squared Omega minus sin squared Omega so it's very similar everything you want to know about hyperbolic trigonometry and the theory of these functions is called hyperbolic trigonometry everything you ever want to know is codified in these simple formulas these in these and they're more or less definitions but there are the useful definitions now yeah go ahead yeah not only is it worth mentioning I was just about to mention it so I squared minus y squared is what hyperbola yeah right exactly so if I were to play the same game that I did here namely plot on the horizontal and vertical axis the values not of cosine of theta and sine of theta but cosine cosine cosh of that of Omega and since Omega what's in other words on the x-axis now we're going to plot cos Omega and on the y-axis cinch Omega then this is a hyperbola not a circle but a hyperbola and it's a hyperbola with asymptotes that are at 45 degrees you can see let me show you why why the asymptotes are at 45 degrees when Omega is very large when Omega is very large then e to the minus Omega is very small right when Omega is very large e to the minus Omega is very small and that means both cosh and cinch are both essentially equal to e to the plus Omega in other words when Omega gets very big cosh and cinch become equal to each other and that's this line here cash equals cinch along this line here so when Omega gets very large the curve asymptotes to to a curve which is a 45 degrees it's not hard to see that in the other direction when Omega is very negative that that it asymptotes to the other asymptotic line here so that's why it's called hyperbolic geometry it the hyperbolic angle the hyperbolic angles the caches the cinches play the same role relative to hyperbolas as sines and cosines do two circles any questions No so cosh Omega equals zero how would you plot that hi purple okay show me hmm Oh cos squared minus sin squared equals zero no that's no no cos squared minus sin squared equals one in the same sense that sine squared plus cosine square it never equals zero I think what I think you want to ask a different question I think oh well since Omega equals zero is the horizontal axis the costume a equals zero is the vertical eyebrows right okay well this is the x-intercept yeah it's it's the vertex I just think here's one point on a minute oh man the x-intercept there is one yeah because Kostroma cost of zero is one to see that just plug one r 0 in here 1 plus 1 divided by 2 is 1 at least it was yesterday yeah stores okay so now we we're sort of starting to cook a little bit we're starting to see something that has that nice minus sign in it but what's it got to do with X and T and X Prime and T prime we're now set up to make let's call it a guess but it's a guess which is based on the extreme similarity between hyperbolas and circles cautions and cosines and so forth he is the guess I'm going to make and then we'll check it we'll see if it does the thing we wanted to do my formula instead of being this has gotten with and we're now going to have instead of x and y we're going to have x and t time and x later on we'll put back y&z we're going to have to put back y&z but they're very easy okay so let's start with X prime X prime is the coordinate given to a point of space-time by the moving observer namely me and I'm going to guess that it's some combination of X and T not too different but not the same as where is it X prime equals X minus VT I'm going to try cosh Omega X let's write X cos Omega minus T sin Omega sort of in parallel with this I could put a plus sign here but you can go back and forth between the plus and the minus by changing the sign of Omega just as you did here so this let's do it this way X cos Omega minus T sin Omega and T prime going to look similar but without the extra minus sign here this you know the relation between sines cosines and cautious and cinches is one of just leaving out an eye you go from sines and cosines the clashes and cinches by leaving out the I well if you track it through carefully you'll find that this minus sign was really an I squared it's not going to matter much I will just tell you it was really came from some I squared and if you leave out I I squared just becomes one squared is no minus sign so here's the guess for the formula connecting X prime T Prime with X and T it equals let's say X since Omega – no – plus T cos Omega in this case there are two minus signs in this case there was only one minus sign okay but but let's check what do we want to check we want to check that X prime squared minus T prime squared is equal to x squared minus T squared your ask you're probably asking yourself what is this Omega what does it have to do with moving reference frames I'll tell you right now what Omega is it's a stand-in for the velocity between the frames we're going to find the relationship between Omega and the relative velocity of the reference frames in a moment there has to be a parameter in the lower end these are the lines in these are the Lorentz transformations connecting two frames of reference in the Lorentz transformations as a parameter it's the velocity the relative velocity that parameter has been replaced by Omega it's a kind of angle relating the two frames a hyperbolic angle but we'll we'll come back to that for the moment let's prove that with this transformation law here that X prime squared minus T prime squared is equal to zero ah is equal to X square minus T squared I'm getting to that point in the evening where I'm going to make mistakes all right this is easy you just work it out you use all you have to use is that cosine squared minus sine squared is 1 you can work that out by yourself but we can just see little pieces of it here X prime squared will have x squared cos squared Omega t prime squared will have x squared sin squared Omega if I take the difference between them I'll get a term with an x squared times cos squared minus sin squared but cos squared minus sin squared is one fine so we'll find the term with an x squared when we square take the square of the difference between the squares of this and this and likewise will also find the T squared the cross term when you square X Prime you'll have XT cost cinch when you square T Prime you'll have XT costs inch when you subtract them it'll cancel and it's easy to check that's our basically one liner to show that with this transformation here x prime squared minus T's prime squared is x squared minus T squared which is exactly what we're looking for let me remind you why are we looking for it if we find the transformation for which the left-hand side and the right-hand side are equal then if x squared equals T squared in other words if the right-hand side is 0 the left-hand side will also be 0 but x squared but x equals T that's the same as something moving with the speed of light in the X frame of reference if this being 0 is equivalent to the left hand side being 0 it says that in both frames of reference the light rays move with the same velocity so that's the basic that's the basic tool that we're using here X prime squared minus T prime squared is equal to x squared minus T squared all right that does follow by a couple of lines using cos squared minus N squared equals 1 but what I want to do let's take another couple of minutes now let's take a break for five minutes and then come back and connect these variables Omega with the velocity of the moving frame of reference somebody asked me a question about the ether and what it was that people were thinking somehow Einstein never got trapped into this mode of thinking um well what were they thinking about when they were thinking about the ether what exactly was the michelson-morley experiment well I'll just spend the minute or two mentioning it certainly Maxwell understood that his equations were not consistent with with Newtonian relativity he understood that but his image of what was going on is that the propagation of light was very similar to the propagation of sound in a material or water waves propagating on water and of course it is true that if you move relative to the atmosphere or move relative to the substance that sound is propagating in you'll see sound move with different velocities depending on your motion if you're at rest in a gas of material isn't there's a natural sense in which is a particular rest frame the rest frame is the frame in which on the average the molecules have zero velocity if you're in that reference frame then first of all light has the same velocity that way as that way number one and it has a velocity that's determined by the properties of the fluid that the sound is moving in okay Maxwell more or less thought that light was the same kind of thing that there was a material and the material had a rest frame and that particular rest frame was the frame in which light would move with the same velocity to the left as to the right and he thought that he was working out the mechanics or the behavior of this particular material and that we were pretty much at rest relative to this material and that's why we saw light moving the same way to the left of the right one would have to say then that Maxwell did not believe that his equations were a universal set of laws of physics but that they would change when you moved from frame to frame just happened by some luck we happen to be more or less at rest relative to the ether to this strange material um of course you could do an experiment with sound if you're moving through the sound you can check that the velocity in different directions is different you do let's not worry exactly how you do that that's what the Michelson Morley experiment was Michelson and Morley I suppose said look the earth is going around in an orbit maybe at one season of the year we just happen to be at rest relative to the ether by accident and some other season six months later we're going to be moving in the opposite direction and we won't well we won't be at rest only at one point in the orbit could we be at rest relative the–this or at any other point in the orbit we wouldn't be so if we measure in November that light moves the same than all possible directions then in what's what's the opposite of November May then in May we should find that light is moving with great with the different velocities in different directions and he tried it and a very fancy and sophisticated way of measuring the relative velocity in different directions but he found that there was no discrepancy that the light traveled the same velocity in every direction at every time of year there were all sorts of ways to try to rescue the ether but none of them worked none of them work and the result was one had to somehow get into the heart of space and time and velocity and mid distance and all those things in a much deeper way in a way that didn't involve the idea of a material at rest in some frame of reference that that propagated the light okay oh where are we I forgotten where we were when we stopped somebody remind me whoo-hah Omega yeah what is Omega forgotten Omega Oh how Omega is really metal speed of light but to the velocity of the moving reference frame here we have two reference frames X T and X Prime and T prime what's the relationship between them well the whole goal here was to understand the relationship between frames of reference moving with relative velocity between them Omega is not exactly the relative velocity but it is closely related to it okay let's say let's see if we can work out the relationship we know enough to do it let's see if we can work out the relationship between Omega and the velocity of the moving frame all right again let's go back to this picture there's a frame of reference moving let's redraw it here's my origin moving along okay what does it mean to say that from your perspective my frame of reference so my origin is moving with velocity V well by definition this is not a law now this is a definition and says that this line here has the equation x equals VT that's the definition of this V here my origin moves relative to your origin it moves with a uniform constant velocity that's an assumption that we're talking about two inertial frames of reference and you in your frame of reference will write x equals VT that's the definition of V if you like what will I call it I will call it X prime equals zero all along there I will say X prime is equal to zero it's my origin of coordinates okay now let's come to this transformation law here and see if we can spot how to identify V well X prime equals zero that's this trajectory moving at an angle with a velocity V X prime equals zero is the same as saying X cos Omega equals T sin Omega X prime equals zero set this side equal to zero and that says that X cos Omega equals T sin Omega all right so whatever the connection between velocity and Omega is it must be such that when X prime is equal to zero X cos Omega equals T sin Omega well let's look at that equation it also says that X is equal to sin CH Omega over cos Omega times T well that's interesting because it's also supposed to be equivalent to x equals VT now I know exactly how to identify what the velocity is as a function of Omega the velocity of the moving transformation the moving coordinate system must just be sin Chi Omega over cos Omega that's the only way these two equations can be the same x equals VT x equals sin Chi Omega over cos Omega times T so now we know it we know what the relationship between velocity and Omega is write it down the velocity of the moving frame now this is not the velocity of light it's just the velocity of the moving frame must just be cinch Omega over cos omega well actually i want to invert this relationship i want to find sin and cos omega in terms of the velocity i want to rewrite these Lorentz transformations where are they i want to rewrite these Lorentz transformations in terms of the velocity that's the familiar form in which you learn about it in in elementary relativity books X prime is equal to something with velocities in it to exhibit that all we have to do is to find Cinch and cosh Omega in terms of the velocity that's not very hard let's let's work it out the first step is to square it and to write V squared is equal to cinch Omega squared over cosh Omega squared that was easy next I'm going to get rid of since Omega squared and substitute where is it I lost it one is equal to cos Omega squared minus cinch Omega squared alright so wherever I see cinch Omega squared I can substitute from here namely cosh squared Omega minus one is equal to sine squared Omega so here we are this is just equal to hash of Omega squared minus one divided by cost of Omega squared or let's multiply by what I want to do is solve for cost Omega in terms of velocity I want to get rid of all these cautions and cinches of Omega and rewrite it in terms of velocity so first x cost Omega squared we have cosh squared Omega times V squared equals cosh squared Omega minus one or it looks to me like this is cosh squared Omega times one minus V squared equals one what I've done is transpose yeah cos squared times V squared minus cos squared itself that gives you cos squared 1 minus V squared equals 1 change the sign can everybody see that the second line follows from the first I'll give you a second yeah yeah yeah it's clear ok finally we get that cos Omega is equal to 1 divided by 1 minus V squared but now I have to take the square root cos Omega / one minus V squared and then take the square root and that gives you cos Omega now we've all seen these square roots of 1 minus V squared in relativity formulas here's where it begins the kayne we begin to see it materializing what about sin Chi Omega let's also write down sin Chi Omega well from here we see that sin Chi Omega is just equal to V times cos Omega this is easy since Omega equals V times cos Omega sorrow sin Chi Omega is V divided by square root of 1 minus V squared let's go back to these Lorentz transformations over here and write them getting rid of the trigonometric functions the hyperbolic trigonometric functions and substituting good old familiar velocities let's get rid of this and substitute the good old ordinary velocities ok so we have here X prime equals x times cos Omega and that's divided by square root of 1 minus V squared then this minus T times sin Omega which is V over the square root of 1 minus V squared or if I put the two of them together and combine them over the same denominator it's just X minus VT divided by square root of 1 minus V squared I think most of you have probably seen that before maybe slightly different let's let's clean it up a little bit X prime equals X minus VT divided by the square root of 1 minus V squared what about T prime T Prime is equal to t minus V X over square root of 1 minus V squared T prime is equal to T times cos cost is just 1 over square root and then x times sin CH that gives us the extra V in other words the formulas are more or less symmetrical and those are all good old Lorentz transformations now what's missing is the speed of light let's put back the speed of light the put back the speed of light is an exercise in dimensional analysis there's only one possible way the speed of light can fit into these equations they have to be modified so that they're dimensionally correct first of all one is dimensionless has no dimensions it's just one velocity is not dimensionless unless of course we use dimensionless notation for it but if velocity is measured in meters per second then it's not dimensionless how do we make V squared dimensionless we divide it by the square of the speed of light in other words this V squared which is here which has been defined in units in which the speed of light is 1 has to be replaced by V squared over C squared likewise over here V squared over C squared now velocity times time does have notice first of all the left hand side has units of length the right hand side this is dimensionless X has units of length but so does velocity times time so this is okay this is dimensionally consistent as it is but over here it's not the left hand side has dimensions of time that's all right 1 minus V squared over C square that's dimensionless this has units of time but what about velocity times X velocity times X does not have units of time in order the given units of time you have to divide it by C square okay let's check that velocity is length all the time times length divided by C squared that's length square R which gets correct but it's correct all right this is probably familiar to most of you who've seen relativity once or twice before these are the equations relating to different moving coordinate systems moving relative to the x axis but you see the deep mathematics or the mathematical structure of it in many ways is best reflected by this kind of hyperbolic geometry here and you know most physicists by now never write down the Lorentz transformations in this form much more likely to write them in this form easier to manipulate easier to use trigonometry or or hyperbolic trigonometry it's a little exercise it's a nice little exercise to use this the hyperbolic trigonometry to compute their to compute the compounding of two Lorentz transformations if frame two is moving relative to frame one with velocity V and frame three Israel moving relative to two with velocity V Prime how is three moving relative to one the answer is very simple in terms of hyperbolic angles you add the hyperbolic angles not the velocities but the hyperbolic angles the hyperbolic angle of three moving relative to one is the hyperbolic angle of three moving relative to two plus two moving relative to one and then you use a bit of trigonometry or hyperbolic trigonometry to figure out how you do the inches and kosh's of the sum of 2 hyperbolic angles very straightforward and I'll leave it as an exercise to see if you can work that out much easier than anything else ok so there there we have the Lorentz transformations yeah oh oh absolutely yes that's that's that's a good point yeah when we that's right if we have frame 1 let's call this x1 and y1 x2 and y2 and finally x3 and y3 well then the angle of – let's call F of 3 relative to 1 let's call it theta 1 3 is just equal to theta 1 2 plus theta 2 3 the angle connecting frame one with frame 3 is just the sum of the angle theta 1 2 plus theta 2 3 so in that respect the Lorentz transformations are much simpler in terms of the Omegas it's the Omegas which combined together to add when you add velocities now how different is omega from the velocity let's work in units in which the speed of light is equal to 1 where is our formula for velocity all right let's take this formula over here what a cinch Omega 4 small Omega let's put the C squared there a let's not put the C square there or not put the C square there since Omega is essentially Omega when Omega is small just like sine is omega where is theta when theta is small the cinch function the cost function looks like like this the cinch function looks like this but it but it crosses the axis with a slope of 1 for small Omega cinch Omega is proportional to Omega for small velocity one minus V squared is very close to 1 if the velocity is a hundredth of the speed of light then this to within one ten-thousandth is just 1 if we're talking about velocities a millionth of the speed of light then this is very close to 1 and so since Omega and velocity are very close to each other it's what's going on here Thanks okay so for small velocities Omega and velocity are the same the actual correct statement is that V over C is like Omega the dimensionless velocity over the speed of light is like Omega for small Omega and small velocity so for small velocity adding velocities and adding omegas are the same things but when the velocities get large the right way to combine them to find relationships between different frames is by adding Omega and not adding velocities when you add Omega like compounding velocities as you've got it there I guess you won't go greater than 45 degrees that guess because that would be faster than light no but Omega no more you see this bit the speed of light is V equals one that corresponds to Omega equals infinity yeah yeah so Omega Omega runs over the whole range from minus infinity to infinity but when it does V goes from minus the speed of light to the speed of light so you can add any omegas and still add any omegas Omega that's right there's no there's no speed limit on Omega is this like we just go on that diagram it looks like it's greater than 45 degrees if here where where I make a and I guess they use the definition of state along the hyperbola yeah that's right sorry where are we right there today I guess that's theta though isn't it this is Theta that's a good oh god yeah right right yeah Omega is the distance along hyperbola that's right distances that's right Omega is a kind of distance along the hyperbola all right now let's let's talk about that a little bit all right now that we've established the basic mathematics structure of the transformations I think we should go back and talk about some simple relativity phenomena and derive them oh one thing which is important which I yeah well let's see we're here are my Lorentz transformations over here I said we should we ought to at the end make sure that our transformations are not too dissimilar from Newton's in particular when the velocities are small they should reduce to Newton that's all we really know that's or at least that's all that Newton really had a right to assume that when the velocities are smaller than something or other that his equations should be good approximations isn't adding velocity good enough isn't velocities adding good enough in fact you're right in fact you're right but let's just look at the transformations themselves all right as long as the velocity is a small percentage of the speed of light an ordinary velocities are what a hundred miles an hour versus 186,000 miles an hour what is that it's small right and it's doubly small when you square it so for typical ordinary velocities even the velocities of the earth around the Sun and so forth fairly large velocities what 60 kilometers per second or something like that 60 kilometers per second is pretty fast that's the that's the orbital earth around the Sun it's pretty fast but it's nowhere near 300,000 kilometers per No yeah looks here on a thousand meters per second we're I'm sorry three times ten to the eighth no three times three hundred thousand kilometers per second right 60 kilometers per second three hundred thousand kilometers per second small fraction and then square it so for ordinary motions this is so close to one that the deviation from one is negligible so let's start with the top equation for the top equation this is negligible and it's just x prime equals X minus VT the bottom equation here you have a C squared in the denominator whenever you have a C squared in the denominator that's a very very large thing in the denominator this is negligible compared to T so here the speed of light is also in the denominator just forget this and it's just T but it's just T prime equals T it's just D prime equals T so in fact Newton's formulas are essentially correct for slow velocities no no significant departure from Newton until the velocities get up to be some some appreciable fraction of the speed of light okay let's talk about proper time proper time and then let's do a couple of relativity examples yeah question the bottom equation when X is very large yes that's right when X is exceedingly large you get a correction but that correction that X has to be very large look let's let's discuss before we do anything else let's let's let's talk about that a little bit X minus VT one minus V squared over C squared yeah let's alright in my drawings I'm going to sitt C equal to one but in the equations you can leave the C there okay this equation we understand apart from this one minus V squared over C squared in the denominator it's just this x equals V T or X minus V X minus X minus VT that's Newton let's look at this one over here okay let's look at the surface T prime equals zero T prime equals zero is the set of points that I in my moving reference frame call T call time equals zero it's what I call the set of points which are all simultaneous with the origin T prime equals zero is just everyplace in space-time which has exactly the same time according to my frame of reference and I will therefore call all those points synchronous at the same time what do you say about them if T prime is equal to zero that says that T is equal to V over C squared X now let's set C equal to one for the purpose of drawing just for the purpose of drawing I don't want this huge number C squared to distort my drawings too much it says the T equals V X what does the surface T equals V X look like it looks like this T equals V X which is also X is equal to 1 over V T so it's just a uniform line like that all of these points are at different times from your reckoning this ones later this ones later this ones later and so forth according to my reckoning all these points are at the same time so we disagree about what's simultaneous this was this was the hang-up incidentally this was the basic hang-up that took so long to overcome that took Einstein to overcome it the idea that simultaneity was the same in every reference frame nobody in fact it was so obvious that nobody even thought to ask a question is simultaneous does it mean the same thing in every reference frame no it doesn't in more in your reference frame the horizontal points are all simultaneous with respect to each other in my reference frame what I call horizontal what I call simultaneous you do not okay so simultaneity had to go let me point out one more thing about these equations I'm not going to solve them for you but I will tell you the solution anyway how do you solve for X and T in terms of X Prime and T Prime well think about it in the case of angles supposing I have a relationship like X prime is equal to X cosine theta what is it plus plus y sine theta and y prime is equal to X minus X sine theta plus or Y cosine theta and supposing I want to solve for x and y in terms of X Prime and Y Prime you know what the solution is just change theta 2 minus theta and write that X is equal to X prime cosine of minus theta but what's cosine of minus theta right cosine theta plus y sine of minus theta what's sine of minus theta minus sine theta times y and likewise for y prime Y prime is equal to minus x times sine of minus theta so that becomes plus X sine theta plus y cosine of minus theta which is cosine theta you don't have to go through the business of solving the equations you know that if one set of axes is related to the other by rotation by angle theta the second one is related to the first one or vice versa the first one is related to the second one by the negative of the angle if to go from one frame to another you rotate by angle theta and to go from the second frame back to the first you rotate by angle minus theta so you just write down exactly the same equations interchange Prime and unprimed and substitute for theta minus theta same thing for the Lorentz transformations exactly the same thing if you want to solve these for X and T write down the same equations replace primed by unprimed and change the sign of omegas to minus the sines of omegas change sinus rgn of all the sign all the cinches okay in other words just send Omega 2 minus Omega and that will solve the equations in the other direction yeah yes it's also the same as changing V 2 minus V yes the way to see that is to go right what was it what do we have cosh Omega yep yeah that's right via sign yes that was correct yeah you just well you change Omega 2 minus Omega it has the action of changing V 2 minus V you can just check that from the equations good alright let's let's talk about proper time a little bit proper time if you're doing ordinary geometry you can measure the length along a curve for example and the way you do it is you take a tape measure and you you know sort of take off you take off equal intervals equal equal little separations you can think of these separations as differential distances DS squared small little differential distances and that differential distance is d x squared plus dy squared with the x squared and the y squared are just the differential increments in x and y DX and dy this is d s alright so that's the way and you add them up you add them up that's the way you compute distances along curves it's quite obvious that if you take two points the distance between those two points depends on what curve not the same for every curve so I'll measure the longer curve you have to know not only the two points but you have to know the curve in order to say what the distance between those points are of course the distance between its longer straight line that's that's well-defined but the distance along a curve depends on the curve in any case D s squared equals the x squared plus dy squared is the basic defining notion of distance between two neighboring points if you know the distance between any two neighboring points in a geometry you basically know that geometry almost essentially completely so given this formula for the distance between two points you can compute if you like the distance along a curve because you've got to take the square root of this and then add them up don't anhedonia the squares add the differential distances all right the important thing is here that square root of DX squared plus dy squared which is the distance between neighboring points doesn't depend on your choice of axes I could choose X Y axes I could choose X prime y prime axes if I take a little differential displacement the X and the y or I just take two points two neighboring points don't even give them labels and measure the distance between them the distance between them should not depend on conventions such as which axes are used and so when I make rotational transformations the X square plus dy squared doesn't change the X and the y may change but the x squared plus dy squared does not change the same thing is true in relativity or the analogous thing we don't measure distances along the paths of particles let's say now that this curve here is the path of a particle moving through space-time there's a particle moving through space-time and we want some notion of the distance along it the notion of distance along it another example would just be a particle standing still as a particle standing still particle standing still is still in some sense moving in time I wouldn't want to say that the distance between these two points and space-time is zero they're not the same point I wouldn't like to say it's zero I would like to say there's some kind of notion of distance between them but it's quite clear that that distance is not measured with a tape measure this point and this point are the same point of space boom here at this point of space and that at a later time boom again at the same point of space two events at the same point of space how do I characterize and some nice way the distance between those two events that occurred in the same place you don't do it with a tape measure all right what do you do with a clock a clock you take a clock and you start it at this point tic tic tic tic tic tic tic a stopwatch you press it at this point tic tic tic tic tic it picks off intervals and then you stop it at that point and you see how much time has evolved that's a notion of distance along a particle trajectory it's not the distance the particle moves in space it's a kind of distance that it's moved through space-time and it's not zero even if the particle is moving standing perfectly still in fact what it is is it's the time along the trajectory what about a moving particle well you can imagine that a moving particle carries a clock with it of course not all particles carry clocks but we can imagine they carry clocks with them as they move and we can start the clock over here and then the clock over here what is the time read off by this moving clock the time read off by a moving clock is much like the distance along a curve measured by a tape measure in particular it should not depend on the choice of coordinates why not this is a question that has nothing to do with coordinates I have a clock made in the standard clock Factory the standard clock Factory and I don't know we're in Switzerland someplace makes a certain kind of clock that clock gets carried along with a particle and we ask how much time evolves or how much time elapses or how much the clock changes between here and here that should not depend on a choice of coordinates it shouldn't depend on a choice of coordinates because it's a physical question that only involves looking at the hands of the clock in fact we can ask it for little intervals along along the trajectory we could ask how much time elapses according to the clock between here and here well the answer again should not depend on what coordinates you use which Lorentz frame you use and there's only one invariant quantity that you can make out of the D X's and DTS describing this point describing these two points there's a little interval DT and there's a little interval DX now we're in space and time not ordinary not ordinary space and the quantity which is invariant there's really only one invariant quantity that you can make out of it it is DT squared minus DX squared it's the same quantity x squared minus T squared for a whole you know for a whole interval the T squared minus DX squared that's the quantity which is invariant it's minus D it's the negative of what I wrote over here x squared minus T squared okay this quantity is equal to the X prime squared minus DT power sorry DT prime squared minus the X prime squared the same algebra goes into this as goes into showing that X prime squared minus T prime squared equals x squared minus T squared incidentally this is the same as saying T prime squared minus X prime squared equals T squared minus x squared doesn't matter which way you write it all right so that suggests that suggests that the time read off the invariant time read off along a trajectory between two points separated by DX and DT is just the square root of DT squared minus DX squared why the square-root incidentally okay you're going to integrate in detail I can integrate DT yeah well alright why not just DT square minus the x squared for the time between here and here is it here's an answer supposing we go to you two intervals exactly the same as the first one we go an interval over here DX and DT and then we go another DX in DT what happens when we double the interval to DT squared minus DX squared it gets multiplied by four because everything is squared well I wouldn't expect a clock when it goes along you know when it goes along a trajectory for twice the the interval here to measure four times the the time I expected to measure twice the time so for that reason the square root is the appropriate thing here okay that's called D tau squared the tau squared the proper time along the trajectory of an object you're right that's just the towel or D tau squared being the x squared minus DT squared the Tau is called the proper time let's go I think we'll let's see the towel is called the proper time and it is the time read by a clock moving along a trajectory it's not just DT that's the important thing it's not just DT the T squared minus the x squared let's do one last thing let's just do the twin paradox in this language I think I think I've had it I'm going to finish you can do the twin paradox in this language all you have to do is to compute the proper time along two trajectories one that goes out with a uniform velocity turns around and comes back with the same uniform velocity versa a trajectory which just goes from one point to the st. the another point along a straight line and it's no more weird it's no weirder really from this perspective than saying the distance from one point to another along two different curves do not have to agree the proper time along two different curves in general will not agree what is a little bit weird is that because of this minus sign the proper time this way is less than the proper time this way that's the consequence of this minus sign here moving with some DX decreases the proper time all right we'll do a little bit more next time but then I want to get to the principles of field theory and and connect some of this with field equations for interesting wave fields the preceding program is copyrighted by Stanford University please visit us at stanford.edu

Integration and the fundamental theorem of calculus | Essence of calculus, chapter 8



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How to Learn Mathematics Fast



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Whether you’re interested in AI or you just want to do some real engineering work, you’re going to need to brush up on your math skills. In this video, I’ll describe my strategy to learn mathematics as fast as possible. Math is a specific, powerful vocabulary for ideas and giving a structure to the way you learn it will empower you to absorb much more of it much faster. I’ll go over my strategies in order.

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why'd the chicken cross the Mobius strip to get to the same side hello world it's Suraj and if you're interested in AI you're going to need to brush up on your math skills in this video I'll describe my strategy to learn mathematics as fast as possible math is a specific powerful vocabulary for ideas imagine a cook that only knows the descriptive words yummy and yucky if he makes a bad meal he has no way to describe it is it too sour too dense too spicy that last one is impossible but all of these critiques are hazy variations of yucky his vocabulary shapes what he's capable of thinking of in the same way math vocabulary shapes concepts and ideas that we are capable of thinking of it uses the rules embedded in our universe to create models and relationships and the process of learning what these rules are has been iteratively refined over millennia take counting for example first there was the unary system which was like drawing lines in the sand then came Roman numerals which had shortcuts for large numbers then decimals and binary than scientific notation and you can bet that in the future we'll have an even better system for counting or take the word quantity for example we understand this concept it's considered common sense but this understanding includes concepts refined over millennia like base-10 notation zero decimals and negatives now imagine improving your vocabulary for structure shape change and tense that's where algebra geometry calculus and statistics come in all math concepts build on other math concepts multiplication and division used to elude geniuses millennia ago and now they are homework for primary school kids newer concepts require them has prerequisites if you're like me you were likely taught math in a very boring way involving blind memorization and speed testing it did improve my ability to fall asleep fast AF though actually memorizing math facts is just a small part of mathematics and being fast at math doesn't necessarily mean you're good at math there's this widespread myth that some people are math people and some people aren't there's so much fear and boredom involved in introductory math curriculums with very little real-world application and if you don't believe you're good at math it will definitely affect your performance neuroscience research shows that there is a strong connection between the attitude that students hold about their own learning ability and their academic performance so the first step before starting to learn any math subject is to believe that you can learn it your brain is adaptable AF it can learn anything if you have the right motivation don't worry if you don't get a concept right away research shows that when you make a mistake in math your brain grows it's that period when our brains struggle to understand a concept that real learning occurs you got worked it out like a muscle like any athlete would to complete a task no pain no gain right when it comes to picking a math topic think the subjects that interest you don't waste your time learning subjects you don't care about like C sharp development just cause if you're like me and are interested in AI linear algebra probability theory calculus and statistics are crucial concepts to understand the field each with a depth of knowledge to learn there are a wealth of great learning resources available to you to learn about any of these textbooks however rarely focus on developing real understanding they're mostly about solving problems with a plug and chug formula for example the Pythagorean theorem isn't just about triangles it's about the relationship between similar shapes the distance between any set of numbers and more math is not a spectator sport while memorizing concepts is useful in order to actually learn you need to be solving problems and you should be solving problems that you actually enjoy solving for it to be fine right for the memorization part just use a cheat sheet for any subjects you want to learn about as a helpful guide as you solve problems I've linked to several in the video description in terms of gamifying problem solving brilliant org has done a great job of making an aging content to learn about different math subjects with a big variety of styles I highly recommend checking that website out you can also find video games that help you learn about a math subject variant for example lets you learn calculus by using it to solve puzzles while playing a young woman who tries to save a planet from imminent destruction while true learn is a game that lets you play the life of a machine learning developer using visual programming to make a living use the internet to find enjoyable ways of learning whatever subject you are interested in that's the best way to motivate yourself to actually learn it make your own curriculum no need to follow existing courses if you don't want to you can use them for inspiration but develop a learning path that works for you on your time that fits your goals whatever subject you're learning make sure to take notes when you're taking notes think of it as a teaching guide for someone who knows nothing about the subject the practice of explaining is the best way to learn any concepts a great methodology to follow for creating explanations for a concept is called adept or analogy diagram example fine English and technical definition this is how you can teach yourself a difficult concept or explain one to others the first part analogy asks the question what else is this concept of light most new concepts are variations of what we already know we've encountered millions of objects and experiences as we've aged surely one of them is vaguely similar to this new topic then make a diagram which engages the other half of your brain dedicated to vision processing if we can create a diagram of imaginary numbers we can see that it lets us rotate around the number line not just move side to side then give an example like what happens after four turns on these axes describe it in plain English once you have an example and lastly use a technical description this final step converts our personal understanding into formal notation it's like sharing a song you made you can hum it but other people need sheet music puppet to play it let's use this method in the context of studying the popular machine learning course by Professor aim on Coursera who also liked a tweet that mentioned me I see you Aang I see you love you when studying if we see an idea that makes sense we can write it down in language that we ourselves would understand if it doesn't we can still write it down and then use the Adept method to decompose it we can write down one sentence explanations for ourselves of any core concept that makes sense to us and then later use these concepts as our own personal cheat sheet to understand later material the course is chock full of formulas that most people have never encountered under advanced optimization for example if we're unclear why the negative sign is used in the formula we can make a note effect then focus on concepts we already understand like derivation it turns out that in derivation the natural log is expected to be negative and sometimes the terminology can get confusing like using the word cost instead of error if we take the time to write out explanations for each term we don't understand we can see that cost captures things outside the model like complexity which error alone doesn't encapsulate it's important to continuously try and create brief easily understandable explanations for everything we learn we could try and summarize the course by saying machine learning is all about creating models with linear algebra then improving them with calculus embrace your confusion it's ok to forget things your notes are meant for you to record what you don't understand what you do understand in the process of how you come to understand eventually eventually you can make technical content for the public based off of your notes and that will improve your understanding even more by making your vocabulary more broadly accessible math is awesome and so is hard work don't let anyone tell you otherwise let your curiosity guide your learning have goals and don't be afraid to be confused it's all a part of the learning process want to become a math genius hit the subscribe button and I will show you the way for now I've gotta solve vehicles and B so thanks for watching

NEET 2020 2021- Mathematical Tools + Motion in 1D – Abhishek Sir | AIIMS | Physics Video Lectures



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Lecture 1 | The Theoretical Minimum



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(January 9, 2012) Leonard Susskind provides an introduction to quantum mechanics.

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“The Beauty of Calculus,” a Lecture by Steven Strogatz



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Calculus is one of the most imaginative and consequential triumphs of human creativity. In this talk, famed mathematician Steven Strogatz will investigate its origins and then show how it, in partnership with medicine, philosophy, science, and technology, reshaped the course of civilization and helped make the world modern. The Franke Program in Science and the Humanities is a new initiative at Yale that aims to foster communication, mutual understanding, collaborative research and teaching among diverse scientific and humanistic disciplines. It is made possible through the generosity of Richard (‘53) and Barbara Franke.
For additional lectures, visit the Franke Program in Science and the Humanities playlist .

hello good afternoon everyone my name is freon at origin I'm an astrophysicist and faculty member in the department's of astronomy and physics at Yale I also serve currently as the director of the Frankie program in science in the humanities I'm absolutely delighted to welcome you all to the frankly distinguished lecture series I would first like to thank the donors for Richard and Barbara Frankie who's generous support has made this possible before I introduce today's star speaker has been a real coup for us to be able to get him to host him and hear him today I need to make a few practical announcements I'm required to give you all notice that this event is being recorded and photographed for educational archival and promotional purposes including use and print on the internet and other forms of media by attending this event today you agree to the possibility of your voice or likeness captured by these means and used for such purposes without compensation to you and hereby waive any related right of privacy or publicity and I also want to announce our continuing first in a practical way that immediately following the lecture please do join us for a reception in room 108 to celebrate and to continue the conversation now to today's speaker Steven Strogatz Steve is a Jacob Gould Sherman professor of applied mathematics at Cornell University after graduating summa laude in mathematics from Princeton in 1980 Steve studied at Trinity College Cambridge my alma mater where he was a Marshall scholar he did his doctoral work in applied mathematics at Harvard followed by a National Science Foundation postdoctoral fellowship also at Harvard from 1989 to 1994 he taught at MIT before joining the Cornell faculty in 1994 where he has been since what is really remarkable about Steve is the originality that he brings to his broad set of research interests in mathematics early in his career he worked on a variety of problems in mathematical biology which was a brand-new field at the time studying the geometry of super coil DNA the dynamics of the human sleep-wake cycle the topology of three-dimensional chemical waves and the collective behavior of biological oscillators such as swarms of synchronously flashing fireflies in the 1990s his work focused more into another exciting and challenging area of mathematics nonlinear dynamics and chaos as applied to physics engineering and biology and many of you probably have used his textbook at some point or the other several of these projects of his dealt with coupled oscillators such as lasers superconducting Josephson junctions and crickets that chirp in unison and I have juxtaposed these things just to show you what the range of the systems to which he has been applying mathematical structures to in each case his research involved close collaborations with experimentalist he also likes boldly branching out into new areas often with students taking the lead in the past few years this has led him to such topics fun topics as the role of crowd synchronization in the wobbling of London's Millennium Bridge on its opening day and the dynamics of structural balance in social systems one of his seminal research contributions is a landmark 1998 nature paper on small world networks that was co-authored with his former student Duncan Watts this is considered to be one of the most influential papers in network science and was you know it was the most highly cited paper on networks for decades as well as the sixth most highly cited paper on any topic in physics it has now been cited more than 38,000 times according to Google Scholar Steve has received numerous awards for his research his teaching and his public communication and enumerated all of them would take up too much time so I will just mention two most recent awards the Lewis Thomas prize for writing about science which honors the scientist as poet and the same george pólya prize for mathematical exposition so one of the unique things about Steve in addition to his working mathematics he's deeply devoted to the public dissemination of mathematics he has been elected to many many prestigious societies for both of these kinds of work both his research work and the work he does in popularizing and making increasing mathematical literacy he's a fellow of the Society of industrial in applied mathematics the American Academy of Arts and Sciences the American Physical Society and the American Mathematical Society and of course he's spoken at all the shishi venues Ted Aspen ideas festival etc etc and he's a frequent guest on radio lab and Science Friday and he is the author of nonlinear dynamics and chaos which is a 1994 textbook which is the standard on in this subject and in 2009 starting in 2003 he started writing much more for the public as well as his research work in 2003 he wrote a book called sink 2009 the calculus of friendship and 2012 book that most of you probably heard about the joy of X which incidentally has been translated into about 15 languages his current book infinite powers will form the basis of his talk today so in closing I want to quote the mathematician Georg Cantor who said mathematicians do not study pure mathematics because it's useful they study it because it delights them and they delight in it because it's beautiful so without further ado it's my pleasure to invite Steven Strogatz who will be speaking to us today about the beauty of calculus welcome thanks for coming I'm very delighted to be here I just drove down from Ithaca and felt good to be back in Connecticut I'm a Connecticut boy grew up in Torrington just go right up Rudy it's an interesting choice of quote you mentioning Cantor's quote because I'm an applied mathematician so I don't know that I really agree with Cantor I do delight in the usefulness of mathematics but I also certainly delight in the beauty like all mathematicians we have the exciting feature today that I don't have a computer in front of me and we're gonna do this old school the Dean of science is here to help me go through my slides so I will I mean it may seem a little bit strange but I'm gonna be saying next next next and he'll be he's gonna toggle through we think this will work so bear with us if it doesn't but hopefully it will we have not rehearsed thank you Jeff for agreeing to do it okay next should I say next what's the coolest thing to say next okay yeah next all right no don't do next well so yeah on this theme of calculus being useful it's it's an unsung hero of our existence it's everywhere and you may not realize it but without calculus we wouldn't have radio television microwave ovens yeah you can kind of go through them fairly quickly because these are all just we wouldn't have been able to put astronauts on the moon unravel the human genome we wouldn't have nuclear energy or nuclear weapons we wouldn't have ultrasound for expectant mothers or GPS for lost travelers and we might not even have the Declaration of Independence now some of these I realize may seem far-fetched I'd be happy to talk at the end if you want me to try to defend any of these some of the connections are tenuous but I think I could make a case for everything I've just said there so it's it's invisible I mean most of us have no idea that calculus is a part of all of these things and many more but it really is next so the take-home message from that is that calculus help make the world modern now of course it didn't do it on its own it wasn't calculus that put people on the moon you needed engineers you need physicists so it's a think of it as like in a drama you know in a theater there are supporting players this is calculus is sort of the supporting player that you've seen in every movie or every play and you never really noticed that it was there but it's a very important player in the drama next now the question is at least the first question here's a branch of math that many people find sort of arcane and abstruse certainly a lot of freshmen taking freshman calculus would say I don't see the point of calculus both before they took the course and after they took the course and a lot of the students taking Advanced Placement calculus and I daresay many of their teachers would be hard-pressed to say why are we doing this exactly you know calculus is in such a big rush there's so much to cover to get ready for the advanced placement test or for whatever there's so many techniques to learn so many theorems to be understood so much jargon to absorb that the larger context is often excluded or just not even you know it no one would even think of including the history the applications the philosophy the connections to medicine and physics and sociology and everything else but I want to try to give you the big picture of calculus and I don't assume that you've had it or that you even know what it is now it could be that you know I don't know you may be ringers out there maybe all of you are mathematicians right but I don't assume that this this lecture series as you may know is supposed to be a meeting ground for the humanities and Sciences and I'm very grateful to the Franky's for you know for sponsoring this colloquium series it's fantastic to bring the humanists and the scientists and the mathematicians together so in that spirit I'm going to be avoiding for the most part technical math because it's really not very realistic to do that if you've never thought about these things before but I do want to give you the flavor of some of the gorgeous arguments and and and the stories of why calculus is so important so the question I'm asking there is you know here's this arcane branch of math how could it be that it helped to reshape civilization and I'm making a little word game there because it grew out of geometry it was originally about shapes and it reshaped the world okay next so in the spirit of humanities trying to find common ground with the sciences and with math I want to tell you a little story first of two men who met I'd say it was probably the late 1940s early 1950s so my parents would have recognized this gentleman maybe a lot of you will not know who he is Herman woke was a great novelist he's actually still alive he's 103 years old but in my parents generation he was the chronicler of World War two for a mass audience and so he wrote these famous books winds of war war and remembrance novels you'll probably have heard of the King mutiny Humphrey Bogart was in the movie that's his first I think was maybe his first book Pulitzer Prize winning book so woke wanted to write a book about World War 2 this was before I mean that book the book II had in mind grew into two big books those two books winds of war and war in remembrance but at the time he had this vision of I'm gonna write a great novel about world war two he was in the war himself as a sailor and he wanted in particular to interview some scientists who had worked on the Manhattan Project on the building of the atomic bomb and he was told that you could find people like this at Caltech and if you go to Caltech there's someone that you should meet but he's difficult named Richard Fineman so let's have the next slide there's Richard Feinman you see him there in a classroom those are the days when it was all boys probably Caltech is still quite I think isn't it pretty bad gender ratio I think so but anyway in those days probably a hundred percent but you know he was a joker you see him there playing the bongos he used to break into safes Los Alamos he's written many books chronicling his escapades he he loved to tell stories about himself he's a joker there's also a great teacher and had a very deep understanding of physics and won a Nobel Prize for his work in an area called quantum electrodynamics but anyway so he's sitting there when he had worked on the Manhattan Project he was very young like maybe early 20s so woke comes to see him they start talking they talk they have a lot in common it turns out they're about the same age they're both Jewish there's a difference you know Fineman is quite an extreme atheist woek is very devout and so they'd get to talking they like each other and they're having fun back and forth talking about things arguing about science and religion and then after they leave just as woek as leaving walking out the door Fineman says i have a question for you do you know calculus and woke admits you know no i don't and finance says well you had better learn it it's the language God talks so I want I want to leave you with that thought right now the language God talks next slide that's a book then that became the title of Herman wokes book about science and religion sparked by his confrontation with I mean if you want to call it confrontation his meeting with Fineman and it's a terrific book it includes his conversations with Steven Weinberg and various other well-known physicists I don't know if you can read it but the subtitle is on science and religion so if you've ever wondered about trying to reconcile your faith with science or can they be reconciled they certainly address a lot of the same deep mysteries this could be an interesting book for you to look at but anyway so the language God talks now woke as I say did not know calculus and he tells a little story in the book about how you know if this is the language God talks maybe he should learn it so he describes his efforts to go out and try to learn calculus and I take that as sort of a metaphor for what I think we're trying to do here today that here's someone from the humanity making a good-faith effort to try to learn what we have to offer in math so next slide lope tells this little story of how he says he went to the bookstore or he skimmed some freshmen textbooks hoping to come across one that might help a mathematical ignoramus like me who had spent his college years in the humanities ie literature and philosophy in an adolescent quest for the meaning of existence little knowing the calculus which I had heard of as a difficult bore leading nowhere was the language God talks so he sets out to teach himself calculus he tries reading books with titles like calculus made easy he hires an Israeli tutor to help him learn his Hebrew better at the same time learning some math he none of this is working he goes to he finally in desperation he goes to a high school and enrolls in the high school calculus class and after a few weeks he falls behind and gives up and leaves and as he's walking out the students clap for him and he says it's like you know when there's a pitiful showbiz act there's this concept of sympathy applause that's what he felt like so he had this really hard – he could not manage to learn calculus and so I want to dedicate my book to Herman Wouk I've never met him I don't know him I sent him a copy he's in a home somewhere I mean he's very very old his wife recently died I mean I think it's probably very hard for him at 103 right now but next slide so yeah so infinite powers is is my attempt to write for people like Herman woke in the humanities what is the reason that anyone would think that calculus is the language God talks why should have so does my wife would tease me for saying like this why should a normal person learn calculus and by the way I don't feel necessarily you need to learn calculus this is a story about appreciating calculus you don't have to learn it to be able to appreciate it so first let me try to address this question of fineman's cryptic comment the language God talks what did he mean I want to try to illustrate what I think he might have meant with an anecdote about electricity and magnetism so electricity here's a picture of my daughter Leah my older daughter I have two daughters but so when Leah was four years old there she is at her grandmother's house and she's playing on the bed and it was winter and you know static electricity made her hair do that and I took a picture of it and you've probably had similar experiences with static electricity so electricity is something we all kind of know about from playing with static electricity from the electricity that we can plug into in the wall there's also magnetism next slide which you know if you take a magnet so there's a magnet with north and south poles the north pole in red and if you sprinkle these so-called iron filings little slivers of iron on paper around the magnet and kind of shaked it so that everything has a chance to move if it wants to the filings form this amazing pattern around the magnet which we take as indication that there's this invisible field of force around the magnet the magnetic field and these filings these iron filings can help us visualize this amazing pattern of force the directions that the magnetic field is pulling on on things depending on where you are so you've got these concepts of electricity and magnetism which in the 1800s when scientists first began to really try to understand them scientifically by doing experiments on you know electrical currents magnets and the relationship between them and here I'm thinking of people like ampere and Faraday Coulomb lens I mean if you've taken freshman physics you wilford these names so this is a great era of discovering the laws of how magnets and electricity work and how they're interrelated the the key concept turned out to be the idea of a field this this invisible pattern of force around magnets and around current-carrying wires and so on and so Michael Faraday had the idea of the field but he was not very mathematical it was a later person named James Clerk Maxwell next slide there's a picture of Maxwell as a young man usually if you know Maxwell's pictures he looks like an old man with a big beard and and that was him later in life but this is Maxwell as a young man Scottish physicist working in the mid-1800s and he realized that these fields that Faraday and others were talking about could be given a mathematical description and when he tried to give a description of all the laws that had been discovered by ampere Faraday and others he found that calculus was the perfect language for describing what had been discovered about electric and magnetic fields but here's the part that gets spooky I mean we're finding uses the term the language that God talks and you always hear people say oh well math is a language yes that's partly true math is a language but it's much more than a language and this analogy is missing something and this is the part that I consider a bit uncanny and I think it's what Fineman had in mind when he said it's the language such as a language it's the language God talks in this sense that when when Maxwell looked at what he had written down his equations that encoded all the known facts about electric and magnetic fields he had done a translation from physics into the language of calculus but then he could do more with calculus which is he could start to operate on the equations he could manipulate them I mean in math we use that word manipulate the equation I want you to think about it literally like you go to the masseuse because you have a kink in your back she will manipulate you she will massage you and she will work on you and you might start to relax and say something and that's what happens to Maxwell's equations as he starts manipulating them by adding one to another or in our language he takes the divergence of huan'er he takes the curl of another he's doing mathematical manipulations on them trying to get them to open up and talk to him and reveal their secrets and he doesn't know what he's looking for he just knows that he wants something to come out and as he's manipulating at some point he sees and by the way what does this mean really you're transforming the equations from one form into another form that's logically mathematically equivalent so in a way you're doing nothing it's just one thing that's I mean if you had Lalaji you could see the chain of reasoning in your mind you wouldn't need the math he's basically constructing a long argument that's what the symbols are doing for him I mean that's why we use this it's not just a language it's a system of reasoning so he's doing his reasoning on the equations and at some point he recognizes a new equation that has come out because it's the same equation that describes the spread of ripples on a pond it's the equation for how waves move and accept that this is a wave of electricity and magnetism where the electricity the electric field is generating a magnetic field which then regenerates the electric field and they're doing this kind of dance a pod to do it together dancing propagating and he calculates we're always interested in a way of how fast does it travel he plugs in the numbers and it turns out it propagates at the speed of light which had been measured around that time for the first time so this I think must have been one of the greatest aha moments in human history that in that moment and I wish I could be there I mean we all wish we could be there watching Maxwell as he realizes what light is I mean humanity had known about light forever but we never knew what it really was now we suddenly knew light is an electromagnetic wave so what was that that happened there that day I mean that was calculus as a language and as a system of reasoning and it was if Maxwell had tapped into something that that is built into the structure of the universe he wasn't just talking I mean he wasn't like being a poet he was actually learning the language that the universe is speaking so there's this prediction that electromagnetic waves exist that they propagate at the speed of light next slide yeah click and so 1860s he predicts this within a few years Heinrich Hertz measures electromagnetic waves they're real they do exist soon Tesla is using them to create the first radio systems and to do wireless transmission of energy and then Marconi is sending messages across the atlantic ocean and you know you have the birth of the Telegraph and very soon after click we have radio we have television we have wireless so it's not correct to say that Calculus created this but calculus was indispensable I mean if it weren't for Maxwell and his calculus these all these things that we take for granted today they may have still been discovered anyway but maybe later or maybe not at all so so that's what I think Fineman is talking about that Maxwell's equations are telling us something about the universe not just it's not just a language it's a language about something very deep next slide so you may have seen these t-shirts you know this is a standard nerdy t-shirt you can get and God said and then there's Maxwell's equations and then there was light so let's see I don't have a pointer but or do I no I don't think I do no all right I'll do all you don't need to be able to read that language just to see that those symbols what's going on on the left where there's an upside-down triangle that's talking about electric and magnetic fields and how they change in space how they change as you move from some point like think of those iron filings around the magnet if you look at one of the filings and then move to the next one nearby the direction may change a little bit how exactly the field is changing in space is encoded by what's going on on the left and then what's going on on the right is you might see that there's a symbol T in the denominator of one of those fractions that T refers to time it's talking about how the magnetic and electric fields are changing in time and how that is influenced by how they're changing in space so there's this interaction between you know things happening in space and time encoded in this and like I say these four equations then imply the existence of electromagnetic waves although it's not obvious you have to be good at calculus and this particular part of calculus called differential equations which is what these are these are for differential equations the word differential meaning that they express how things change from one point to another or from one moment to another those are differences in time or space so that's the difference in differential okay let's move on so next slide now so the larger point though I mean Maxwell is just supposed to illustrate this broader point why calculus matters this is my claim sort of the thesis of the whole book so if you want to just get the punchline they say it's good to give the punchline at the beginning because people will space out and get tired so I'm giving it to you right now the rest you can start checking your email or whatever you do but why does calculus matter here's the argument first because the laws of nature happen to be written in this particular language in the language of calculus second because calculus is more than a language it's a system of reasoning that taps into something about the structure of the universe it's as if the universe runs on calculus if you want me to say it that way I mean I'm getting a little mystical here but I think it's true in that we see this again and again not just with the laws of electricity and magnetism but if you look at Einsteins laws about how gravity works encoded in general relativity which I was just listening to in Priya's book mapping the heavens which I highly highly recommend if you want a great book about astronomy and astrophysics but so Einstein's equations or differential equations Newton's laws of motion and gravity differential equations the laws of how fluids move like air and water differential equations everything under the Sun that we understand in mathematical terms we understand through differential equations ie calculus it is the language God talks next by learning to speak this strange language I mean it took humanity thousands of years to learn this language but by learning to speak it and discovering that it is somehow tapping into this deep structure of the laws of nature we've been able to use calculus you know in combination with science and technology to remake the world that's the argument now how did this all happen I mean we didn't start with Maxwell remember Maxwell was at the time of the Civil War that's 1860 something we didn't begin with that we began thousands of years or it's commonly you know many books will tell you calculus was invented by Newton and live nets in the 1600s I don't think that's the right point of view I think that's way too parochial of view of what was really happening I would much rather say we we started calculus a few thousand years before that next slide so I'll try to explain why and to do this let me try to personify calculus I think it's the easiest way to think about it think of it as if it were a person um that is not really a person of course it's in it it's a subject but when I say what calculus wants I guess you could then if you're very literal-minded you should hear that as what the practitioners of calculus want the people who do calculus what is it that we want the books textbooks about calculus are like a thousand pages long they're very heavy they weigh like cinderblock wait what what is in there what are we trying to do it looks very complicated but that's misleading what calculus wants is simplicity it the reason it looks so bulky is it's tackling very hard complicated problems and it wants to make them simpler calculus is all about what do you do to make difficult problems easier and it has a grand strategy and this strategy is buried under all the minutiae of derivatives and integrals for those of you who have taken calculus you learned about doing substitutions in every other possible trick that stuff is distracting that here's the real point what is calculus really about I will now tell you next here's how it gets what it wants it has one big idea and this idea runs like a theme you know in a musical theme and variations there is one idea in calculus and if you get this one idea which for some reason we never tell the students this idea but everybody who knows calculus knows it Jeff the mathematician will tell you I'll check this on you you tell me if you agree this is the one big idea the big idea is that we're gonna make problems easier by slicing them into smaller problems now that is an idea everybody has who knows how to solve problems to make a hard problem easier make it smaller break it into parts and work on the parts that's an ancient idea the crazy idea of calculus is stop do that forever do that ad infinitum next this is the big idea I'm gonna call it the infinity principle nobody else calls it that I'm just making it up but but this is what I think is the heart of calculus is that you can make problems easier by slicing them and slicing them and slicing them all the way to infinity next so you keep doing that forever and then what are you left with your left next you're left with infinitely many infinitesimally tiny pieces and the philosophy is that those pieces whether they're short pieces of a curve or little patches of a surface or instants in time or tiny bits of material whatever it is that tiny thing is going to be much more manageable than the original big thing and so the strategy is a two-stage strategy first you cut the thing down to the smallest infinitesimal bits next you solve the problem for the tiny pieces that turns out usually to be fairly easy that's why you did this next then you have to put the answers back together to get the original hole that tends to be hard very hard there's no free lunch so putting the pieces back together is what makes calculus hard but it turns out it will be easier than dealing with the original problem without this strategy so this is the best thing we've ever thought of it's one of the great ideas in human history up there with human rights democracy evolution quantum mechanics I mean this one idea the infinity principle has had as big of a consequential impact on the world as I think just about any other idea next so the jargon you've heard of differential calculus that's what we call the operation of slicing and taking the tiny pieces calculating derivatives amounts to this putting them back together that's the hard part that's called integral calculus and so if you've taken calculus you know that you learn derivatives before integrals that's why because differential calculus is easier you do that first now there's a key assumption in this when we say that we're gonna chop problems into their tiniest bits we're assuming that we can do that forever this is the importance of being continuous next the infinity principle says that will work I mean this strategy will work only on those objects that can be infinitely subdivided endlessly okay if that's not gonna work well then you can't use calculus so the calculus only works on these things that are infinitely sub divisible we call those things continuous meaning from the old Latin roots what con plus tener a would mean holding together so a continuous thing is that which holds together in that sense that it's it's all touching itself so continuous objects are grist for calculus next but notice the creative fantasy in case you think math is all very objective and rigorous I want you to want to disabuse you of that idea that's only half of math that's the second half the first half is creative fantasy you have to have imagination and creativity all math is like that then you tidy up and make things nice and logically pristine but at the beginning it's wild fantasy imagination and desire and in this case the desire is to pretend that the world is like this pretend that everything can be infinitely divided as much as you want why is it fantasy because that's not really correct the world is not like that we know today depending who you talk to but but you know this was a debate going back to the you know the ancients that when you think about atoms what's the word atom mean it means literally a Tom it means uncuttable right the things though things that cannot be cut the smallest bit of matter is an atom that cannot be cut any further and so the Greeks you know in the time of Democritus were arguing to atoms exist is the world grainy is it made of tiny things or is it infinitely sub divisible forever and not just matter but space and time is there a smallest interval of time is there a smallest amount of space this is a live question today if you talk to the string theorists or people doing quantum gravity they will tell you yes there is a smallest thing it's called the Planck scale after Max Planck there's the smallest unit of space something like 10 to the minus 35 meters way way way smaller than the smallest particle that we know there's a smallest unit of time which is the time it would take light to travel that distance and there's nothing smaller that we know of so it is not true according to modern physics that you can infinitely subdivide time and space and matter do you think we care about that in calculus yes and no okay we want to get things right but we also want to make progress and so we're gonna pretend that's what I mean by creative fantasy we will pretend that the world is infinitely sub divisible next so calculus then in my definition and essentially nobody else's but I think it's it's really the heart of it is calculus should be thought of as the use of the Infinity principle to solve or to shed light on anything that's continuous whether it's a shape an object something moving any kind of phenomenon that's the strategy and so if I had to boil it down to what calculus has really been obsessed with it's three things next oh okay the three are curves and curved shapes so geometry but specifically about things not made of straight lines and flat planes but that are curved and sinuous that was a big problem in ancient geometry calculus came in to solve that and that's where it began dealing with curves so that let me go back sorry to let's just go back one so yeah so we're gonna start with curves but calculus has you know the three things that it has focused on throughout its like twenty five hundred year history curves motion and change so if you want a little mantra that's your mantra curves motion and change some people like to define calculus as the mathematics of change that that works pretty well actually because a curve you could think of as changing direction and motion you could think of as changing position so if you had to say it in one word calculus is the mathematics of change but specifically continuous change never-ending ongoing change okay next so let's begin with curves now here's an ancient problem figuring out properties of a circle you might think that's trivial I learned that in high school geometry it's not trivial it's not easy to figure out like think of how weird the number pi is right everybody is fascinated by pi with its infinitely many don't show any pattern why is pie so weird because papaya is a creature of calculus it's not a creature of geometry the fact that it has these infinitely many digits that don't repeat and don't you a pattern is already a clue that there's something with the Infinity principle at work here you know when high school you just memorize pie maybe you didn't think why is that the number how does anyone calculate that number you have to use calculus that's how so let me remind you pi is defined as the ratio of the circumference of the circle to its diameter okay circumference the distance around diameter the distance across but a big question if that's a definition of pi fine that's that's one property of a circle what about the space inside the circle the area inside the circle you may have memorized a formula for the SATs PI R squared the area of a circle R is the radius that's half the diameter the distance from the center of the circle out to the edge there where does that formula PI R squared come from that's a calculus result you will not find it in Euclid by the way I mean if you think it's Euclidean geometry you don't understand you can look at Euclid you will not find PI R squared and Euclid Euclid does say the area of a circle is proportional to the square of the radius but he doesn't have PI for him pi is not even a number that's you know okay so you have to wait till the like two more generations for Archimedes and that's when we start to really begin calculus he's the great maestro of the Infinity principle our community so let's see the next slide here's an argument this is not really Archimedes argument it's closed in spirit to his argument for calculating the area of the circle but if you've never seen it I want to show it to you because I think it's very dramatic and it's an example of what it feels like to have an aha moment in math okay so what I'm doing there is I'm taking a circle and I've chopped it into four pieces you could think of them as slices of pizza if you want to an abstract mathematical pizza and I'm rearranging those slices into this funny shape on the bottom which I'll refer to as a scalloped shape because you know it's got these bulbous curves on the bottom and I've put on the bottom PI R to indicate that the amount of rust if you thought of it as a pizza the length of the crust on the bottom is pi times the radius why am I saying that the whole crust is 2 pi times the radius right that's the circumference pi times the diameter the diameter is 2 times the radius so 2 PI R is the circumference half of the circumference half of the curvy part is on the bottom half is on the top so it's PI R on the bottom and then the edge of the pizza that straight piece this piece that was that's just the radius of the pizza so at the moment the strategy is if we could figure out the area of that thing we would then know the area of the circle the trouble is we've looked like we need the problem worse that shape is harder than the circle but what we're gonna try to do is somehow with the use of the Infinity principle change the circle into a shape whose area we know that's the grand strategy all right so next slide the thought is maybe the reason that shape on the bottom was bad is we didn't take enough slices so instead of taking 4 slices what if you take 8 and you arrange them like this you can you see that the shape is getting better it looks like it's trying to turn into some shape you recognize anyone want to volunteer what shape it looks like it's trying to become what but looks like it's trying to be a parallelogram right it's almost a parallelogram because this side is parallel to that side the bottom is not straight like it's supposed to be in a parallelogram a little bit curvy but um here's I want to test your dexterity for saying it Jeff can you toggle back and forth where what I mean is when you go to the optometrist they will sometimes say better worse better or worse can you do that between the two slides better worse which is better better okay go one more forward better ok that's was 16 that's better now can you see what's happening actually keep your eye on the tilt of this side can we do that go back 1 see it's more tilted and more so now going forward it's getting less tilted if you did this infinitely often you'd have infinitely many infinitesimally thin slices but they would be standing upright and you would get that isn't that cool and notice the time the bottom was always length half the circle PI R and this was always the radius so now you know how to find the area of a rectangle the length times the height ya PI R times R that's PI R squared that is basically the real reason the area of a circle is PI R squared because it can be morphed into a rectangle with the help of the Infinity principle now if you're really a tight mathematician out there you know I've done some fudging here just relax then this is basically correct but you know there's some loose moments in the park but remember what would the venue it is come on give me a break but that's the gist of it okay so that's an example of the power of infinity that they also notice that the shape becomes best at infinity you know the optometrist test keep going things get better at infinity that's a key principle in in calculus things are better at infinity okay next so as I said earlier there are these three obsessions curves motion and change and for the sake of accessibility I'm probably going to mostly talk about curves it's the least technical but we can talk about how calculus has revolutionized our understanding of things that move on the earth and in the heavens and anything that changes whether it's traffic on the highway whether it's the level of HIV virus in the bloodstream of patients who are infected I mean it's been applied to everything under the Sun next so back to curves just because I want to show you one more masterpiece of Archimedes to show you another a little bit of how the infinity this is actually more honestly what Archimedes did take a circle you can make shapes with a circle I mean they started with circles that was the fundamentally first mysterious shape and then you could make the shapes like a cone and if you slice a cone and with planes of different tilts you can make a shape like that called an ellipse or you can make a shape like this if you slice parallel to the side of the cone you make a shape called a parabola so originally these were thought of as sections of a cone and they're still called conic sections ellipses and parabolas and also hyperbola so let's not bother with them for now so next so one question that Archimedes wondered about this this is like 250 BC he's in Syracuse on the island of Sicily a part of the larger Greek Empire the Romans would like to invade and the fact they do invade and try to take over Archimedes you know take some time away from his math to build war machines that according to legends you know Plutarch tells us in his history that Archimedes made these fearsome machines that could grab the Roman ships out of the ocean giant cranes lift them up shake them so that the soldiers would you know awed the sailors would come out like if you were shaking sand out of your sandal you know I don't know if it's really true but but there's all kinds of stories about Archimedes but we do know this because he wrote these treatises that we still have about how he found areas of curved shapes so one was what's the area of a shape like that so-called parabolic segment next this is amazing here's his strategy to figure out the area of this curved shape he is going to regard it as a sequence of shapes made of straight lines so you know in cubist painting you draw pictures of people or other things Braque and Picasso they're using all these rectilinear shapes Picatta Archimedes is doing that in his vision I think it's an amazing vision to see this curved shape as this combination of triangles so what he does is he puts a big triangle inside and then there are two smaller triangles like a shade of gray sitting on the side of the original big triangle and then you can see there's a little bit of empty space left under the parabola he could wedge in more triangles in there and keep doing that he's gonna exhaust all the area until it's all nothing but triangles so that's his concept but now which triangles exactly should he put in there next oh I should say before I get to that here's the big punchline he's going to show that the area of that parabolic segment compared to the area of that first big triangle there in the ratio four to three and that's not obvious but that's what he's going to show the parabolic segment is four thirds as big as the big triangle and if you are interested in music you should think about the numbers four and three do you have any musical people here yell is great at music you do music do you but I'm wondering oh you did wow there's a musical family are you brother and sister yeah cheese but this is a hard question I'm asking and do you know anything about why four and three are related to music you do why oh that's interesting could be a time signature you have a fun oh good this is now you're on the right track so an octave is two to one tell us more what you mean what do you mean two to one for an octave like if I take two strings suppose I was playing a guitar and I put my finger on a fret halfway up the string and I pluck one string and if it was the same other the same string but with the same tension and everything but not with my finger on the fret the one that's long and the one that's half as long they will sound an octave apart right the one that's short would be octave higher twice the frequency if the frequency if you do this with strings but in the ratio four to three Pythagoras and his people legend according to legend discovered laws of musical harmony the three to two ratio makes do you know so 3 to 2 would be 1/5 4 to 3 would be called a fourth I think so and I don't know music but my daughter who's my younger daughter told me think about the Star Wars theme that's a third no that's a that's a fifth that's those are separated by a fifth sheep – alright but then she said but here comes the bride I didn't do it well bah bah bah bah bah that's a fourth that's four to three so a 4 to 3 ratio the point ok the Florida 3 ratio is something every self-respecting Greek knew 4 to 3 was considered beautiful because it's connect to music and so you could imagine how excited our communities would have been that the parabolic segment to the big triangle it's in a 4 to 3 ratio okay but why why is it 4 to 3 so let's see the argument he takes this triangle and now the way he constructs that triangle is he takes the line that defines the bottom of the segment he slides it up until it's just touching the parabola at one point that's in the jargon the tangent line try to be tangent to the parabola at one point so if he then that defines a unique point and then he builds the triangle touching that point then he's going to use that parallel sliding trick to build new triangles next slide so you see then he builds those little triangles the same way slides the sides parallel till they touch at one point builds new triangles and what he can prove with the geometry that he knows and properties of the parabola that he knows is that those new triangles will have 1/8 the area of the original big triangle not obvious but he can prove that and that and he shows that that rule is true at every stage whenever he creates a new triangle by this technique it will always have 1/8 the area of the triangle it came from and so if you total up the areas so far we had one for the big triangle 1/8 plus 1/8 makes 1/4 so we have 1 plus 1/4 and then if you believe me about this rule that you're always going to get a quarter of what you had before you're led to next an infinite series which is 1 plus 1/4 plus 1/16 plus dot dot and that turns out to add up to 4/3 so if you've had high school algebra you've learned a formula for that but Archimedes did not have high school algebra because algebra is a product of the Middle East right I mean algebra is gonna be invented about like eight hundred years later in places like Baghdad and I mean there's a kind of geometric algebra that the that the Greeks know but still for them geometry is the thing so he does his algebra this way when he calculates this infinite series he draws a picture which has four squares here's a square of size one in area the whole picture is four units of area right because it's four big squares so that's the 4 in 4/3 watch the see what's going on you take one that's one big square then you take a quarter of that square that's the thing marked a quarter then you take a quarter of that that's the thing marked a sixteenth and so the total of the grey is the infinite series he wants but he says by inspection I can see that that's one-third of the whole shape right because what is it it's a big square plus a second size square plus a third sky square but that's also copied over here big square another one of size a quarter the white ones you have three copies of the same structure in this picture of size four so four thirds is occupied by the gray so that's what we would call a proof without words but you know what's really nice about Archimedes here too now speaking to the rigorous mathematicians for a second is he actually calculates the error term correctly he doesn't just say dot dot dot I mean you're talk when you teach calculus properly you don't say dot dot you say calculate the error term and make sure that the error term goes to zero and the error term is that little tiny square up in the corner and he cannot see exactly how big it is you know he could another words he can calculate the finite series not just the infinite series okay so anyway that's a start that's an example of Archimedes ingenuity with the tools he has which is geometry and ingenuity but he doesn't have algebra he doesn't even have decimals right decimals are being created over in India so this is a big world story the story of calculus it's not like don't think of it as just a european greek thing it's not but but this part of the story is okay next so what about Archimedes today this is another case of this invisible presence of calculus you are using Archimedes all the time when you go to the movies and possibly when you go to the doctor but you don't know it so what am I talking about you know how he built up that curved shape the parabolic segment out of triangles well you can make a shape in computer graphics out of triangles that can approximate any smooth surface so you can make a picture of a mannequins head by just triangulating it more and more finally and that is the technology that goes into something like the next slide this kind of movie Jerry's game which you can find on YouTube don't do it right now but but so Jerry's game was the first animated movie completely computer-generated movie that had a human character that it was emotionally expressive we had earlier movies like Toy Story but that was you know those are toys that this is a real human being who acts like a person except he's made of polygons in fact he's made of triangles but millions of them so you don't see the triangles but they're there and and the people at pixar who made this you know animate all those polygons but how to make a polygon that cut up sir how to make a sequence of triangles that will make the shape that you want an old man's face with his wrinkles under his eyes and so on there's a lot of calculus in computing the right triangulation to make whatever you want next so you know this is probably a movie you know better Shrek telling donkey that onions have layers and his little trumpet like ears is round belly these are all smooth surfaces that required millions of triangles and so this is the technology that Pixar and DreamWorks and everybody uses nowadays it's an Archimedean idea that you can represent any smooth surface with triangles next now here's a little more medically oriented example if you just think it's kind of frivolous for kids movies it's not here's a gentleman who you can see on the left this is him before surgery I've his eyes have been you know pixelated because they for a privacy but anyway this guy you can see his jaw is sticking out in a way that it's not just cosmetically on pleasing for him but but I think was causing him medical problems this malformation and then there's a scan of his bones taken in a medical scanner so that's him before the surgery now what doctors would like to do is figure out if they would cut out certain bones in his jaw and then reattach everything surgically you know sew him back up what's his face gonna look like after surgery and the issue is there's a lot of other tissues than just the bones bones are pretty rigid but there's soft tissues the face is made of skin there's all the tissues behind the face there's cartilage there's you know a lot going on tendons all kinds of things to think about so the branch of math that that's involved here is we want to make a mathematical model of all the soft tissues and bones in a person's head such that when we change the conditions by cutting out bones and reconnecting things they're going to be many elastic forces they're going to be parts of the skin and cartilage and bone that are pulling on each other with this new configuration and things are going to shift obviously that's the whole point of doing the surgery so people have made these elasticity models now okay elasticity elasticity is a branch of engineering and applied math that uses calculus if you take a course in elasticity in the engineering department at Yale or anywhere else you will be doing tensor calculus you'll be doing partial differential equations it's calculus okay it's been applied everywhere and so I'm trying to say that when this poor guy had his face redesigned the doctors were able to tell him what his new face would look like the computer model said it would be that third image from the left they did the surgery and that's his new face and you can see that the prediction basically got it right now this has been quantified by referred to the study in the book you can check the original paper if you want to see the data but I mean with with calculus and computer modeling and lots of careful scanning and all this other stuff you can essentially build like a flight simulator for surgeons the surgeons can do the study because you know it's very serious to cut someone's bones you don't want to do it wrong you're not gonna get a second chance so they can practice on virtual patients before they do the real thing with the help of calculus and computer model so that's an example but now what does it have to do with Archimedes well what when I talk about this elasticity model what is the model next slide it's the kind of thing that is built on triangulation it's old Archimedean idea except that instead of just triangulating the face I don't know if you can really see but they've gone behind the face and they have three-dimensional analogues of triangles that are tetrahedra that are modeling the soft tissues and they have different stiffnesses and elastic properties and so they have this gigantic model of billions of polygons and and simplices tetrahedra and other things so that they can figure out all the forces and how everything's gonna rearrange itself after you do whatever you're gonna do with the cutting of bones so this is just to give you an a sense of how today through computers and calculus we are living Archimedes legacy all over the place okay next all right so I don't remember quite when we start I'm just about ready to wrap up I'll give you can I have like five more minutes I just want to talk a little bit about what you might think of as calculus because everything I did so far probably doesn't look like the calculus you had I understand that and that's because you know calculus was going on for thousands of years before algebra came along the the calculus you'll learn nowadays is all algebra they're all formulas it's a tremendously powerful thing using symbols that came from the east that came from India and the Islamic world and gradually made its way into Europe at the beginning of the Middle Ages so like 1200 1300 you start getting algebra in Europe and algebra then collides with geometry and that's when differential calculus is soon born in the middle of you know like say the 1600s or so but so 1800 years after Archimedes algebra and geometry collide but before Newton and Leibniz you have people like firma and Descartes creating the subject of analytic geometry this is this fusion of taking a curve like here I'm showing a parabola just like Archimedes was studying but whereas Archimedes thinks of it as a section of a cone now to Descartes and firma it's an equation like y equals x squared the relation between these symbols in this very familiar picture that we think of as Cartesian coordinates with x and y-axes all the classical curves of the Greeks can now be thought of as equation and so you have this great you can play two ways with them you can visualize them or you can work on them as equations so analytic geometry is a great breakthrough that then sets the stage for calculus in that now there's all kinds of new curves you can make you can write down an equation there's a new curve it doesn't have to be a section of a cone you could do anything you can create a whole jungle of new curves and start asking questions about them what's the area under the curve or we saw how Archimedes use a tangent line in his constructions what's a tangent line gonna be like you know now that we're doing it with algebra so that became a big question for our confer a Descartes and for MA and I just want to give you the intuition about how to think about tangent lines imagine a microscope and I'm gonna zoom in on this point so let's see the next slide if i zoom in you don't see what looks like a parabola anymore but I'm now just sort of this is the part in my microscope field of view the curve has gotten straighter sort of like what we saw when we were doing that Archimedean pizza proof right in the bottom got flatter and flatter this curve is going to start getting straighter next slide if we zoom in more it really looks very straight I mean you can see from the numbers Oh point five point five oh five we have zoomed in a lot so under great magnification a curve starts to look like something made of straight pieces and this is the great idea behind differential calculus that you can approximate curvy things with straight things or what in the jargon we would say local linearization but it just means that you can think of a curve as made of lots of straight pieces but they have to be infinitesimally and this is again the infinity principle that to deal with the problem of a tangent line to a curve defined by any equation we can solve it by zooming in enough and then figuring out slopes of straight lines so I want to end with something that is more modern still you know for firma and Descartes curves were just curves they still wanted to do geometry but nowadays we think of curves as meaningful about the world curves tell us about stock prices going up and down they show your blood pressure your heart rate you know everything can be graphed and so for us curves are now visual representations of certain kinds of data so let's see the next slide they can represent in particular things that are moving and changing the dynamic world that calculus describes began with the study of things that that could be represented as curves so just to give you a taste of what that's Mike let me you look at the next slide I want to just remind you of something so don't start this video yet this is a video taken in 2008 it was a at the Olympics when Usain Bolt was running the hundred meters so that long ago only 11 years ago Usain Bolt is a sprinter from Jamaica who doesn't look like a sprinter he's 6 foot 5 very tall very gangly he's also very mischievous Joker he loves to joke around and he never really ran the hundred meters that was not his race he was known as a 200 meter actually used to run the 400 meters a longer race and he was great at that because he wasn't so great at starting but once he got going he was very fast and but he sort of wanted to try the hundred meters and his coaches said you're never going to be any good you know you don't look like a hundred meter sprinter those guys are short and muscular and he said I could you know I could get muscular so they started pumping him up he got more muscular he had five races in the hundred meters competitively at this time that he was entering the Olympics so think of him as a beginner ok don't we mutt done it five times in competition but he was already able to contend with the best in the world and so I guess there is nothing to do but just oh are we gonna have audio I guess we'll find out let's see if this will work because it's sort of fun to hear the announcer let's try go ahead no I guess not well anyway you can see the runners are getting themselves ready and watch what happens now do you see what he did at the end where he put his arms down that is not normal he's coasting he's so far ahead he's coasting that he's in fact he's slapping his chest I mean you saw there was this whole other group of runners the world's best runners with clear daylight between him and them and he's fooling around going like that sorry banging his own chest so and that's like I say he's not being disrespectful he's just mischievous he's celebrating a little bit he's so far ahead so there was a question let's see the next one I mean that shows you at the end of the race and you won't be able to see it from where you are but if you look very very closely I can tell you that that's his shoelace he has an untied shoelace so he's way way ahead and that question among sports aficionados was if he had run hard what time he set a new world record of I think this was nine point six nine how fast could he have run if he had really run all the way to the end so that's a calculus question that we could answer so let's see the next slide this is the way that they record performances like this is that there are detectors every 10 meters and so we know where at what time he crossed 10 meters 20 meters 30 meters down the track so we have this split times those are the dots we don't have his position in between the dots but with calculus we can fit a curve in an optimal way two dots and you know what's the smooth curve that has zero velocity because we know it when he began he wasn't running it he's in the block you have to start with zero velocity zero velocity would mean this curve has to have a flat tangent it has to that the slope tells you how fast he's run when it gets steeper he's moving faster so you can see at the beginning he's not moving that fast but then he gets faster and faster now what we could do would this having fit the curve through there we can then calculate the slope at each point using what I just said you zoom in as if with a microscope and then record the slopes at every point and if you do that you get the next slide which shows his velocity as a function of time in meters per second so you see he speeds up and then somewhere after like about 8 seconds he's very visibly slowing down which we saw by just watching him as he's goofing around at the end so you can answer questions like what was his maximum speed you can just read that off the graph when did he achieve his maximum speed and so on but I haven't actually done the calculation of how fast could he have gone if he kept running hard because actually we don't need to do it the next a few months later there was the World Championships in Berlin and then he didn't fool around so I don't have the video of that although it's on YouTube you could watch it if you want but people who thought that something might happen in Berlin so they went with radar guns like the type that police would use to you know first catching a speeding car so they they put little reflectors on the back of every runner their aiming their lasers at the runners and these things could measure data you know like 100 times a second or something like that and so the next slide shows his speed as a function of time as detected by the radar gun at Berlin and you'll notice some interesting thing his instantaneous speed is that wiggly curve and the average that's sort of going through the Wiggles is shown dotted do you have any idea why there were all those Wiggles oh I'll take another young you're a Brit here so faster should be on Jeopardy you want to tell what do you think the Wiggles are do what time for his feet to leave the ground he'll though yep you're seeing his individual strides you're actually seeing the fact that when when his I mean what is running running is a series of leapings and landings right and when he lands he's slowing down a little bit and when he's leaping he's going a little faster so that is all being resolved in that Wiggly curve and you'll I think there should be 41 because it's known that he takes 41 steps every time he runs the Hun everybody else does 44 but he's tall so he always does 41 steps 41 strides and what I find interesting about this picture is that we don't care about the Wiggles having measured them we're not interested in I mean they're sort of interesting but if we really I mean what we really want is the trend and I think that there's a kind of metaphor here that if you start getting too precise sometimes you're picking up information you're not that interested in and so this has been a recurrent theme in the story of calculus that like I said it has creative fantasies in it where it ignores certain things and focuses on others and it seems to me that this is a case where you know we could have just drawn a smooth curve through his data and we would have gotten really just as much information as this more microscopic view so yes it's true if we start going very microscopic calculus may break down but maybe we don't care maybe we've really gotten the essence by looking at this bigger picture so this was said better than I'm saying it now by Picasso can I have the next slide yeah art is a lie that makes us realize truth right I mean no picture is really realistic and yet if it's if it's great art it often captures the essence of the truth so I would like to suggest that the calculus has done the same thing for us in science it's a kind of a lie that has helped us realize the truth thank you thank you so much for fantastic talk so I hope I'm open Americans okay happy to take any talks or criticisms compliments you name it okay yes declaration yeah sure that's that as a teaser isn't it okay here's what the story I tell in that in the book infinite powers but it's not my idea this is an idea that goes to back to the historian of science I Bernard Colin a great Newton scholar so so Colin points out that the founding fathers in the u.s. all you know these Enlightenment thinkers we're very much interested in in Newton and Euclid and logical reasoning so in particular think about the structure of the the preamble in the Declaration right there's this famous ringing line we hold these truths to be self-evident that all men are created equal etcetera this this it's interesting phrase we hold these truths to be self-evident where does that come from if you look at Euclid so Euclid who gave us the first geometry textbook begins with the self-evident truths in math we call them the axioms right the axioms are the things that are supposed to be self-evident you just accept them of course they're true and then starting with the self-evident truths and commonly agreed-upon rules of reasoning you erect this edifice of propositions and theorems building on the axioms now if you look at the rhetorical structure of the Declaration it has a Euclidean structure because now what that is it starts we hold these truths to be self-evident then the axioms are listed the right to you know pursuit of happiness and all that and then there's an argument and then there's an inescapable conclusion a theorem which is essentially that the colonies have the right to separate themselves from from the tyrant king so why would Jefferson have written this Euclidean doc can I call it Euclidean why because I mean because rhetorically it goes back to Euclid but if you look at Newton in the Principia so-so Euclid is writing about geometry but Newton is using this geometrical style of argumentation to write about the world because the system of the world where he begins with the proposition he begins with axioms which are his laws of motion and then using logic derives how comets move how the planets move the tides I mean in other words if you want to make an argument that no one can argue with to the thinkers of the Enlightenment the best way you could do it was to use a geometrical style self-evident truths first theorems later and I'm not making this up I mean you can see it in Spinoza so in Spinoza's ethics he the actual title of Spinoza's ethics book is ethics demonstrated in geometrical order he tries to derive the rules of ethics with geometrical reasoning that's considered irrefutable so I really think it's no accident and like I say but I Bernhard Cohen had the idea first that the declaration is this quintessential Enlightenment document and just to top it off we know that Jefferson had a Newton Euclid fetish so for instance he had a death mask of Newton that's the kind of thing people used to do right someone is dead you make a big wax or I don't know what you do you make some kind of impression of their face and then it's like whoa cool I have Newton's face so so Jefferson had one of those and he also wrote to John Adams back when they were no longer presidents and were just kind of shooting the breeze in 1812 I think it was he's writing to jet to John Adams and he says you know like I'm really happy to be done with politics and he says I've given up newspaper good up reading newspapers for Herodotus and Thucydides Newton and Euclid and I find myself much too happier so so that's the argument but but those we also see Jefferson using Newtonian principles like he designs a plow that will cut the soil in the optimal way he uses a branch of math called calculus of variations what's the correct shape of a plow minimum resistance as it's going through the dirt and to put that you know remember he's interested in agriculture to put to plow as much dirt as fast as you can as easily as you can he doesn't know calculus of variations Jefferson but he knows that that's what he needs to know and he gets the help of a professor at Penn to help him design this optimal plow which you can still see down and monta so what was that I pronounce it cello yeah what have you been there Monticello have you been yeah Jefferson's all place yet yeah it's there anyway yes good so you have a question too or comment yes hi so you talked a little bit about how calculus / geometry and algebra were developed separately and I've noticed that I guess quantum mechanics comes to mind that you can look at it and either kind of a differential calculus way or a linear algebra way with the calculus being more intuitive but computationally more difficult and the algebra being less intuitive but computationally easier and I was wondering if you could kind of speak to the relationship between the two sides I think I missed one word early in what you said are you asking for the relationship between algebra and geometry and the two ways of thinking was that yeah yeah okay I mean I feel like I missed a key word but that okay that's just the question okay well yeah I mean they sort of represent two styles of thinking right that that algebra has this virtue of being very systematic you can as I said earlier could massage equations manipulate symbols you in a way don't have to think like you don't have to have insight to do algebra you just have to do the bookkeeping correctly and that's one of the great strengths actually live notes when he's doing calculus his version of calculus says and actually he's the one who gives us the word calculus he says you know what the virtue of my calculus at first he calls it my calculus my way of thinking my style of reasoning Newton doesn't refer to it as calculus by though he calls it fluxions he's interested in the world in flux so but anyway so Leibniz says something like the virtue of my calculus is that it frees the mind frees the imagination you don't have to spend any effort thinking you're just it's like knitting you know you just move the symbols around it's all good whereas geometry although it's extremely intuitive and visual requires tremendous ingenuity and you know this is partly why it takes two thousand years after Archimedes for anyone to really advanced calculus much beyond what he did because he was such a towering genius that and there were no tools for everyone else you know I mean if you couldn't reason like Archimedes geometrically or invent I mean exaggerating because there were things that were done after him but it was very progress was very slow when it was just geometry so algebra really did speed things up but I mean to me algebra is quite sterile I find al Jabar is not about anything whereas geometry is about something it's just very hard and so when you put geometry and algebra together now you're really cooking and that's what happened in the 1600s at one thing I feel a little bad about and I want to I can't really properly correct it but I just want to show you that I'm woke and I mean I really am I'm aware of this is that women are part of the story and so are people in India and China and Japan and in the Mayan civilization I mean there's a lot of calculus being done around the world I mean with women honestly it's only sort of around the 1800s that women start to be allowed to don't even go to university and hear lectures and stuff but but as soon as they're able you get great people Sophie Germain and Sophia kovalevskaya and certainly in the 20th century we've got lots of tremendous women mathematicians all over the world so so although I've told it a certain kind of dead white male European way I don't mean it that way and it's not that is not an accurate picture of what really happened so I do try to tell it the right way in the book including Katherine Johnson you know who we all know now from hidden figures you know who helped put astronauts on the moon and bring them back safely using calculus of course yes just very quickly thank you so much and also if the internet had been around when Luke was looking he would have found out that in 1910 Sylvanas Thompson wrote calculus made easy he didn't know he looked at that book he absolutely looked at that book yeah yeah that's one of the books he looked at it's a famous book calculus made easy that's the title no he looked and found it I'm sorry if I had meant it it sounded like I said you looked for it but didn't find it no he did find it he read calculus made easy by Thompson and many people say that that is still the best book to learn calculus from so you may if you're interested you may want to look at that book it is quite funny and quite skillful pedagogically but it's sort of an old book it's a hundred years old and I think it's you know it probably has some deficiencies but for whatever reason it didn't really work for Herman Wouk no yes oh wait okay who's next I have two related fast questions at first thank you for the talk I hope even like high school students or first-year college students get exposed to calculus this way and my first my first take is the first idea you were talking about that nature speaks calculus yeah what how would you think if we reframe this saying that calculus is our only tool to measure only what we can measure and only what's quantifiable because that sounded to me that it eliminates so many things in life that are not quantifiable and they're very abstract and they exist in nature and relatedly which is very fast and related my second question is you seem to embrace modernism to a great extent on the talk which is understandable because we're talking about calculus but if you can have a brief answer to that like what's your take on post modernist thought that question the rigidity and the privilege stone of modernism okay so the first question is is interesting let's see if I can rephrase it so you're saying that calculus does well at measuring at quantifying the things we can measure to some extent but the things we can't measure or whatever doesn't have anything to say about those and I think I probably agree with you I do I I'm not sure maybe I mean because the state of not being able to measure something is often a temporary state right sometimes something can't be measured now but can be measured later and my claim would be that when it can be measured but maybe your claim too would be that once we can measure it calculus will turn out to be just the right thing for it would you agree with that not really uh-huh so no so there are things that we can't measure do they affect us in any way these other things okay right so things like that uh-huh yeah right so calculus of course would have it be hard-pressed to say anything about that so I think I accept that first point I mean just that there's something I wonder if though if I'm agreeing too quickly because well just that there's some in towards the end of the book I mentioned one thing that I regard as very spooky to me that makes me think but I don't think it will refute or even contradict what you're saying because it's still in the realm of what can be measured I'm just thinking of a case where there are certain properties of subatomic particles of electrons that have been measured and accurately predicted to eight decimal places with this fineman's part of physics quantum electrodynamics that that we can understand something in the gyromagnetic ratio of the electron can be calculated eight digits after the decimal point why am I making a big deal about that I mean that's that is so unbelievably accurate but it wasn't built into the theory we didn't know it would agree it does agree to eight digits now you're not saying it's some kind of like equivalent to faith or social construction or are you I mean I'm saying it can't possibly be because it works so well and we didn't know that it would work that well ahead of time that to me it's capturing something that I want to call the truth or close to the truth I wouldn't say it's the end-all be-all truth because truth is always provisional but it's a very close to the truth and if you want to tell me you don't believe there is such a thing as truth that anything can we really do disagree but but I don't hear you saying that I think you're saying there's some domains where we have sort of almost hear you saying the boring domains the things we can measure or no you're not saying that you don't want to go that far okay okay you're just raising the issue yeah well certainly the things that can't be quantified I don't see what calculus would have to say about them because as I say it's all about this principle of regarding problems it applies to problems that can be viewed as continuous in some way blah blah blah so so for religion or whatever it does I don't think it will have anything to say about that but then you asked about modernism and I don't feel well enough equipped to even know what that I don't know enough about post-modernism to know how to respond to that okay maybe I should try someone else's comment or a question yes let's hold on there someone is gonna try to pass a microphone over to you that's okay that everyone will be happy if you just talk in there um you mentioned the title of the book that calculus is the language of God right this is what Feynman said I think I personally don't believe in God in case you're curious okay but both anyway well then okay good you could ask I could pretend and if god thank you if God if the reason for our feelings is a result of God you know everything is permeated through God then calculates it should be able to mention or anything let's see or or Fineman to know when he was talking about and calculus really isn't the language kind what is it what is this a comment or a question you're saying it's more of which is like a rhetorical question that if I if calculus is God's language and God should be all powerful and most conceptions of God and anything again well a couple of big if statements there yeah I don't know I I don't know I don't okay so the word God gets used a lot in different ways right Einstein talks about God but he was careful to say not the God who cares about you know whether you've been naughty or nice it's it's the God the creator of the universe God who makes the laws of nature etc or maybe just the laws of himself like he always used to say it's talking about Spinoza's God meaning the God that is nature not to God not a personal God who cares about sin and things like that so that if we just mean nature I mean debate it's still a big mystery I do feel this is a mystery that's running throughout the story as I tell it that why why can math I do actually didn't really raised this question clearly let me try to raise it now why why does math work in yeah okay let's suppose we restrict ourselves to the domain you're talking about the domain of things that can be measured it's still a question why does nature obey logic in particular human logic that's it's not obvious to me that that should be possible we're not the best thing imaginable where you know a few generations removed from primates and and we have logic that can get the gyromagnetic ratio of the electron to eight digits that's weird that our puny logic matches what really goes on in the universe this well it didn't I don't know why that worked why does it work this is the question that Eugene figner famously asked you know the unreasonable effectiveness of mathematics and lots of people have wondered about this some would say it's because God made the world very orderly others say if the world were very disorderly we would not be able to evolve to even our puny level of intelligence like only universes that support enough order to allow intelligent life was the only universes we can observe so it sort of that's the anthropic view of why the universe makes sense yes oh okay oh yes please go ahead um so I was like umm so I've been wondering like I've researched about the black hole picture like like I wanted to one like I was wondering did calculus contribute to that like in a big level or because I know that calculus contributes to stuff like does it contribute like greatly to the black hole picture or to the black hole picture do you mind talking about that we have a put like one of the world's experts right here she actually helped make the black hole picture I think it's fair to say well you tell us yes absolutely calculus is really fundamental because as you saw there were curves that needed to be computed to be put together that's a great question and probably a great place to stop why don't we go to the reception thank you oh the reception yes please join us at the reception

Calculus by s.m yusuf Exercise 1.1 Lecture 01-Q.1 to Q.5



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A famous inequality is derived in this video …..
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question number one if a and B they both belong to set of real numbers and you're given that a plus B is equal to zero you have just to prove that is equal to minus B this is not a very difficult however we will solve it out you're given that a plus B is equal to zero and this is given and you're also given that a and B belong to set of real numbers they belong to set of real numbers and every real number has its additive inverse in the set of real numbers and therefore there exists the additive inverse of this number B that is minus B and it must belong to set of real numbers you can add this additive inverse on the both sides of this equation our age the result is a plus B plus additive inverse of P is equal to zero plus R is even worse offbeat next you'll make the use of property that when an additive inverse is added into its corresponding number the result is an additive identity that is zero plus zero plus minus V next when an additive identity is added into a real number when an additive identity of set of real numbers is added into a real number the number is not acted by that additive identity therefore therefore you will get the same number back a is equal to and by using the same property here you get minus speech this one what you have to prove touch number two is quite similar to it and it is not very important form paper point of view therefore I will actually move to number three in Custer number three you're asked to prove that absolute value of a absolute value – be absolute value is less than or equal to absolute value of a minus B where a and B are both in the set of real numbers you can write the absolute value to be equal to a minus b plus three and you can rewrite it as more of a is equal to what of a minus b plus b next if we will make use of this inequality a plus B is less than or equal to mod of a plus mod of T by making use of this inequality here we get mod of a is less than or equal to mod of a minus B plus mod of B you have to remove this plus B from here therefore you will have to add its additive inverse on both sides again the result of plus B and minus V is an additive identity therefore we will simply cancel them the result is what of faith minus mod of B is less than or equal to mod of Z minus P and this is that we term it as equation number one the day we will make the same procedure to absolute value of P absolute value of B is equal to B minus a plus a again by using by making use of this inequality on on this equation we get B to be less than or equal to mod of B minus a that's more than fair you have to remove this from here and you get mod of D minus mod of fee is less than or equal to mod of B minus a plus mod of a minus mod of fee again the result of this is 0 therefore we get mod of B minus mod of P is less than or equal to mod of B minus e next you have to find out a form like this one a minus B and a minus B on both sides therefore you will have to take – as a comment from here we get and you can also make – comment from here next note that mod of minus a is equal to mod of 8 these all of inequalities and this relation are turn in the articles of exercise 1.1 so therefore you will have to review them before understanding before watching out this lecture this minus sign is immaterial to this absolute value therefore it is removed simply you will simply remove it from here we get a minus 3 mod of fair minus mod of B next you have to remove this minus sign from here therefore you will have to multiply a minus 1 from on both sides and bind to by doing so you get a minus B is greater than or equal to negative of a minus v note that when an equality for example a is less than minus B is multiplied by a negative number for example minus 1 the Equality sign simply reverses name this equation as equation number two look out at equation number one it is more of a minus mod of B is less than or equal to a minus B this equality is inequality is also similar to it mod of a minus mod of B here it is greater than or equal to and there was negative or sorry less than or equal to and it is negative here and it is positive a minus B mod here okay by combining them you get this equation is mod of a minus mod of B is less than or equal to a minus B absolute value absolute value of a minus B and in this equation it is written that what of a minus mod of B is greater than or equal to negative times of a minus B or you can also write that minus a minus B is less than or equal to mod of a minus mod of B this can also be written as negative of a minus B is less than or equal to mod of a minus mod of B now come you can by combining this equation and this equation is simply cat this is greater than or equal to negative of a minus B I think that it is quite clear to you now we will also make a use of final relation and this relation is given at the page number 4 and it is property number 4 that this is very useful and very important property that if absolute value of x is less than a then minus a is less than X and X is less than e so we'll make use of this relation mod of X is less than a then minus is less than X is less than positive way this is the similar expression it is written that this is less than this and this is less than positive of this like this one therefore we will get the central the central element this one it's absolute value to be less than or equal to this one therefore we get absolute value of a minus B is less than or equal to absolute value of absolute value of a minus absolute value of B is less than or equal to absolute value of a minus B this was what you had to prove question number four expressed we less than X less than seven in modulus notation this is quite important and very simple question given that a is less than X which is less than seven note that X less than a implies minus a is less than X less than a this is property number four on the page number four of your text so by making use of this inequality you get another inequality for the blacks minus a is less than B what of X minus a is less than B this implies by making use of this relation you get X minus a greater than minus B and this one less than B you need X in the center therefore we will remove minus F from here which can be done by adding B in all of the sides of inequality sorry by adding a in all of the sides of this inequality so minus B plus a is less than X minus a plus C is less than B plus in region a minus B is less than X because and they are canceled out less than a plus B limit by inequality number one and damaged line aconitine number two from this we have seen that when inequality one holds then we get inequality number to compare your original inequality let it term is an inequality number capital a compare this with this you get a minus B is equal to three a plus B is equal to 7 so we were right comparing inequality a and inequality 2 we get a plus B is equal to 7 and a minus B is equal to 3 from here is equal to three plus P for this value of way into this equation you get three plus B plus B is equal to seven this means two B is equal to 7 minus 3 is equal to 4 that is B is equal to 4 4 divided by 2 which is 2 so mu is equal to 2 but this value of B is equal B that is 2 here you get is equal to 3 plus 2 that is a is equal to 5 so you have taken out the values of and and B by solving this equation and this equation simultaneously now put these values back into equation number inequality number one you simply get mod of X minus a that is 5 is less than B that is true this was what you needed answer of question number 4 now we will solve the question number five this is quite similar to this equation so in this question in question number five here given that that Delta be greater than 0 and a is in the set of real numbers you have to prove that a minus Delta is less than X is less than a plus Delta a friend only if what of X minus a is less than Delta for this equation because there is if and only if so way you will first suppose test that this is true and you will prove this one and then you will suppose this one and you will have to prove this one first we will suppose that a minus Delta is less than X is less than a plus Delta [Applause] was my solution let P minus Delta is less than X is less than a plus in turn down from this since you need X minus a in the absolute value therefore we will subtract minus a in all of the side of this equation a minus Delta minus a less than X minus a and a plus Delta minus a we have subtracted a from all of the sides of this inequality previous inequality you get this is cancelled by this and this one is cancelled by this you simply get minus Delta is less than X minus a is less than Delta what is this this is mod of X minus C is less than Delta because from property four on page number four more of X is less than a implies minus a is less than X is less than a we have simply used this inequality this is what we needed to prove now we will suppose this is true that is X minus a is less than Delta now by making use of this definition you get X minus a rather than minus Delta and less than plus Delta you have to remove minus F from here therefore you will add a on all of the sides of this inequality X minus a plus C Delta is a and the result is a minus Delta is less than X is less than Delta plus a or a plus Delta so you have done both of the things you have first supposed this assertion and you prove this one and next you proved you you supposed this one to be true and you prove this one therefore you have done settings and this one is called the converse this this implies this this was what you needed to proof

The Holographic Universe Explained



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We live in a universe with 3 dimensions of space and one of time. Up, down, left, right, forward, back, past, future. 3+1 dimensions. Or so our primitive Pleistocene-evolved brains find it useful to believe. And we cling to this intuition, even as physics shows us that this view of reality may be only a very narrow perception. One of the most startling possibilities is that our 3+1 dimensional universe may better described as resulting from a spacetime one dimension lower – like a hologram projected from a surface infinitely far away.

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The holographic principle emerged from many subtle clues – clues discovered over decades of theoretical exploration of the universe. Over the past several months on Space Time, we’ve seen those close clues, and we’ve built a the foundations needed to glimpse the true meaning of the holographic principle. We’ve moved from quantum field theory to black hole thermodynamics to string theory. We’ve made a background playlist if you want to start from scratch, and I especially recommend catching last week’s episode. But this is tough material, so let’s do a review.

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سلطان الخليفي

we live in a universe with three dimensions of space man one of time up-down left-right and forward back and past and future three plus one dimensions or so our primitive Pleistocene evolved brains find it useful to believe and we cling to this intuition even as physics shows us that this view of reality may be only a very narrow perception one of the most startling possibilities is that our three plus one dimensional universe may be better described as resulting from a spacetime one dimension lower like a hologram projected from a surface infinitely far away the holographic principle emerged from many subtle clues clues discovered over decades of theoretical exploration of the universe over the past several months on space-time we've seen those clues we've built the foundations needed to glimpse the true meaning of the holographic principle we've moved from quantum field theory to black hole thermodynamics to string theory we've made a background playlist if you want to start from scratch and I especially recommend catching last week's episode but this is tough material so let's do a review the story started with black holes and with jacob bekenstein who derived an equation to describe their entropy a black hole entropy represents the amount of quantum information of everything that ever fell into it this bekenstein bound represents the maximum possible entropy / information of any volume of space oddly that maximum is proportional to the surface area of that space not its volume and that was surprising surely the information in a volume of space depends on that volume like one bit per infinitesimal voxel not one bit per pixel on the surface Stephen Hawking confirmed that bekenstein bound by calculating the amount of information leaked by a black hole as it evaporates in Hawking radiation his discovery of Hawking radiation led to the black hole information paradox because this radiation was expected to erase the quantum information of everything that fell into the black hole but destroying quantum information would break the foundations of quantum mechanics hence the paradox this conundrum inspired Dorada truth to show that the information of all material that fell into the black hole could be imprinted on that outgoing Hawking radiation and while it's waiting to be radiated that information should be encoded on the event horizon of the black hole nice solution the new paradox things that fall into a black hole do actually experience crossing the event horizon and being inside the black hole so the interior of the black hole has a dual existence from the point of view of outside observers its contents are smeared into 2d on that surface but from the POV of anyone falling in they are definitely inside the black hole plummeting to their doom in full 3d glory this is the first glimpse of a holographic space-time a 2d surface that encodes the properties of the 3-d interior at woofed along with leonard susskind extrapolated this to propose that not only is any surface sufficient to describe the locations of all particles in its volume but also the full machinery of that volume can exist on the surface all degrees of freedom needed to describe the behavior of everything within but it's one thing for this stuff to fit on the surface but how is it actually encoded how does the 2d surface store information about that extra dimension and how do interactions on that surface correspond to interactions in the volume leonard susskind laid out the first steps to how this could be achieved using string theory but ultimately it was one Melda sena who figured out a concrete string theoretic realization of the holographic principle with a d' s CFT correspondence though I'm getting ahead of myself let's ignore string theory for the moment and just think about how to create an extra dimension let's say we start with a plane a flat 2d space-time now grid it up into a lattice of cells and make a set of rules about how those cells interact with each other those rules are a field theory the lattice itself is the field and the cells are some elementary component of the field but perhaps they're not the smallest possible component for now let's just say the size of those cells depend on how we're looking at the grid for example the resolution of our microscope or the power of our particle accelerator probably the rules between cells the field theory depends on this scale focus on a very small scale and we see a very fine grid that interacts according to one set of rules zoom out and we see a coarser grid with cells that are the average of smaller cells but which presumably interact via different rules or so you'd think but we're gonna add something weird we're gonna say that our field theory is scale invariant we'll say that the rules are the same for small pixels or big pixels we see this scale invariance in fractal patterns where the rules defining structures repeat to infinitely large or small scales we also see it in string theory which I'll come back to a field theory with this property is called a conformal field theory in the last episode I said that a conformal transformation is one that leaves all internal angles unchanged a conformal field theory has this property for example you can change the scale at every point on the grid separately and not change the internal angles or the shapes of the pixels which corresponds to not changing the rules of interaction by making this conformal field theory we've added a symmetry invariants under local changes in scale also known as a vial invariants this adds a degree of freedom everywhere like a new infinite number line at each of the 2d grid points objects on the 2d grid also have values on that number line they exist at a certain scale if objects at different scales don't tend to interact with each other then this new degree of freedom behaves like just any other dimension our 2d grid behaves like a 3d volume and we can treat it like one at least mathematically you might say is not a real 3d grid because the 3rd dimension is fake but is it what is a dimension that a number line of possible values which a exists alongside the other dimensions but is independent of them B over which the rules of physics stay the same and C imposes some kind of locality for example elements of that number line need to be next to each other to interact crudely this is how an extra dimension can be coded in a holographic universe but for the details we need string theory even from the beginning string theory had hints of this scale invariance and dimensional weirdness the first iteration of the theory around 1970 tried to model the strong force between pairs of quarks maisons and this strand of gluons that behaves like a vibrating string a nice feature of this model is that changing the length of the Strand which defines the energy in the bond doesn't change the basic physics that means you can pretend string length / energy is a separate dimension as a calculation trend the weird thing is that when you write the quantum wave equation for the gluons strand with a length expressed as a separate dimension you get the wave equation for the graviton the quantum particle of gravity which is ridiculous given the puny energy scale of the Maison gravitons shouldn't even exist there this and other glitches led to string theory being abandoned as a model for the strong force but it was quickly rejigged to make it a theory of quantum gravity and the scale invariance of the strings became a central feature of string theory fast forward a couple of decades to the 90s we now have several versions of Street theory that try to explain how vibrating strings can lead to the familiar particles of the universe these were tentatively united by ed Witten's and theory which showed that different types of string and string theories were all related by dualities a duality is when two seemingly different theories proved to represent the same underlying physical reality these arose from the way string size and energy scales could be rescaled but the strangest string duality was still to come with AD s CFT correspondence proposed by argentinian physicist one maldacena in 1997 strange because it provided the first concrete description of a holographic universe Melda Sen imagined a set of string theory objects called brains these are like multi-dimensional strings the conservers start and end points for strings but also as spaces embedded within higher dimensions Melda Center considered in geometrically flat 3d brains these brains are extremely close together basically overlapping the strings connected to these brains are scale invariant so their length and energy can vary without changing the physics under certain assumptions he found that the resulting brainy structure looks just like a man coughs ki space time of 3 plus 1 dimensions on which there lived a field theory that arose from interactions between brains in itself that field theory wasn't stringy rather it was a quantum field theory like the ones that give us our standard model of particle physics a young Mills theory but with supersymmetry added in it was also a conformal field theory a CFT so it was invariant to the scaling of grid sizes this quality came from the energy scale invariance of the strings embedded in the construction of this space in good ol string theorist style Melda Center to find a new spatial dimension that incorporated that invariant scale factor the 3d space became a 4d space well the original space was flat the new space had negative curvature who is a hyperbolic anti-de sitter or a D s space the conformal field theory in the original space included no gravity but in the higher dimensional space gravity emerged revealing a full quantum theory of gravity this is the ad s CFT duality as with the other dualities in string theory this one was extremely useful for calculations when interactions in the lower dimensional field theory are extremely strong we would say the fields are strongly coupled then the corresponding high dimensional gravitational structures would be weak and solvable conversely strong gravitational fields in the higher dimensional space like in black holes look like solvable configurations of particles in the low D space among other things this provided a new resolution to the black hole information paradox the information lost in a black hole persists perfectly comfortably in the lower dimensional space and the techniques of a DSC FD correspondents are even extended to disparate fields like nuclear and condensed matter physics but the more startling implication of a DSC of T is that it's the first concrete realization of the holographic principle the low dimensional CFT space is the surface of the ad space because the field theory exists where that new dimension becomes infinite infinitely far away that's tough to imagine so let's go back to our depiction of an infinite hyperbolic space from the last episode represent a 2-d hyperbolic plane as a compactified map and it has an edge at least a mathematical one anyone inside the hyperbolic space still has to travel infinitely far to get to that edge now stack many maps to represent slices in time the resulting column has a geometrical flat and finite surface that is a space-time all on its own the rules of interactions between cells on that surface is a quantum field theory but those rules translate to interactions in the volume in the bulk where it's a theory of gravity a DSC ft is a hint do we may live in a holographic universe now a DSC ft doesn't represent this universe because our universe doesn't appear to be negatively curved ad r space nor does it have four spatial dimensions as in Melville Senna's calculation but there are efforts to generalize this to a universe more like our own the question we now wrestle with is this a series of mathematical clues indicate that our universe may be holographic or at least have a dual representation in a lower dimension can these just be crazy mathematical coincidences maybe but perhaps our familiar three plus one universe has an alternative perhaps a more true representation out there an abstract mathematical surface infinitely far from our location and from our intuition projecting inwards our familiar holographic space-time Before we jump into comments I want to let everyone know that there's new merch in the merch store including the return of our Game of Thrones inspired shirt the heat death of the universe is coming it's a great way to support us s is joining us on patreon leaks in the description so last week was the warm-up to today's episode in which we looked at how infinite space-time can have a finite boundary and first up no psychedelics were involved in making that episode despite what people thought then the universe is just that weird a few of you asked whether our perceived universe is just the surface of a higher dimensional space so that's actually the opposite of the proposition behind the holographic principle we suggest that our perceived universe is the volume but it can be encoded on its lower dimensional surface in a DSC ft correspondence the volume exhibits gravity via type of string theory while the surface exhibits no gravity only a quantum field theory similar to the field theory behind the standard model part of the confusion comes from the fact that mel decenas derivation is for a volume with four spatial dimensions which would then have a three-d service so obviously that doesn't directly correspond to our universe but there's work to generalize it to the case of a 3d volume with a 2d surface related to that music always asks whether according to a DSC ft correspondence can we say that there would be no gravity on the surface of the 2 plus 1 min cops key space-time so first the surface in the current a DSC of te space-time is three plus one three spatial one temporal dimensions as I just mentioned that surface contains only a conformal field theory and no gravity the strange miracle of a DSC of T is that gravity arises naturally when you add the extra special dimension which ends up looking like the volume contained by a 3d surface ki9 asks whether the things we learn from a DSC of T are applicable to our universe even that our universe doesn't have negative curvature well we don't know for sure that it doesn't have negative curvature just that any curvature negative or positive is very weak compared to our current ability to measure it measurements of the geometry of the universe indicate flatness but we may never know whether it's truly flat or just flat as far as we can see several people were offended that I dissed Chronicles of Riddick well I want to be on record as saying that pitch black was an artistic masterpiece real mammal will summarizes my position well chronicles is the third best Riddick movie but is still better than any Marvel movie and I'm sure saying this will cause no further comments to Alito notices that it's looking more and more likely that Roger Penrose might literally be a Time Lord in a separate comment midplane wonderous states that sir Roger Penrose is an unsung wizard so apparently we can't agree on what genre Roger Penrose belongs to personally I've always thought of him as a Jedi Master especially with all of that dubious quantum consciousness stuff midi-chlorians microtubules potato potato anyway perhaps we need to accept that Penrose is beyond genre like if Gandalf had a TARDIS and a lightsaber by the way if anyone feels like drawing Roger Penrose dressed as Gandalf with a lightsaber and a TARDIS you would win the Internet

Economics with Calculus 1



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First of a series on Economics with Calculus. This Series Assumes that you took calculus, but did not really understand it. I’ll show you the way! Video 1: Introduction and integration to find variable costs, total benefit, consumer surplus, and producer surplus.

okay in ten minutes I'm gonna try to teach you calculus and teach you some ways that it is used in economics don't believe it watch it's gonna go fast there are two parts to calculus there's the derivative and the integral or antiderivative in economics if in and this is true in general if you take the derivative of a total it tells you the slope or the rate of change in economics we call a slope for a rate of change a marginal normally the antiderivative is the opposite of a derivative if you take the integral of a function it's basically a marginal kind of function it adds things up to get a total or part of a total at least so here we have some function of Q suppose F of Q is 10 plus 12 Q minus one-third Q squared what's the derivative of this function with respect to Q well even if you don't remember much about calculus the derivative of a constant 10 is 0 the derivative of 12 Q is 12 and the derivative of minus 1/3 Q square multiplied by 2 to get 2/3 Q and subtract 1 from the exponent and so you get minus 2/3 q is the derivative of this function what is the derivative well it tells you the slope of this function at any point here is that function I've graphed it for you it's a function a parabola quadratic equation the slope at any point can be gotten by plugging Q into this function for example where is the slope going to be equal to 0 solve this function 12 minus 2/3 Q equals 0 for Q and you're going to get Q equals 18 where is the slope going to be equal to 0 where Q equals 18 is where it's flat on top that's where the slope is equal to 0 so you see the derivative of this parabola gives us the slope if this is some kind of total function it would give us the marginal function if this was total benefit it will give us marginal benefit marginal utility if it was total utility you get the point now here's another function P equals 5 plus 2 Q plus one-half Q squared what's the derivative again the derivative of five is zero the derivative of two Q is 2 and the derivative of 1/2 2q squared bring down the 2 multiplied by it two times one half is one little subtract one from the exponent and you get cute so two plus Q is the derivative same thing it tells you the slope or the marginal now what if we wanted to go to the in the opposite direction what if we had a demand curve that was P equals 12 minus two-thirds Q then if we wanted to do the integral we do the opposite rather than subtracting one from the exponent and multiplying by the exponent we're going to add one to the exponent and we're going to multiply sorry we're going to divide by the exponent now this demand function 12 minus two-thirds Q that's the same thing we got here 12 minus two-thirds Q is the derivative of this function and so the antiderivative of 12 minus two-thirds Q we're just working in the opposite direction is going to give us this function except there's no way to figure out where that 10 came from so instead of the integral being 10 plus 12 Q minus one-third Q squared let me just copy that the integral of this demand function is going to be exactly that except we don't know the ten we call that C some unknown constant similarly if we had a supply function that was 2 plus Q well that's the same thing we had here the integral is moving in the opposite direction to the original function except there's no way to tell where that 5 would come from because it's nowhere in this part so with that's just an unknown constant C so again let me copy that down there we have the indefinite integral except this is an unknown constant C okay so what can we do with this information well as I said if you have a total function and you take the derivative if you get a marginal if you have a marginal and you take the integral you get a total function so let's look over here at this demand and the supply curve I've graphed them over here on these axes and you see that the demand is the marginal benefit that's what we call it an economics and the supply curve is marginal cost so if we take the integral under the demand curve instead of marginal benefit we get total benefit which is the largest amount and somebody would be willing to pay in order to get a certain number of units and instead of supply here if you integrate it you get the area under the supply curve which we don't call total costs we call it variable costs because there's a cost that's missing what kind of cost is missing well that's the fixed cost if you have a total cost function it might look like this P equals 5 plus 2 Q plus one-half Q squared if you take the derivative derivative of a total cost function you get the marginal cost function 2 plus Q where's the fixed cost well the 5 would be the fixed cost for something like rent or other fixed costs but when you take the derivative the fixed cost goes away and you can't see it on a supply curve graph so if we integrated this it would be the the variable costs not the fixed costs included there so let's suppose you know we see here the equilibrium quantity is 6 so if I plug 6 into this integral function of the demand function then you get C something we don't know so let's let's just ignore it now plus 12 Q 6 12 times 6 minus 1/3 times Q squared which would be 36 well 12 times 6 is 72 and so the integral under the demand curve for 6 units would be 72 plus 12 times 6 sorry we already did that part minus 1/3 times 36 sorry which is going to be minus 1/3 times 36 so that's minus 12 and so we'd get the area under the demand curve equals 60 and we call that the total benefit which tells us that the most this person would be willing to spend for six units would be $60 and not a penny more now if we take the integral under the supply curve we're gonna get the variable costs so we'd plug in 6 units into this curve to get the area under the blue supply curve here for a variable cost so 2 times 6 is going to be 12 plus 1/2 times Q squared 36 so we get 12 plus 1/2 of 36 is 18 equals 30 dollars would be the variable costs now I know we I said that this is a total function but it's the total variable costs that's what you get if you add up all of the marginal costs so once you have all of these numbers we can just draw a graph so that we know what all the numbers are where they come from the variable costs right here is going to be this area under the supply curve and we said that that's going to be 30 and so that yellow is the variable cost now the total benefit the $60 is going to be the total area under the demand curve down here and let's give that a different color here and make it a little transparent so we can see the difference here so now the variable costs are orange and everything else the total benefit will be the pink plus the orange part now just the pink part there that is total surplus the total amount of surplus that goes to the consumer and the producer so let me draw another area here so we can see the consumer surplus on the top here and we'll give it a different color and all these colors on top of each other okay so that's green so the green is the consumer surplus pink is the producer surplus and the variable costs is orange and just by knowing that the total benefit the total area there is sixty and by knowing that the variable cost part is equal to 30 the orange and by calculating very easily that 6 times 8 is 48 is the total revenue then you know that total benefit the 60 minus the 48 gives you that the consumer surplus is $12 in consumer surplus the variable cost down here of 30 the total revenue is 48 48 minus 30 gives you the 18 and that is producer surplus and they're in I've got time to spare under 10 minutes is what calculus is all about how you can use it in economics and I'll come back with a second and third and fourth video and show you a little more

Work, Energy, and Power: Crash Course Physics #9



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When you hear the word, “Work,” what is the first thing you think of? Maybe sitting at a desk? Maybe plowing a field? Maybe working out? Work is a word that has a little bit of a different meaning in Physics and today, Shini is going to walk us through it. Also, Energy and Power!

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عندما أقول "العمل"، ما هو أول ما يخطر
ببالكم؟ ربما مكتب؟ أو حقيبة أوراق؟
أو إمتحان التاريخ قريب الموعد؟ ولكن إن كنت فيزيائياً، العمل له معنى محدد
جداً – معنى لا يتعلق بالجداول أو انهيار الإمبارطورية الرومانية. اليوم، سنستكشف ذلك التعريف
– وارتباطه يأحد أكثر مبادئ الفيزياء أهمية : صون الطاقة. سنتكلم أيضاً عمّا يقصده الفيزيائيون
عندما يتكلمون عن فكرة أخرى كثيراً ما تبرز في الحياة اليومية:
القدرة. إذاً لنباشر… العمل. حتى الآن في هذه الدورة، أمضينا معظم وقتنا
في الكلام عن القوى وطريقة تحريكها للأشياء. وعلينا فهم القوى قبل أن نفهم العمل. لأن العمل هو ما يحدث عندما تطبق قوة
لمسافة معينة، على نظام. النظام هو أي جزء من الكون يكون موضوع
حديثنا. كمثال، إن كنت تستخدم حبلاً لسحب صندوق
على الأرض، نستطيع القول أن الصندوق هو نظامك، والقوة التي تستخدمها
لسحب الصندوق هي قوة خارجية. إذاً، لنقول أنك تسحب هذا الصندوق-النظام
خلفك بحيث يوازي الحبل الأرض. إن استخدمت الحبل لسحب الصندوق متراً
واحداً، نقول أنك تطبق عملاً على الصندوق وكمية العمل الذي تطبقه
يساوي جداء القوة التي استخدمتها لسحب الصندوق مع المسافة التي قطعتها به. كمثال، إن شددت الحبل – وبالتالي الصندوق –
بقوة شدتها 50 نيوتن، وحركته لمسافة خمس أمتار، نقول أنك طبقت
عمل قيمته 250 نيوتن-متر على الصندوق. أو بشكل أكثر شيوعاً، على أي حال،
يعبر عن العمل بواحدة الجول. والآن، أحياناً، القوة التي تطيقها على جسم،
لن تكون في نفس الإتحاه تماماً كالإتجاه الذي يتحرك فيه الجسم. مثلاً، لو حاولت سحب الصندوق عندما تكون يدك
أعلى من الصندوق، بحيث يشكل الحبل زاوبة مع الأرض، في تلك الحالة، يتحرك الصندوق
بموازاة الأرض، ولكن القوة ستشكل زاوية معها. وفي هذه الحالة، ستضطرون لإستخدام حيلة من
التي تعلمناها عندما تكلمنا عن حوامل القوى. تحديداً، يجب عليكم أن تفصلوا القوة التي
تطبقونها على الحبل إلى مكوناتها الرئيسية: قوة موازية للأرض، وأخرى
عمودية عليها. لإيجاد جزء القوة الذي يوازي الأرض،
وهي التي تسحب الصندوق إلى الأمام فعلياً عليكم فقط أن تضربوا شدة القوة بكوساين
الزاوية بين الحبل والأرض. ستتذكرون أننا عادةً نشير للزاوية
في نظام ما بحرف ثيتا. إذاً، لحساب العمل الذي طبقتموه على
الصندوق، عليكم أن تضربوا الجزء الأفقي – وهو جداء F بكوساين الثيتا –
بالمسافة التي قطعها الصندوق. هذه إحدى الطرق التي غالباً ما يكتب
بها الفيزيائيون معادلة العمل، سيجعلونها تساوي القوة
ضرب المسافة ضرب كوساين ثيتا. وتلك المعادلة تنطبق على أي قوة ثابتة
تطبق على مدى مسافة معينة. ولكن ماذا لو كانت القوة غير ثابتة؟ ماذا لو، فرضاً، شددت الصندوق بقوة،
ولكنك أصبت بالتعب، ولذلك شدة القوة التي تطبقها على
الصندوق تناقصت كلما سحبته أكثر. لحساب العمل الذي طبقته في تلك الحالة،
عليك حساب شدة القوة التي طبقتها على كل مسافة ضئيلة. وإن شاهدتم حلقاتنا عن التفاضل،
فستعلمون أنه توجد طريقة أسرع لجمع التغيّرات الصغيرة بشكل لا متناهي:
التكامل. إذاً، لإيجاد العمل الناتج عن قوة متغيرة،
عليكم مكاملة نسبة الحركة إلى المسافة التي قطعها الحسم.
والذي سيدو هكذا. ولكن (القوة ضرب المسافة) هي واحدة فقط
من الطرق التي يحسب بها الفيزيائيون العمل. بسبب، هل تذكرون عندما قلنا
قبل قليل أن الجول تستخدم كواحدة العمل؟ حسناً، الجول هي أيضاً واحدة شيء آخر:
الطاقة. والعمل يستخدم نفس واحدة الطاقة،
لأن العمل مجرد تغير في الطاقة. هو ما يحدث تطبق قوة خارجية على
نظام وتغيّر طاقة ذلك النظام. في الواقع، هذه إحدى طرق تعريف الطاقة:
القدرة على إنجاز العمل. توجد أنواع مختلفة من الطافة، ولكن في هذه
الحلقة، سنتكلم بشكل رئيسي عن اثنتان: الطاقة الحركية والطاقة الكامنة.
الطاقة الحركية هي طاقة الحركة. عندما كان الصندوق ساكناً على الأرض،
نقول أنه لم يكن لديه أي طاقة حركية. ولكن حالما تطبق عليه قوة ويبدأ بالحركة،
يصبح لديه طاقة حركية. وطاقة الصندوق تغيّرت،
مما يعني أنك طبقت عملاً عليه. بشكل أدق، الطاقة الحركية لجسم ما هي
نصف كتلته مضروباً بمربع سرعته. إن كان هذا يبدو مألوفاً، قذلك لأنه
ناتج عن تطبيق كل من قانون نيوتن الثاني والمعادلات الحركية على فكرة أن العمل
يساوي القوة ضرب المسافة. إذاً، إن كانت كتلة الصندوق 20 كغ،
وفي إحدى لحظات تحريكك له، وصلت سرعته إلى 4م/ثانية،
نقول أن طاقته الحركية في تلك اللحظة تساوي 160 جول. ثم توجد الطاقة الكامنة، وهي في الواقع
ليست ما تبدو عليه. الطاقة الكامنة ليست طاقة كامنة،
إنها عمل كامن. بكلمات أخرى، الطاقة الكامنة هي طاقة
يمكن استخدامها لأداء عمل. أحد أنواع الطاقة الكامنة الشائعة
هو الطاقة الكامنة الثقالية. جوهرياً، الطاقة الكامنة التي تنجم
عن واقع أن الجاذبية موجودة. إن حملت هذا الكتاب على ارتفاع متر فوق
الأرض، نقول أن لديه طاقة كامنة ثقالية. لأنك إن تركته، ستطبق
الجاذبية عملاً على الكتاب. طبقت الجاذبية قوة تحركه نحو الأرض. ولكن حالما يرتطم الكتاب بالأرض،
نقول أن طاقته الكامنة الثقالية صفر. لأن الجاذبية لم تعد قادرة على تطبيق
عملٍ عليه بعد الآن. حساب الطاقة الكامنة الثقالية سهل: إنها قوة الجاذبية على الجسم
– إذاً، فهي كتلة الجسم ضرب g – ضرب ارتفاع الحسم.
أو mgh للإختصار. مما يعني أن، بمعرفة أن كتلة الكتاب
هي كيلوغرام تقريباً، وأنه على ارتفاع متر فوق الأرض،
نحسب طاقته الكامنة : وهي 9.8 جول. نوع شائع آخر من الطاقة الكامنة
هو الطاقة الكامنة المرونية. وبالرغم من إسمها، فهي ليست موسمية،
وأجل، أنا من اخترع تلك النكتة. في الواقع، إنها نوع من الطاقة الكامنة
الخاصة بالنوابض! قوة النابض تساوي المسافة
التي إما يكون فيها مشدوداً أو ممدوداً، ضرب ثابت نرمز له بحرف k. تسمى هذه المعادلة قانون هوك، تيمناً
بالفيزيائي البريطاني روبرت هوك، والذي اكتشفها في عام 1660. والآن، الثابت k
– والذي يسمى ثابت النابض – يختلف في كل نابض وهو قيمة صلابة النابض. والمعادلة منطقية تماماً،
عندما تفكرون بها: كل ما بعد سحبك للنابض، وكل ما زادت صلابته،
كل ما اشتدت مقوامته. تستطيعون أن تختبروا ذلك بأنفسكم بتفكيك
قلم ضعط واللعب بالنابض داخله. بجمع قانون هوك، مع فكرة أن العمل
يساوي القوة ضرب المسافة، نستطيع إيجاد الطاقة الكامنة في نابض: إنها نصف ضرب k ضرب مربع المسافة. مثلاً: إن كان لديكم نابض بثابث نوابض
قيمته 200 نيوتن على متر، وتضعطه قرميدة مسافة نصف متر،
فستكون طاقة القرميدة الكامنة 25 جول. إذاً، عندما يطبق شيء ما عملاً
على نظام، تتغير طاقته. ولكن كيفية تغير الطاقة تعتمد على النظام. بعض الأنظمة تفقد طاقة. تسمى هذه
أنظمةً غير حافظة. والآن، هذا لا يعني أن الطاقة المفقودة
تختفي من الكون… وليس لها علاقة بسياسة النظام الشخصية
أيضاً، إنها متعلقة بواحد من أكثر المبائ
رسوخاً في الفيزياء: وهو أن الطاقة لا تفنى ولا تنشأ من العدم. ولكن يمكن للأنظمة أن تخسر الطاقة، مثل
عندما يوّلد احتكاك الصندوق بالأرض حرارة. بالنسبة للأنظمة غير الحافظة، تستطيعون
الكلام عن طاقتهم الحركية أو الكامنة في أي لحظة، ولكن الأنظمة الحافظة
تمكنكم من قعل أكثر من ذلك بكثير. النظام الحافظ هو الذي لا يفقد
الطاقة من خلال العمل. مثلاُ، بندول بسيط. عندما يكون البندول في أقصى مطاله، يتوقف
عن التحرك للحظة خاطفة قبل تغيير حهته، مما يعني أن طاقته الحركية في تلك اللحظة،
صفر. ولكن لديه الكثير من الطاقة الكامنة، لأن
قوة الجاذبية تستطيع أداء عمل على البندول، بسحبه إلى الأسفل حتى يصل إلى أدنى مطاله. في مطاله الأدنى، تلك الطاقة
الكامنة تصبح صفراً. لأن الجاذبية غير قادرة على سحبه
إلى الأسفل عندئذٍ. ولكن الآن لدى الندول الكثير من الطاقة
الحركية لأنه يتحرك على مطاله. واكتشفنا أنه في أي لحظة من حركة
البندول، يكون مجموع طاقته الحركية وطاقته الكامنة هو نفس الرقم. إن تناقصت طاقته الكامنة؟ ستزيد طاقته
الحركية بنفس القيمة، والعكس صحيح. إذاً، بما أننا نعرف الآن كيف نعرّف العمل،
نستطيع إستخدام ذلك التعريف لتفسير مصطلح آخر شائع ويعني به الفيزيائيون
أمراً محدداً جداً: القدرة. أو بشكل أدق، القدرة الوسطية. تعرّف القدرة الوسطية على أنها العمل عبر
الزمن، وتقاس بالوات، وهي طريقة أخرى للتعبير عن
الجول في الثانية, تستخدم لقياس كمية الطاقة التي تتحول
من شكل لآخر عبر الزمن. إذاً، أتذكر الصندوق الذي كنت تسحبه؟ وجدنا أنك تطبق 250 جول من العمل
على الصندوق عندما تحركه مسافة خمس أمتار. إن استغرقك الأمر ثانيتان لتحرك الصندوق،
فإن قدرتك المتوسطة هي 125 وات. أي أنك في الواقع مصباح! والآن، نستطيع أيضاً أن نصف القدرة
بطريقة أخرى، بجمع حقيقتين مختلفتين: الأولى، أن العمل مساوٍ للقوة ضرب المسافة. والثانية، أن السرعة المتوسطة مساوية
للمسافة على الزمن. بمعرفة ذلك، نستطيع القول أن القدرة هي
محصلة القوة المطبقة على شيء ذو سرعة متوسطة. إن حرّكت الصندوق خمس أمتار في ثانيتين،
تكون سرعته المتوسطة هي 2.5 متر في الثانية. وقلنا مسبقاً أنك إن كنت تسحب الصندوق
بقوة شدتها 50 نيوتن. إذاً، القوة التي تسحب بها الصندوق،
ضرب سرعة الصندوق المتوسطة، تعطيك أيضاً قوة وسطية قيمتها 125 وات. معادلتا القدرة الوسطية تصفا نفس العلاقة؛ ولكنهما تستخدمان صفات أخرى
لفعل ذلك. سنتكلم عن القدرة كثيراً عندما
نناقش الكهرباء في حلقاتٍ قادمة. إنها أفضل طريقة لحساب
حركة الطاقة في دارة. ولكن تلك قصة ليوم آخر.
للوقت الحالي، عملنا انتهى. اليوم، تعلمتم المعادلتان اللتان نستطيع
استخدامهما لوصف العمل، وأن الطاقة هي القدرة على أداء العمل. كما تكلمنا
عن الطاقة الحركية والطاقة الكامنة، وكيف تلعبان دورهما في الأنظمة الحافظة
وغير الحافظة. أخيراً، أوجدنا معادلتين مختلفتين للقدرة. Crash Course Physics ينتج بالتعاون مع
PBS Digital Studios. تستطيعون التوجه إلى قناتهم لمتابعة برامجهم
مثل The Art Assignment، PBS Idea Channe، و PBS Game Show. صورت هذه الحلقة من Crash Course في استديو
Doctor Cheryl C. Kinney Crash Course Studio وبمساعدة هؤلاء الأشخاص الرائعين
وفريق بصرياتنا هو Thought Cafe.

Derivatives in Economics



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This video shows how derivatives are used in economics.

you're probably wondering why it's important to know about derivatives and one important application of derivatives is in economics marginal revenue and marginal cost are two derivatives the marginal revenue function in economics is the derivative of the total revenue function it's the change in total revenue over the change in quantity which is a derivative we know and this is the same thing as the derivative of the total revenue function similarly the marginal cost function in economics is the derivative of the total cost function or the change in total cost over the change in quantity which we just said is the derivative of total cost now why would you need to know this well if you are running a business these derivatives would allow you to determine how much of a certain good you should be producing to maximize your profit the profit function is defined to be your total revenue function minus the total cost function so if you wanted to maximize your profit you would want to take its derivative so the derivative of the profit will just say profit prime would equal the marginal revenue function minus the marginal cost function which we just said is the I apologize for that the derivative of the total revenue function minus the derivative of the total cost function we won't get into exactly how to optimize profit how to maximize it that will be taught in another lesson but this could be used in real life if you want to determine how much you should produce to maximize profit

Mathematics Gives You Wings



Views:1096531|Rating:4.83|View Time:52:28Minutes|Likes:6622|Dislikes:228
October 23, 2010 – Professor Margot Gerritsen illustrates how mathematics and computer modeling influence the design of modern airplanes, yachts, trucks and cars. This lecture is offered as part of the Classes Without Quizzes series at Stanford’s 2010 Reunion Homecoming.

Margot Gerritsen, PhD, is an Associate Professor of Energy Resources Engineering, with expertise in mathematical and computational modeling of energy and fluid flow processes. She teaches courses in energy and the environment, computational mathematics and computing at Stanford University.

Stanford University:

Stanford Alumni Association:

Department of Mathematics at Stanford:

Margot Gerritsen:

Stanford University Channel on YouTube:

Stanford University thanks very much I'm a very tickle that you're here because I know I'm competing with a football game the start of a baseball game so I'm keeping a very close eye on the time because I know the first pitch is at 457 as I want to be out before then and I'm also competing with the rain and then the first thing is I'm talking about mathematics which generally is something that people want to escape from after college not revisit at a homecoming weekend but my purpose for this afternoon is really sharing with you a bit of the passion that I have for computational mathematics and as Adam was saying I'm directing quite a large Institute now on campus is the Institute of computational mathematical engineering and you may not have heard of that because we haven't actually been in existence all that long there was a precursor to this institute which was a program in computer science and I actually did my PhD in that program so I'm an alumni myself from 96 and I was fortunate enough to come back here in 2001 as a faculty member after getting my PhD in computer science in energy resources don't ask me how that happened but they allowed me back in and since for weeks I'm director of this Institute but we have 140 graduate students 70 masters and 70 PhDs we have no undergraduate students but we're teaching 20 courses in the School of Engineering and for our Sciences for Applied Mathematics and so on in the whole for the whole University 4,000 student units and so most of the harassing the mathematical harassing of undergraduate and graduate students is led by my Institute and so that's a wonderful feeling to be able to control that for some people very painful experience but I love it and I've always really loved mathematics and I want to show you a little bit today why and one of the reasons why I absolutely love this is on the board right now and that's a whole bunch of complicated Asians and those are the equations we're going to be talking about today but my idea is that I make this much simpler for you because really when you look at it very carefully all of the equations that govern fluid flow processes be it climate models where the models optimization of SIL design for competitive yacht races one of the things I've done another thing I've done is optimizing wings for pterosaurs of course they don't exist but we just made up some for a National Geographic I've also done fluid flow in the oil and gas reservoirs aquifers groundwater models all of these processes that may seem completely different are all governed by these equations there's all the same stuff and then they look very complex right and this is of course what we what we like we like to use Greek symbols we like to use long mathematical expressions so we can impress people at Friday beers and so on now sometimes we even put it on t-shirts and have something like that and now we understand this and you don't but really when you start looking at it and breaking it down it's all relatively simple and it's really quite wonderful when you see how all of these fields know coastal oceans wind turbine optimization you name it is all connected in exactly the same way and this is one of the reasons why I love computational mathematics and why in the 25 years or so that I've been doing this professionally I've worked on 10 12 13 different projects all sorts of different fluid flow problems and sometimes when people look at my CV or my publication say you're crazy you know you're all over the place they know I'm always doing just this and so that's what I'd like to share with you so by the end of this hour you're either going to tell yourself that you will never ever do a math course classes without quizzes again at a homecoming weekend or you're going to be so excited that you'll come back next year and apply to do a master's in my program all right it's normally either-or okay so I'm hoping to see you all back next year for masters it actually happened once about five years ago I gave a talk at Homecoming weekend and there was somebody in the audience who was 84 years old and decided at the end of the talk that he wants to do another PhD at the PG already from Stanford in physics from 1954 I think it was and he came back and he started it but unfortunately it was a little bit too fast for him but he did take some of my courses which is absolutely fantastic anyway so let's go and and look at these equations so so here they are and they like I said these equations govern fluid flow no matter where the fluid flow is no sometimes the equations look slightly different but in concept they're all the same so I wanted to tell you a little bit about it and I won't raise the screen right now I'm just just going to write here because you know I am just right so let me start here um there is a couple of things in these equations a couple of terms when we understand these terms that you put one other down here you've understood all these equations and then I can start talking about how to solve these on a computer because ultimately that's what we'll do okay so and then we'll move between blackboard and screen for a bit because you know this is a map or so obviously I have to use this blackboard okay now what are all these things here you is stuff that we're interested in knowing now when you think about fluid flow what could you be you is just a name for something we're interested in when we're looking at fluid flow something that describes ocean flow wind what could you be up for example energy or velocity volume we're leaving pressure if we had all of that right if we had we're assuming that density does not change so we're not taking that into account right now but if we knew how the velocity changed as I'm traveling through space or as time changes if I know how the pressure behaves in the system and if I knew how energy behaves or what the temperature was like then the main regulus thing is I can really describe this whole process and moreover I can predict it because think about for example air flow like this in the room ok so we open some windows and there was a nice breeze outside we could maybe feel the air move in this room what drives that air now what is causing it to move difference in pressure now you can say well it's the wind outside right but ultimately that wind is driven by pressure differences low pressure high pressure what else ultimately by the summer we're not going to go that far I want to keep the computations a little bit less than than the solar system because then maybe we have a chance to actually simulate it sure size of the opening so configuration right now how things are moving is dependent on the configuration of my room and other words on the boundaries that I have in in that flow domain you know if you understand that lingo well how about gravity no you can say gravity doesn't really play a big difference if I have an air molecule flowing around it's not going to just drop down right but gradually gravity will also influence things so things move because there is a force acting on them right and as a result of this force things change velocity changes when things start to heat up somewhere can maybe have a density change I can have energy change because things can start flowing faster so kinetic energy changes and so on so all I really need to understand of any fluid system is in that system what's going to be moving or changing that I'm interested in and most of the time that is just that's my you that is things like pressure velocity and most of the time we don't think just about velocity but we actually think about something called momentum which is nothing but a mass times the velocity but if my mass is constant but if I say our air is always the same density weighs the same God so a certain percentage of nitrogen certain percentage of oxygen all these sort of ways the same per volume then I can just think about velocity and I can maybe write something down for energy but that's coupled to pressure so very often these things are taken together so I think about momentum I think about energy ok and then if I need to I think about mass that's all so that my you and these things are changing because of forces and most of fluids processes have very few forces acting on them there's there is a pressure gradient right pressure difference pressure differences there is gravity sometimes there are forces at boundaries and you think of a force at the boundary of something that may influence a flow friction right or sometimes we like to call that skin friction because it's friction that happens at the boundary for example on an airplane wing the actual airplane surface is holding back air flow you don't realize that when you're sitting in the airplane when you're sitting in in an airplane and you're looking outside over the wind and you know that you're going 500 miles per hour the funny thing is that at the surface of the wing the air is actually not moving at all it's just changing very very quick from 500 miles per hour to zero in a very thin layer over that wing which we call a boundary layer like a very very strong pressure or velocity change in that very thin layer there's a lot of force on that wing as a result of it hey that wing is really stopping that flow and you can just imagine if you're trying to stop a runaway train and there's a lot of force on you it's the same on that wing right but luckily we know how to engineer these things so we have maybe some boundary forces okay and all of these things here they determine how you changed so let me just write on the determines how you changes they'll be it pressure momentum energy whatever as a function of time right so how it changes in time how fast it changes how slow it changes know how long does it take for it maybe doesn't change at all and also as a function of space because not everywhere in my domain I may have the same velocity or the same pressure right so if I look at whether we have nice low pressure here now and somewhere else maybe high pressure so it changes also as a function of space now this is really all we need to do and all we need to know we not just need to set up some equations that describe this that says equations that say if I have a strong pressure gradient so my pressure somewhere is high and very close to it it's low what do you think will happen to the air velocity it's going to be high right because I have a strong force pushing that air from high pressure to low pressure to have a strong wind so I need an equation that says this high pressure gradient gives me a strong high velocity I need to have things like if gravity is strong then my downward velocity my vertical velocity will grow know things will start to fall if I have skin friction at the boundary then near that boundary my flow will need to slow down that's all we do it's just a bunch of rules the govern the behavior very natural intuitive behavior of this fluid flow so we look at the fluid problem we write down the sort of things that we want to understand about this flow we write down all of the forces that we can think of that may drive that flow sometimes we throw some out because not all of them may be important right for example in this room if I did air flow in this room I would leave out gravity sure it has some influence but it's so tiny compared to any drafts or other winds that I may feel in this room that I can throw that out so we look at it we simplify and then we decide how this changes and the ratio you know these rules that say if my pressure gradient my pressure force is this big then it starts to flow this fast or that fast well that I just know from experience that comes from experimentation because that depends on things like viscosity of a fluid and so on but really all of these rules the general behavior is very intuitive pressure gradient higher flow stronger but how much stronger what that ratio is that is something we just get from experimentation there's no rocket science involved with that ok so we look at tables and other things and we just write down an equation now let's look at this a bit it determines how you changes as a function of time or function of space bad stuff function of time function of space is in symbols like this so if I see D u DT what it says is how much does you change with respect to a change in time the little Delta is just the change in so if I translate this in words this just says change in u in a certain change in time that's all right what would this say then yeah so if X my X Direction you know we are in three dimensions so I have an X Direction I've got a y direction of God the Z Direction this is says if I walk in this direction so I'm changing X this is how much you is going to change now right now here nothing much is changing but it could be that if I'm walking here all of a sudden and I'm picking up a draft from one of the open windows on that side and I start feeling a breeze and then obviously the velocity of the air in this direction changes right so that would then be you so here it's zero and here start picking up the breeze and say oh the udx must be positive because as I'm changing X all of a sudden that velocity starts to grow so that's the UDS the u dy is the same in the Y direction I can do this for pressure I can do this for velocity I can do this for mass and that's it so as a physicist and mathematician we just set these things up looking at these physical rules and they're come the equations so that's the first step it's not so hard can we load the screen go back to the presentation yep be Thanks so now let's look at these equations that are a little bit fuzzy which I said before is some by choice because that's how we sometimes feel about them no it's just because copy and paste didn't work very well from leh Tech I don't know if you can see this but do you see all these DTS the X's device and what do you see you see V's you see little peak that's what that stands for pressure what do you think little you would be that's a velocity but it is one of the real how many velocities do I need to describe flow three right an X and y in the Z direction so we're so incredibly creative as mathematicians we generally call them uvw okay so we have X Y Z uvw so you recognize it the u DX this is the change in velocity in the X direction it's multiplied by something that's the fist causa T that makes a lot of sense if I start to push on something whether or not that goes real fast or not well that depends on whether I'm pushing air or whether and pushing peanut butter well what's the difference between them well one is a lot runnier than the other no one is very sticky we call that very viscous so obviously that viscosity needs to come in somewhere well there it is now let's see if I can find some forces right because these are all equations that tell me how velocity changes or momentum changes and they can only change as a result of applying a force okay so where are my forces they're hidden a little bit but you can see some forces here's a little guy this is G G is the gravitational constant it's multiplied by the density this is just gravity mg now so there's a gravity force and then I had one other force that I talked about there was the pressure gradient so what would be a pressure gradient that tells me how much a pressure changes over space right so how what would that look like what sort of term would the pressure gradient the pdy dpdx a DP DX and the P DX and the U DX are in the same equation no makes perfect sense okay so that's really it so now we have the equations and then the only thing that we need to think about is how do we model these now you know from algebra that if you have an equation like x squared is 9 you can solve for x I can you still do that plus or minus 3 right or a more complicated one to X is 16 X is 8 right so here in this case though the equations are a little bit more complicated first of all we've got these DD X's DDT's all put together then as you can see we have a bunch of equations because how many unknowns do we have here well we have P we need to know you we need to know V we need to know W right so these are already four equations if I have four unknowns how many equations or four unknowns how many equations do I need to solve for four unknowns for okay that was the easy one about algebra it's always the same okay and so I have four equations and are these equations independent can I take one and just solve it and then I have the solution and solve the next one now they're all related to each other and that makes a lot of sense because I can start moving things in this direction in the air without impacting flow in that direction of why would that be suppose I was god it's always like the fantasize about so this is my room right it's filled with air and I point my finger there and they tell these air molecules there to start moving in this direction what do you think will happen with the rest of the room will I just see that flow only in this direction I would have a vacuum there then right that would not be sustainable so what what happen things would rush in from this side so if I flow in this direction I'm immediately causing a flow in that direction so I cannot solve an equation for this direction only it will all be coupled together right and so all of these systems are coupled the very complex and I cannot just find a formula that gives me the solution at any point in space at any point in time that's impossible in other words there is no analytic solution to this I cannot do what you did in algebra ok so when I first saw this I thought for all these years I have been taught algebra and now I finally come to grad school and the first thing they tell me it's I can't use it I have to go do something else so what do we do well let me go on and show you first a couple of pictures of things that I've worked on these equations that you just saw can do things like this okay so this is a vertical takeoff and landing aircraft and you see the jets coming down now what these streams are here these are for DISA T or velocity lines but you can sort of imagine Highfill ASSA T coming down okay so this would be W in my equation will be very high here right this is the vertical velocity and what is driving that well a jet and in this jet I'm really just creating huge pressure difference that pushes that air out right it's just a pressure gradient that is doing this so if I were to put the pressure here then I see really really high pressure somewhere there and immediately a lower pressure as that flow is pushed down and then here it hits the boundary and then there is a boundary force right that I talked about skin friction and other things and the flow is forced away in this direction and then becomes turbulent and all of these things are governed by these equations okay I don't need anything more for turbulence or all of that it's all just in those equations that I showed you with the DD exes the DD Weiss the disease it's all the same okay but obviously I didn't do this algebraically this is another example one of my colleagues it's just air flow past a race car and then here you see pressure so these these are streamlines to indicate how the air is actually moving and how the particles move over this over this car and here you see the pressure build up at the surface some areas at the surface experience a much larger force than than others as this air is hitting it that depends on how aerodynamic this this car is but again all of this is just governed by these equations that I showed you and they're all solved in exactly the same way in fact these two could be solved by the same software this is another thing that I've done it's looking at flow 4-cyl design I worked for T New Zealand for a while for the America's Cup if you know who knows about the America's Cup okay good some sailors in the audience it was wonderful to work with them I worked on developing the gennaker and if you remember well in 2000 that was the only thing that didn't break right but that may be because we didn't design it very well so that we overloaded all the other systems you don't know that either right but here in in seal flow and I just show you this again exactly the same equations that we just had but I show this because the behavior of the flow is very different depending on this boundary configuration all right so these equations are all exactly the same but what you see in observe can change a lot no way the vertical takeoff and landing aircraft we had all the turbulence coming up but sometimes flows are very smooth so for example if I'm sailing upwind very close to the wind with my with my sails my sails just act like a wing of an airplane it's the same sort of thing but you can see here with smoke this is all smoke so you can visualize how the airflow okay so these are just how the smoke particles are moving and you see that especially on the on the jig there the the headsail it's very very smooth but there's nothing turbulent there is everything is smooth almost as if it is attached here you see a little bit of a wake a little bit of a turbulent wake where the flow is coming off but on the downwind leg when I have my gennaker when the flow hits that I see big eddies and turbulent flow again forming very very different behavior but exactly the same equations and you would think the same configuration because I have a boundary like a seal in both of these cases it's just that the uncoming flow is from a different direction and in one case it's from a direction that is very directly aligned with the seal and then the flow stays very smooth is not really him that so much by that seals so we call this attached flow and then the other case it's more like a parachute coming down where the flow just hits the seal that on is going around it and then the seal with its friction on the sill surface cannot keep that flow close to it it's almost as if you're on your bike your motorbike and you're trying to go around a sharp bend and you're going too fast and you're flying out of the bend what is causing that well you don't have enough centripetal force right not enough friction to in other words to balance the centripetal force that's the same here airflow hits this sail needs to go around the bend goes really really fast and the seal just doesn't have enough skin friction now it's a very simplistic way of looking at it to keep that flow attached to the seal and flies off it detaches and it starts twirling around okay but the same sail the same equations just different behavior and it I found it amazing that with that set of equations I can do all of these different things I'll show you a little simulation now this is just a if it works yeah this is just a simple did you see it move isn't it amazing this is my my little very simple little simulation but here I've have a downwind sill like a like a gennaker the wind is coming in goes around and here are the touches and I can just on my little laptop this just plays on this little thing do these type of simulations with these equations and then I could do more complicated things like going into three dimensions and actually doing both of these sails but now this particular simulation here that I that I ran for Team New Zealand would take like a week to run it's not very good right so these sometimes these equations they describe everything but they're very very expensive to simulate so we need to do something about it and this is why when we're doing things like silh design we generally just look at two dimensional cross-sections and when they do wing design they generally look at two dimensional cross-sections of a win yeah did my work change the look of the seal not really see when you are working on something like this for competitive team what they're looking for with Team New Zealand is looking for shaving off two or three seconds of a lag of the sailing race that maybe not 20 minutes long they just want to shave off a few seconds that's all and so when you think about this the changes that are looking for really are very subtle changes and I can design a beautiful sail but if the crew is not behaving now then they won't see the two or three seconds the errors that are making because I'm using approximations right I'm ignoring some stuff even when I write these equations down here I simulate things in windy conditions while wind is gusty and I can predict what it's going to be over 20 min a times so I make some approximations to the win they're probably off by more than the two or three seconds so here we always say these simulations give me bread and butter right on the shelf in my cupboard but they don't help because I'm working in the error margin if you'd like but the nice thing about doing these calculations is that with a simulator like this I can look at some really strange shapes and display now this is the way I see it this is not to give them the final design that is exactly a little bit better and it's the same in the aircraft industry you don't use these computers nowadays anymore to get this final design that is just right you always do that with Windtunnel tests and then you have to go and actually fly these things to see how it really behaves right ultimately but what you can do in this it's like a virtual laboratory right sort of at skill and I can put in all sorts of funky designs be creative and see if they do anything I can explore in other words much more than I can if I have to build a skilled model and put it in the wind tunnel test which is much more expensive so what we do with a lot of this stuff is just play trying out new ideas and with the ginika for Team New Zealand we did that we tried out some new ideas and then the other there's a really interesting thing because now I can come up with a new Jenica design what does that mean I can come up with a new shape saying if you had this shape of sail you could seal a little bit faster then I give it to the SIL designer and say I want this shape of civil and they say are you kidding me how can I build a sail that retains that shape when I put it in the wind because of course it's a lot more complicated when you actually need to build this right and so there's all sorts of stuff going on but see this more as a virtual laboratory where you can play okay all right now I will let's just leave this up but I'm going to scribble a little bit here on the side because we had those equations and I told you we cannot compute this with an algebraic equation right so and I've shown you results so obviously we must be doing something to be able to compute this but it's not a formula and so what do you do well the very first thing that you do as an engineer when you have an equation that is way too complex to solve exactly right or algebra Utley is what you compromise right I mean I'm call myself a computational engineer I'm not really an applied mathematician as I look at these equations to say I cannot find one formula I'm not interested and simplifying the physics so much that I can do this analytically I could do that could look a very simple domain very simple flow maybe just two dimensions or even just one dimension they' people do that one dimensional flow that's and funny right and so that they can find real algebraic equations that model this but I don't want to do that I want to have the three-dimensional feeling of this flow but I can't find it exactly and I think who cares if I can't find it Klee I'll find it approximately I don't need to know exactly that it is one point three seven four five six nine meters per second I just want to know it's around one that's probably enough right or maybe I want to get a few floating points in there but I don't need this completely accurately so here is the general idea of my field they say okay suppose that I want to simulate something and it's on this domain we're looking down on the Pacific Ocean okay and I want to simulate wind here maybe the wind sort of goes like this okay and all of these equations that are just showed with all the DD X's and DDT's simulate model this and I'm going to say to myself I don't need to know the solution everywhere that is the trick right so I want to know things only approximately and the way I'm going to compromise is to say I don't need to know the solution at every single XYZ and that every time T I'm going to be happy if I know the solution in some set of points okay and this was a fantastic idea in the whole field of computational fluid dynamics that I'm talking about is based on it so what they do is they divide this domain into a grid okay and the idea is that at each of these intersections of these grid lines are called grid points you find an approximate solution so in one dimension but would that look like so one dimension do something like this now say we have a domain between 0 & 1 I don't find a solution everywhere but I just find it add a bunch of points all of these points here and if I was looking for you of X but now I'm saying no I'm not going to look for U of X as a function but I'm going to find u 0 u 1 u 2 all of these points this is X 0 this is X 1 this is X 2 through two well maybe this is X capital n and all I'm doing is finding those solutions UN the UN minus one I just find n values that's all somehow we still need to talk about how but somehow I find them and then I start to think hey this the solution here or maybe the solution here in this Pacific Ocean through the equations that I have of course depends on the solutions they on there right for example the velocity here will depend on the pressure I have there and the pressure I have there but it also depends on the velocity I have there right because as I said earlier all of these things are coupled together so when I start translating these complex equations to relationships between the use at each of these points I will probably get a large set of coupled equations out of it right but it's now just a system of equations for the use in each of these grid points and even if there are million grid points and every great point says five unknowns I don't care it's only five million unknowns with five million equations my computers are big enough that I can solve this no problem right on the computer so the first step is that we do this that we say okay instead of wanting the solution everywhere we just wanted on the grid and then my question to you is what if I now one they have the solution right here because this specifically maybe Hawaii you know I'm interested in the wind on the Big Island and it's not in the grid point what would I do I could take a finer grid but I don't have enough computer power for that maybe I could shift the grid I heard that so that the grid point overlaps with that exactly but that may be hard because then the next thing I know somebody in Tahiti wants to know it and has just shifted my grid and Tahiti is no longer on it okay so I keep shifting don't want to do that either but I could just interpolate right so if I have a solution here here here and here I can probably find a pretty decent approximation to something in the middle and take some sort of average of them interpolated in other words if I have the solution at the bunch of points I can probably find a solution that is okay and that's all we're looking for in between those points if I want my accurate solution I got a computed for more points is a simple thing no such thing as a free lunch right you want accuracy you got to do it for more for more points now this opens up fantastic area several one is there are graders that look at the domain and create a nice grid for that domain and there's all sorts of ways you can do this I'm a very simple grid or I look at the domain and I want to put in straight lines like this because I like working with these types of grits because computationally they lead to simpler things but there are people that like using triangular ization so instead of having these straight lines they grid something like this anybody from near puts itself so you recognize this so this is simulation we they don't push it sound on for tidal flow modeling and everywhere did there is a vortice vortex of a triangle a point in a triangle is one of these grid points they're not as organized as nicely but these points is where we compute a solution and guess what the closer we get through the land to boundaries where things are changing faster the more points we want okay out here not much is happening really sort of steady flow we sort of know what it is it doesn't change very rapidly it just changes only with the tides I don't have all sorts of small disturbances or turbulence or small eddies of a river coming in all of these things happening at small skill don't have any of that so I can take what we call a coarse grid here there's only a couple of points in which you need to know the solution and then I know pretty much everywhere but here where there's a lot of stuff happening with the river coming in and all sorts of you know Jets and Eddie's here I need to have a lot of points so much in fact that you can't see them anymore a lot of points are together so this actually took one of my students about two months to create this grid to get it just right it seems very simple to do this triangular ization but there all sorts of things you have to be aware of for example if I do a grid here and the triangle is completely skewed almost squashed together like this then my equations don't behave so well this is just too skewed and and when I actually look at the equations and then how these equations solve themselves on the computer I get problems now I may have too big of an error now and so there's a whole science behind is that when you start translating these equations on these grid points how they behave okay so but that if they look very fancy I think we like looking at these things we've done one for Elkhorn Slough here in Monterey Bay right this is the grid and I gave you the answer but I always ask what's this thing when I show that to people that's actually the train track it's a little bit higher so we there's no flow going there it's just flow here in this little area here under the Train okay so that's Alchemist loop again takes a long time to think of these types of grids years agreed on the airplane now obviously this is not just the airplane itself that we're grading but also the flow around it but I won't show you that because it's so hard to see these things grow in three dimensions but in three dimensions just little tetrahedra so not my angles on the surface but tetrahedra volumes in three dimensions and you can see something's here near this edge here the leading edge of the airplane wing very very dense grits because a lot of stuff is happening there see huge differences in pressure and so on you can actually see the pressure distribution on the wing here blue is very low pressure I will be the top of an airplane wing have very low pressure well the better if it wasn't we'd be in real trouble because it's the pressure difference between the top of the wing in the bottom of the wing that keeps this plane up and so we like seeing very very low pressure here and these low pressure areas to true low pressure areas where most of this lift is is actually near the front of the airplane wing okay so if the back of the airplane wing damages a little bit don't worry too much when you're looking out and you see that fall off or break but if the front falls off you better get out okay now you can do other things like this is just a little simulator very simple thing but as you're moving something this is gas going through a reservoir you see as the gas is moving you see what the grid is doing can you see that very hard to see you but as this gas is moving through if you looked at this really carefully you can see the grid change so locally we're putting in more grid points because it's much harder to calculate some of this this gas flow in those regions where the gas and the oil come together so we can we do what we call adaptive grading so do all sorts of wonderful tricks that you can play to get good solutions the question I still haven't entered and I'm going to take three minutes for this and that's the end of your math class is how do I do the translation from these equations on to the grid okay and they're whole courses at Stanford called numerical analysis courses finding different sources final volume courses they give you ways to do it and there's all sorts of different methods but they're basically all the same thing so that's what we're going to talk about and it's something like this but I'd like to use the board so if we can just raise this close this and let's just give you a quick idea of how this is done and then you can probably use your your creativity to understand that this is a little bit more complicated in multiple dimensions but it doesn't matter it's all the same thing really nice let me just use this B I think yeah okay so here we are in one dimension okay just one dimension and we have a bunch of points in which we want to compute a solution okay and this point I want to compute you and that point is point number three or four number four point number five but I call it point number I okay so I give it a little number I and what point is this then I plus one see you get an A for this class and this is UI minus 1 okay now maybe you look something like this I don't know I don't know what that solution is right but suppose that this is my real solution now in my equations what do I have floating around I have things like D DX right the U DX right that is the change in U as X is changing what is that Billy it's a derivative remember that from way past so in this on at this position here now I'm interested in the U DX in other words I'd like to replace the U DX in all of these equations that I had with an approximation right that involves only Solutions at these points right because I cannot compute the true the udx because I don't have all that information about you the only information that I want to compute the only thing I want to know about the solution is these guys just in those points okay so it's this one this solution and that solution so now my task is find the udx an approximation to it in this point using only discrete values like this again I don't know how big the U is and the UI minus once and so forth are but it doesn't matter because I'm going to compute it all I want to find is how to replace this in my equation by an approximation and we call this a discrete approximation because I'm only using the discrete points yeah doesn't make sense so how do I do that I want to find an approximation to the slope in this point right the slope of this line the tangent line using only those points what would be really nice approximation well the slope is nothing but the change in U corresponding to some change in X so how about if I take a change you this minus this for example right so you I plus 1 minus UI minus 1 divided by this distance well I like these grids that have the same distance between no grid points ok because this would then simply be that so now imagine all these complex equations I had in the beginning and every D DX and every D dy and every D DZ is simply replaced by an approximation like that what would I get I would get a whole bunch of equations with use phase WS and PS in I plus 1 I minus 1 all of these grid points that I have and and for every grid point I would get a bunch of those equations so altogether if I have a million grid points and five unknowns foregrip one to five equation together a system of five million equations that all contain the solution values UI minus 1 UI plus 1 and so on and I can solve them all together but it's a large coupled system of equations how many of you have solved systems of coupled equations ok so it just takes a big computer and the smart algorithm I may maybe some of you who have heard about a Gaussian elimination to solve a matrix vector equation this is just a make big huge matrix vector equation that you get but for those of you know about it it's very large a fully coupled often nonlinear so I don't know if you notice this but sometimes we don't have UD UD X but we have something like u times 2 u DX it comes up in the momentum equation for example when I derive a momentum equation and so when I start discretizing that I get something like UI times UI plus 1 minus UI minus 1 divided by 2 H and that is part of my equation so now I have a product a multiplication of two unknowns and becomes nonlinear maybe it's quadratic okay but I have to solve a large system of nonlinear equations ultimately I get a huge matrix vector equation and I need to solve this on my computer but that's all ok so if you know how to do this so you take a couple of my courses next year I think you need with rusty calculus you need to and then you know everything there is to know about this so two quarters worth of investment and you can go and use commercial packages and do your simulations it tricks and the reason why I get money for consulting the the hard parts of this is to have good boundary conditions that are realistic know what parts of the physics you can ignore so that means that you need to know a little bit about the underlying physics I do need to understand it well enough to be to know what parts to throw out you need to be a reasonable gritter you need to be able to create nice grits and that's actually more of an art than a science this takes a long time to play with this especially if you have complex domains like put it sound or Monterey Bay but sometimes for a room like this would be very very simple because you kind of square very easy to put a good grid in this and then once you have all of these equations you really need to know a little bit about matrix calculations to do this fast okay so that's maybe another course so let's say three courses right – numerical analysis one matrix computation and you can solve almost any fluid flow problem okay and you can make the bit money you're looking for another career not as much as if you do search engine design which is also just a matrix calculation okay anyway that's my job and that's what I love and hopefully you learned a little bit for more please visit us at stanford.edu

Basic Concept to Understand One Sided Limits in Calculus



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One side limits Playlist:

I'm Ali Kemal and in this video I'll discuss one-sided limits basic concepts let me sketch few functions so that we can talk about this topic a bit let's say we have a function kind of like this let me fill this out and let me draw a straight line from here and let's say the line is kind of like this so so I've sketched up function which seems to be of two different pieces and at some value on the x-axis let's say this value is 2 we find that there is a break there is a jump discontinuity we'll call this function as f of X now the question here is what is the limit of this function when X approaches 2 from the left side and we'll call this as the left side okay from the left side which we write with negative exponent so what is the limit of this function when X approaches 2 from the left side now since we are approaching only from one side we call it one-sided limits is it ok the answer for this is when you approach from this side you actually hit this part of the graph and where you reach is its limit let's say this value which you're approaching is 3 then the answer will be 3 is that ok I could have asked a different question what is the limit of this function when X approaches 2 from the right side and that right side is indicated by plus sign so this is your right side so if I approach from the right side I'm going to hit this part of linear curve and I'll approach this value as I come closer and closer to 2 right let's say this value is minus 2 so the value of this function or the limit of this function as X approaches 2 from the right side will be minus 2 so that is what we consider one side limits limit from either side now you have been doing limit when X approaches a value let's say 2 now in this particular case if you approach true from left side you reach at 3 if you are approaching 2 from the right side you're approaching minus 2 they are not equal so when they are not equal we say the limit does not exist the reason is these two limits are not equal I hope you got the concept right now from the graph at times you may be asked to find the value of the function at 2 which let me write it as what is f of 2 equal that filled in hole here gives you the value which is 3 is it ok so that is how you're going to answer one-sided limits based questions if a graph is given to you right now let me take another example so let's take an equation this time so so we have a function f of X wedges X square divided by X as you can see from here that X should not be equal to 0 because then it is not defined is that ok so for this particular function you need to find limit as X approaches 0 from the left side that is – you need to find the limit when X approaches 0 from the right side for the same function you need to find the limit when X approaches 0 right let's check it out well the best way is to really sketch it and then write down the answers so let's sketch it here and we'll write down the answers so of course at 0 it is not define so this function could be written as if I cancel one of those X's so you read this as f of X is equals to X where X is not equal to 0 it does make sense to you yes it does right so basically at 0 we have a hole and the function could be graphed as X perfect so that becomes the given function and now it's easy for us to answer all these questions so sketching is a good way of doing things for finding one-sided limits okay so when you're approaching zero from the left side you hit this portion and you're approaching zero itself so this answer is zero if you are approaching from the right side you again approach origin so it's zero since these two limits from left and right are same the limit exists at as X approaches zero and this is also 0 however what is the value the function at 0 does not exist do you see that it is not defined at origin so there could be cases when limit may exist however the function may be discontinuous as in this particular case now as an exercise I'd like you to do this question which is f of X equal to absolute x over X right find left-side limits at 0 and the value function at 0 you can see this function is also not defined at 0 is it okay but does the limit exist at 0 not figure it out if you sketch this function you're going to get a sketch like this and that will help you to answer your questions I wasn't Kumar and I think you got the concept of one sided limits in this list of videos I'll consider a couple of examples well we'll explore the topic with the help of graphs equations piecewise functions and word problems I will look more and I hope you'll appreciate this journey of learning about one-sided limits you can always share and subscribe to my videos and feel free to like them thank you and all the best

Newton's Laws: Crash Course Physics #5



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I’m sure you’ve heard of Isaac Newton and maybe of some of his laws. Like, that thing about “equal and opposite reactions” and such. But what do his laws …

Anyone Can Be a Math Person Once They Know the Best Learning Techniques | Po-Shen Loh



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Po-Shen Loh is a Hertz Foundation Fellow and Carnegie Mellon mathematics professor who thinks that history is a much harder subject than math. Do you agree? Well, your position on that might change before and after this video. Loh illuminates the invisible ladders within the world of math, and shows that it isn’t about memorizing formulas—it’s about processing reason and logic. With the support of the Fannie and John Hertz Foundation, Po-Shen Loh pursued a PhD in combinatorics at the Pure Math Department at Princeton University.

The Hertz Foundation mission is to provide unique financial and fellowship support to the nation’s most remarkable PhD students in the hard sciences. Hertz Fellowships are among the most prestigious in the world, and the foundation has invested over $200 million in Hertz Fellows since 1963 (present value) and supported over 1,100 brilliant and creative young scientists, who have gone on to become Nobel laureates, high-ranking military personnel, astronauts, inventors, Silicon Valley leaders, and tenured university professors. For more information, visit hertzfoundation.org.

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I think that everyone in the world could be a math person if they wanted to. The keyword though, I want to say, is if they wanted to. That said, I do think that everyone in America could benefit from having that mathematical background in reasoning just to help everyone make very good decisions. And here I’m distinguishing already between math as people usually conceive of it, and decision making and analysis, which is actually what I think math is.

So, for example, I don’t think that being a math person means that you can recite the formulas between the sines, cosines, tangents and to use logarithms and exponentials interchangeably. That’s not necessarily what I think everyone should try to concentrate to understand. The main things to concentrate to understand are the mathematical principles of reasoning.

But let me go back to these sines, cosines and logarithms. Well actually they do have value. What they are is that they are ways to show you how these basic building blocks of reasoning can be used to deduce surprising things or difficult things. In some sense they’re like the historical coverages of the triumphs of mathematics, so one cannot just talk abstractly about “yes let’s talk about mathematical logic”, it’s actually quite useful to have case studies or stories, which are these famous theorems.

Now, I actually think that these are accessible to everyone. I think that actually one reason mathematics is difficult to understand is actually because of that network of prerequisites. You see, math is one of these strange subjects for which the concepts are chained in sequences of dependencies.

When you have long chains there are very few starting points—very few things I need to memorize. I don’t need to memorize, for example, all these things in history such as “when was the war of 1812?” Well actually I know that one, because that’s a math fact—it was 1812—but I can’t tell you a lot of other facts, which are just purely memorized. In mathematics you have very few that you memorize and the rest you deduce as you go through, and this chain of deductions is actually what’s critical.

Now, let me contrast that with other subjects like say history. History doesn’t have this long chain, in fact if you fully understand the war of 1812 that’s great, and it is true that that will influence perhaps your understanding later of the women’s movement, but it won’t to be as absolutely prerequisite. In the sense that if you think about the concepts I actually think that history has more concepts than mathematics; it’s just that they’re spread out broader and they don’t depend on each other as strongly. So, for example, if you miss a week you will miss the understanding of one unit, but that won’t stop you from understanding all of the rest of the components.

So that’s actually the difference between math and other subjects in my head. Math has fewer concepts but they’re chained deeper. And because of the way that we usually learn when you had deep chains it’s very fragile because you lose any one link—meaning if you miss a few concepts along the chain you can actually be completely lost. If, for example, you’re sick for a week, or if your mind is somewhere else for a week, you might make a hole in your prerequisites. And the way that education often works where it’s almost like riding a train from a beginning to an end, well it’s such that if you have a hole somewhere in your track the train is not going to pass that hole.

What is Calculus? (Mathematics)



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What is Calculus? In this video, we give you a quick overview of calculus and introduce the limit, derivative and integral.

We begin with the question “Who invented Calculus?” Next, we talk about the two main tools you’ll study: derivatives and integrals. To understand both of these you’ll first learn about limits. After you learn how to compute the derivative and integral for basic functions and apply them to real-world problems, you’ll move up to higher dimensions and study things like “partial derivatives” and “multiple integrals.”

What to watch next:
The Tangent Line & the Derivative

Product Rule for Derivatives

Quotient Rule

How to Study For a Test

How to Study Physics

How to Study Programming

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Michael Harrison received his BS in math from Caltech, and his MS from the University of Washington where he studied algebraic number theory. After teaching math for a few years, Michael worked in finance both as a developer and a quantitative analyst (quant). He then worked at Google for over 5 years before leaving to found Socratica.

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what is calculus this isn't something we inherited from the ancient Greeks like geometry this subject was discovered much more recently in the late 1600s by Isaac Newton and Gottfried Leibniz they didn't work together they both discovered calculus on their own and as a result there was a huge argument over who should receive credit for its discovery but we're going to save that story for another video today let's talk about what they discovered in calculus you start with two big questions about functions first how steep is a function at a point second what is the area underneath the graph over some region the first question is answered using a tool called the derivative and to answer the second question we use integrals let's take a look at the derivative the tool that tells us how steep a function is at a point another way to think about the derivative is it measures the rate of change of a function at a point as an example let's use the function f of X equals x cubed minus x squared minus 4x plus 4 suppose we want to find the steepness of the graph at the point negative 1/6 how would we do it and what do we even mean by steepness in algebra you find the rate of change of a line by computing the slope the change in Y divided by the change in X but this is a curve not a line so we get a good look let's zoom in a bit here's the idea pick a second point nearby how about the point negative 0.8 6.0 4 8 next draw a line through these two points the slope of this line is a good approximation for the steepness of the curve at the point negative 1/6 if you compute the slope you get zero point two four this is a good approximation but we can do better what if we pick a different point that's even closer how about negative zero point nine six point zero six one if you compute the slope of the line through this point and negative 1/6 you get zero point six one if you keep picking closer and closer points and computing the slopes of the lines you'll get a sequence of slopes we start getting closer and closer to some number the lines are getting closer and closer to the tangent line and the slopes are approaching one so we say the slope of the curve at the point negative one six is one we call this number the derivative of f of X at the point where x equals negative one this is the slope of the tangent line through the point negative one six luckily you won't have to do this every time you want to measure the rate of change at a point in calculus you'll learn how to find a function which will give you the slope of any tangent line to the graph this function is also called the derivative next let's take a look at the integral this is the tool that lets you find areas under curves as an example let's look at the function G of x equals sine X what if we wanted to find the area under this curve between x equals 0 and x equals pi how would we do it we know how to find the area of simple shapes like rectangles and circles but this is much more curvy and complicated let's zoom in to get a closer look here is the idea slice the region into a bunch of very thin sections let's start with ten slices for each section find the area of the tallest rectangle you can fit inside there are ten thin rectangles the width of each rectangle is PI over ten and we can find the height using the function G of X next add up the areas of all ten rectangles we get a combined area of one point six six 936 this is a pretty good approximation to the area under the curve but we can do better what if we do this again but use twenty-five slices instead this time we get an approximate area of one point eight seven one nine five let's do this again and again using thinner and thinner slices fifty slices a hundred slices one thousand slices you get a sequence of areas that are getting closer and closer to some number it looks like the area is approaching two we call this area the integral of G of X from x equals zero to x equals pi so we have these two tools the derivative and the integral the derivative tells us about a function at a specific point while the integral combines the values of the function over a range of numbers but notice there is something similar to how we found the derivative and the integral in the case of the derivative we found two points that were close to each other then we let one point get closer and closer and closer to the point that we're interested in in the case of the integral we took the curve and we chopped it up into a bunch of rectangles to approximate the area under the curve then we took thinner and thinner rectangles to get better and better approximations in both cases we're using the same technique in the case of the derivative we're relating the points get closer to each other in the case of the integral we're letting the rectangles get thinner in both instances we're getting better and better approximations and we're looking at what number these approximations are approaching the number they're approaching is called the limit and because limits are key to computing both the derivative and the integral when you learn calculus you usually start by learning about limits a lot of your time in calculus will be spent computing derivatives and integrals you'll start with the essential functions polynomials trig functions sine cosine and tangent exponential functions and logarithmic functions these are the building blocks for most of the functions you'll work with next you'll make more complex functions by adding subtracting multiplying and dividing these functions together you'll even combine them using function composition in calculus there are a lot of rules to help you find derivatives and integrals of these more complex functions the derivative rules have names like the product rule quotient rule and chain rule the integral rules include a u-substitution integration by parts and partial fraction decomposition when you first start calculus your focus will be on basic functions functions with one input and one output but we don't live in a one-dimensional world our universe is much more complicated so once you've mastered calculus for basic functions you'll then move up to dimensions for example consider a function with two inputs and one output like f of X y equals e to the negative x squared plus y squared earlier we computed the derivative by computing slopes of tangent lines but in higher dimensions things are a bit more complex this is because on a surface instead of a tangent line you'll have a tangent plane to handle this you'll compute the derivative both in the X direction and in the Y direction we call these partial derivatives these two partial derivatives are what you need to describe the tangent plane we'll also need to generalize the integral the region below a surface is three-dimensional it has a volume not an area to compute the volume will approximate it using a bunch of skinny boxes to sum up all the volumes you'll need to use a double integral because the boxes are spread out in two dimensions but don't forget we live in three spatial dimensions so you'll also need to learn calculus for functions with three inputs XY and Z if a function has three inputs and one output we call it a scalar field an example would be a function returning the temperature at a given point in space and the outputs of functions don't have to be simply numbers that can also be vectors a function with three inputs and a vector output are called vector fields an example would be a function which gives the force vector due to gravity at every point in space to recap the two main tools you'll learn about in calculus are the derivative and the integral the derivative tells you about a function at a specific point namely tells you how quickly the function is changing at that point the integral combines the values of a function over a region you'll start your study of calculus by learning how to compute the derivatives and integrals of a wide variety of functions next you're going to take these tools and apply them to higher dimensions by using things called partial derivatives and multiple integrals and along the way you'll learn how to apply derivatives and integrals to solve real-world problems so now that you've seen the big picture it's time to start learning the details so let's get to work we'll be releasing many more calculus video soon the best way to find out when we release a new video is to text a friend each morning and asked if Socratic ax has published a new video and if you would like to help us grow in to release videos more quickly please consider supporting us on patreon

Probability Part 3 || GATE Questions || Gate lectures for engineering mathematics



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Probability Part 3 GATE Questions
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The Map of Mathematics



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The entire field of mathematics summarised in a single map! This shows how pure mathematics and applied mathematics relate to each other and all of the sub-topics they are made from.

If you would like to buy a poster of this map, they are available here:
North America:
Everywhere else:

I have also made a version available for educational use which you can find here:

To err is to human, and I human a lot. I always try my best to be as correct as possible, but unfortunately I make mistakes. This is the errata where I correct my silly mistakes. My goal is to one day do a video with no errors!

1. The number one is not a prime number. The definition of a prime number is a number can be divided evenly only by 1, or itself. And it must be a whole number GREATER than 1. (This last bit is the bit I forgot).

2. In the trigonometry section I drew cos(theta) = opposite / adjacent. This is the kind of thing you learn in high school and guess what. I got it wrong! Dummy. It should be cos(theta) = adjacent / hypotenuse.

3. My drawing of dice is slightly wrong. Most dice have their opposite sides adding up to 7, so when I drew 3 and 4 next to each other that is incorrect.

4. I said that the Gödel Incompleteness Theorems implied that mathematics is made up by humans, but that is wrong, just ignore that statement. I have learned more about it now, here is a good video explaining it:

5. In the animation about imaginary numbers I drew the real axis as vertical and the imaginary axis as horizontal which is opposite to the conventional way it is done.

Thanks so much to my supporters on Patreon. I hope to make money from my videos one day, but I’m not there yet! If you enjoy my videos and would like to help me make more this is the best way and I appreciate it very much.

Here are links to some of the sources I used in this video.

Links:
Summary of mathematics:
Earliest human counting:
First use of zero:
First use of negative numbers:
Renaissance science:
History of complex numbers:
Proof that pi is irrational:
and

Also, if you enjoyed this video, you will probably like my science books, available in all good books shops around the work and is printed in 16 languages. Links are below or just search for Professor Astro Cat. They are fun children’s books aimed at the age range 7-12. But they are also a hit with adults who want good explanations of science. The books have won awards and the app won a Webby.

Frontiers of Space:
Atomic Adventure:
Intergalactic Activity Book:
Solar System App:

Find me on twitter, instagram, and my website:

Matematika që mësojmë në shkollë nuk i bën drejtësi mjaftueshëm fushës së matematikës. Ne shohim vetëm një kënd të saj, por matematika si tërësi është lëndë e madhe dhe me një llojllojshmëri të mahnitshme. Qëllimi im me këtë video është të ju tregoj juve të gjitha ato gjëra mahnitëse. Ne do të fillojmë që nga më e para pikë e fillimit. Origjina e matematikës bazohet në numërim. Në fakt, numërimi nuk është vetëm aftësi e njeriut, kafshë të tjera kanë aftësi të numërojnë gjithashtu dhe dëshmi për numërimin e njeriut shohim që në kohët pre-historike me shenja të bëra në eshtra. Përgjatë viteve ka pasur risi të ndryshme, me ekuacionin e parë nga Egjiptianët, Grekët e lashtë bënë hapa të mëdha në shumë fusha si gjeometria dhe numerologjia, dhe numrat negativ ishin zbuluar në Kinë. Dhe zeroja si një numër ishte përdorur për herë të parë në Indi. Pastaj në Kohën e Artë të Islamit matematicientët Persian bënë hapa të mëdha mëtutje me librin e parë të shkruar për algjebër. Pastaj matematika lulëzoj në renesancë përgjatë shkencave të tjera. Tani, ka më shumë për historinë e matematikës se sa ajo çka unë sapo thashë, por do të kërcej në kohën moderne dhe matematikën ashtu si ne e njohim. Matematika moderne për së gjeri mund të ndahet në dy fusha, matematika e pastër: shkenca e matematikës për hir të vetes, dhe matematika e aplikuar: kur ju zhvilloni matematikën në ndihmë të zgjidhjes së disa problemeve reale në botë. Por, ka shumë mbikalime. Në fakt, shumë herë në histori dikush ka kaluar përgjatë shkretisë së matematikës motivuar thjesht nga kurioziteti dhe sikur i drejtuar nga një sens estetik. Dhe pastaj ata kanë krijuar një tërësi të re të matematikës që ishte e mire dhe interesante por nuk sjell diçka që mund të jetë e dobishme. Por pastaj, le të themi pas qindra vjetësh, dikush do të punoj në ndonjë problem në këndin e fundit të fizikës apo shkencës kompjuterike dhe do të zbulojnë që kjo teori e vjetër në matematikën e pastër është pikërisht ajo çka ju nevojitet për të zgjidhur problemin e tyre real të botës! Që është e mahnitshme, mendoj! Dhe kjo lloj gjëje ka ndodhur kaq shumë herë përgjatë shekujve të fundit. Është interesante sa shpesh diçka kaq abstrakte përfundon të jetë shumë e dobishme. Por do duhej të përmendja dhe se matematika e pastër në vetvete është ende një gjë shumë me vlerë për tu bërë ngase mund të jetë interesante dhe në vetvete të ketë bukuri dhe elegancë të vërtetë që gati bëhet si art. Në rregull mjaft me këtë shpjegim të entuziazmuar, le ti hyjmë punës. Matematika e pastër është e përbërë nga disa pjesë. Studimi i numrave fillon me numrat natyral dhe çfarë mund të bësh me ata me operacione aritmetike. Dhe pastaj shikon lloj të tjerë të numrave si numrat e plotë, që përmbajnë numrat negativ, numrat racional me thyesa, numrat real që përfshijnë numrat si pi që vazhdojnë në pika të pafundme decimale, dhe pastaj numrat kompleks dhe një tërësi tjetër. Disa numra kanë veti interesante sikur janë numrat e thjeshtë, ose numri pi ose eksponenciali (treguesi). Ka edhe veti të këtyre sistemeve numerike, si për shembull, edhe pse ka një sasi të pafundme të të dyjave numrave të plotë dhe real, ka më shumë numra real se sa ka numra të plotë. Kështu disa pafundësi janë më të mëdha se të tjerat. Studimi i strukturës ka të bëjë me ku ti fillon të marrësh numrat dhe ti vendosësh në ekuacione në formën e ndryshoreve. Algjebra përmban rregullat se si pastaj manipulon këto barazime. Këtu do të gjesh dhe vektorë dhe matrica që janë numra shumë dimensional, dhe rregullat se si këto bashkëveprojnë me njëra-tjetrën është kapur në algjebrën lineare. Teoria e numrave studion karakteristikat e të gjithave në pjesën e fundit në numrat si vetitë e numrave të thjeshtë. Kombinatorika shikon vetitë e strukturave të veçanta si pemët, grafet, dhe gjëra të tjera që janë të bëra nga copa diskrete që mund ti numërosh. Teoritë grupore shikojnë tek objektet që janë në bashkëveprim me njëra-tjetrën pra në grupe. Shembull i njohur është kubi i Rubikut që është një shembull i grupit të permutacioneve. Dhe teoria e renditjes heton se si objektet renditen duke ndjekur rregulla të caktuara, si diçka është sasi më e madhe se sa diçka tjetër. Numrat natyral janë një shembull e një strukture objektesh të renditura, por gjithçka me çfarëdo lidhje binare mund të renditet. Një pjesë tjetër e matematikës së pastër shikon tek figurat dhe si ato sillen në hapësirë. Origjina është në gjeometri që përfshin Pitagorën, dhe është e afërt me trigonometrinë, me të cilat jemi të njoftuar në shkolla. Gjithashtu ka gjëra argëtuese sikur gjeometria fraktale që janë modele matematikore me shkallë të pandryshueshme, që do të thotë mund të zmadhoni modelin pafundësisht dhe gjithmonë do të duket e njëjtë. Topologjia shikon tek pjesë të ndryshme të hapësirave ku jeni të lejuar që vazhdimisht ti deformoni ato por jo të i ndani apo ti bashkoni ato pjesë. Për shembull rripi i Mobiusit ka vetëm një sipërfaqe dhe një kënd pavarësisht çka i bëni. Dhe gotat e kafes dhe petullat në formë gjevreku janë e njëjta gjë – në aspekt topologjik. Teoria e matjeve është mënyra e caktimit të vlerave në hapësira apo struktura të lidhura së bashku me numrat dhe hapësirën. Dhe së fundmi, gjeometria diferenciale shikon vetitë e formave në sipërfaqet e lakuara, për shembull trekëndëshat kanë kënde të ndryshme në sipërfaqe të lakuar, dhe kjo na sjell neve në pjesën e radhës, që është ndryshimet. Studimi i ndryshimeve përmban kalkulusin që përfshin integralet dhe diferencialet që shikon tek fusha e hapur nga funksionet apo sjellja e shkallëve të funksioneve. Dhe kalkulusi vektorial shikon të njëjtat gjëra për vektorët. Këtu gjejmë dhe një tërësi të fushave të tjera si sistemet dinamike që shikojnë sistemte që evukojnë gjatë kohës nga një gjendje në tjetrën, sikur rrjedhja e lëngut apo gjërave me cikleve me reagim sikur ekosistemet. Dhe teoria e kaosit që studion sistemet dinamike që janë shumë të ndjeshme ndaj kushteve iniciuese. Së fundi analiza komplekse shikon vetitë e funksioneve me numra kompleks. Kjo na sjell në matematikën e aplikuar. Në këtë pikë është me vend të përmendim që çdo gjë këtu është më shumë e ndërlidhur se sa që unë e kam vizatuar. Në realitet kjo hartë duhet të duket më shumë si një rrjetë që lidh të gjitha subjektet e ndryshme por ju mund të bëni kaq shumë vetëm në një rrafsh dy dimensional, kështu i kam shpërndarë këto më së mirti si kam mundur. Në rregull do të fillojmë me fizikën, që përdor gati gjithçka në anën e djathtë në një shkallë. Fizika matematikore dhe teorike ka një lidhje shumë të ngushtë me matematikën e pastër. Matematika gjithashtu përdoret në shkenca të tjera natyrale me kiminë matematikore dhe biomatematikën që merren me shumë gjëra prej modelimit të molekulave tek biologjia evolucionare. Matematika është po ashtu e përdorur dukshëm në inxhinieri, ndërtimi i gjërave ka marr shumë matematikë që nga koha e Egjiptianëve dhe Babilonasëve. Sisteme elektrike shumë komplekse sikur avioni apo rrjeti i energjisë përdorin metoda në sisteme dinamike të quajtura teori të kontrollit. Analiza numerike është mjet matematikor zakonisht i përdorur në vende ku matematika bëhet shumë komplekse për tu zgjidhur plotësisht. Kështu që në vend të saj, ti përdor shumë
të vlerave të përafërta dhe i kombinon ato së bashku për të marr përgjigje të përafërta të mira. Për shembull, nëse ti e vendos një rreth brenda një katrori, hedh shigjeta në të, dhe pastaj e krahason numrin e shigjetave në pjesët e rrethit dhe katrorit, ti mund të përafrojsh vlerën e numrit pi. Por në botën reale analiza numerike bëhet në kompjutera të mëdhenj. Teoria e lojërave merret me se cilat janë zgjedhjet më të mira në rastin e një strukture rregullash dhe lojëtarëve racional dhe përdoret në ekonomi ku lojëtarët mund të jenë inteligjent, por jo gjithmonë, dhe fusha të tjera si psikologji, dhe biologji. Probabiliteti është shkenca e ngjarjeve të rëndomta sikur hedhja e monedhave apo zaret apo njerëzit, dhe statistika është shkenca e koleksioneve të mëdha të proceseve të rëndomta apo organizimi dhe analizimi i të dhënave. Kjo natyrisht është e lidhur me financa matematikore, ku ju doni sisteme të modeleve financiare dhe të marrë një avantazh për të fituar të gjitha ato shtresa të trasha. Lidhur me këtë është optimizimi, ku ju provoni të kalkuloni zgjedhjen më të mirë përgjatë një strukture të shumë opsioneve të ndryshme, të cilat ju mund ti vizualizoni normalisht duke provuar të gjeni pikën më të lartë apo më të ulët e një funksioni. Problemet optimizuese janë të natyrës së dytë tek ne njerëzit, ne i bëjmë ato gjatë gjithë kohës: duke provuar të marrim vlerën më të mirë për para, apo të zmadhojmë harenë tonë në një mënyrë. Tjetër fushë që është shumë e lidhur me matematikën e pastër është shkenca kompjuterike, dhe rregullat e shkencës kompjuterike në fakt janë derivuar në matematikën e pastër dhe është një shembull tjetër i diçkaje që është punuar shumë përpara ndërtimit të kompjuterëve programues. Mësimi i makinës: krijimi i sistemeve kompjuterike inteligjente përdor shumë fusha në matematikë sikur algjebra lineare, optimizimi, sistemet dinamike dhe propabiliteti. Dhe në fund teoria e kriptografisë është shumë e rëndësishme në kompjutim dhe përdor shumë matematikë të pastër sikur kombinatorikën dhe teorinë e numërave. Pra kjo mbulon pjesët bazike të matematikës së pastër dhe të aplikuar, por nuk mund të përfundoj pa shikuar në themelet e matematikës. Kjo fushë provon të punoj vetitë e matematikës vet, dhe pyet se çka është baza e të gjitha rregullave të matematikës. A është ndonjë strukturë komplete e rregullave bazike, të quajtura aksiomë, nga e cila vjen e gjithë matematika? Dhe a mund të vërtetojmë që e gjitha është e qëndrueshme në vetvete? Logjika matematike, teoria e vendosur dhe teoria e kategorisë mundohen ti përgjigjen kësaj dhe një rezultat i famshëm në logjikën matematike janë teorema e pakompletuar e Godel, që për shumë njerëz, do të thotë që Matematika nuk ka një strukturë të kompletuar dhe të qëndrueshme të aksiomave, që tregon që e gjitha disi është e bërë nga ne njerëzit. Që është e çuditshme duke parë që matematika shpjegon kaq shumë gjëra në Univers kaq mirë. Pse do duhej një gjë e bërë nga njerëzit të ketë aftësi të bëjë një gjë të tillë? Po aty, kjo është një mister i thellë. Gjithashtu kemi teorinë e kompjutimeve që merret me modele të ndryshme të kompjutimit dhe se sa me efikasitet ata mund të zgjidhin probleme dhe përmban teori të kompleksitetit që merret me atë se çka është dhe nuk është e llogaritshme dhe sa memorie dhe kohë do duhej, që për shumicën e problemeve interesante, është një sasi e çmendur. Kështu ajo është harta e matematikës. Tani gjëja që kam dashur më së shumti kur kam mësuar matematikë është ajo ndjenja që ti e ke kur diçka që dukej kaq konfuze më në fund klikon në trurin tënd dhe gjithçka ka kuptim: sikur një moment pagëzimi, disi sikur të shihje nëpërmjet matricës. Në fakt disa nga momentet më të kënaqshme intelektuale kanë qenë kuptimi i disa pjesëve të matematikës dhe pastaj të ndjerit sikur unë kisha një shikim të shkurtër në natyrën themelore të Universit në të gjitha çuditë e saj simetrike. Është e mrekullueshme, e dua atë. Punimi i hartës së matematikës ka qenë kërkesa më e popullarizuar që kam marrë, për të cilën isha shumë i lumtur sepse e dua matematikën dhe është e mrekullueshme të shikosh kaq shumë interesim për të. Kështu që shpresoj që ju ka pëlqyer. Natyrisht është vetëm kaq shumë sa mund të fus në këtë pjesë kohe, por me shpresë që i kam bërë drejtësi subjektit dhe se ju e keni vlerësuar të dobishme. Kështu do ketë më shumë video që do vijnë nga unë shpejt, këtu janë të gjitha gjërat e zakonshme dhe ishte një kënaqësi, shihemi herën tjetër.

The Austrian Cult and Mathematical Economics with Ash Navabi



Views:305|Rating:3.50|View Time:33:55Minutes|Likes:7|Dislikes:3
This podcast episode was first posted on August 22nd, 2014.

In this episode, Ash Navabi discusses whether the Austrian School of Economics is a cult and the value of mathematics in economic theory. Ash is an economics student at Ryerson University.

Ash wrote an article responding to recent criticisms of the Austrian school by Keynesian bloggers Noah Smith and Paul Krugman. Krugman approvingly referenced Smith’s attacks on the “hermetic system that is Austrians.” Just a week later he made the following telling comment about the economics mainstream:

“And modern academic economics is very much an interlocking set of old-boy networks; to some extent this has become even more true since the decline of the journals, with most discourse taking place via working papers long before formal publication. I used to refer to the international trade circuit as the floating crap game — the same 30 or 40 people meeting in conferences all over the world, reading and citing each others’ work; it’s the same in each sub-field. And to some extent it’s inevitable: there’s so much stuff out there, and you have to filter somehow, so you mainly read stuff by people you know and people they tell you are worth reading.”

Ash was quick to point out that, by the logic of the people who deride Austrian economists as “cultish” because they interact mainly with one another, each of the “old-boy networks” Paul Krugman refers to (that is, each sub-field of mainstream economics) must also be a cult.

Gary Becker, another Nobel Laureate, referred to the Austrian school as a cult in a letter to Walter Block. Becker’s definition of a cult was “a small number of dedicated followers who speak mainly to each other, and interact little with let us call them mainstream economists.” This definition is problematic, to say the least. When people hear the word “cult,” they don’t think of Becker’s dry definition but of animal sacrifice and mass suicide. The word “cult” also implies unquestioning devotion to the cult leaders, but modern Austrians frequently criticize Mises and Hayek, in highly un-cultish fashion.

Ash also wrote an article on mathematical economics versus so-called “literary” economics. John Cochrane recently referred to non-mathematical economics as “literary,” a mild slur that goes back at least as far as the 1940s when Mises responded to it in Human Action. The Austrian method is not “literary” in the sense of using airy prose and fuzzy logic, rather it uses a highly rigorous form of verbal logic to derive causal chains from the basic axioms of human action.

Mathematical economics forces economists to start their analyses from unrealistic assumptions in order to put all problems in mathematically tractable terms. However rigorous the mathematics itself is, the foundation is flawed so the conclusions are flawed.

Austrians conceive of economic theory as a descriptive science rather than a predictive one. That is, pure theory cannot tell you how the future will turn out, nor is a theory tested by its empirical predictions. An entrepreneur can have a true theory of how the economy works, and yet he can still make wrong predictions if he misjudges the actual factors at play.

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you're listening to economics detective radio today I'm talking with Ashe Navabi about whether or not Austrian economics is a cult and about the use of mathematics in economic theory Ashe welcome to economics detective radio thanks for having me so first a question about yourself how did you come to join the Austrian School of Economics and did the process involved cloaked figures chanting and animal sacrifice know how I came about it is pretty interesting other people have told me I was searching for Peter Sellers videos on YouTube Peter Sellars was a comedian and actor in the 70s I was searching for his videos on YouTube and I typed in Peter s and the first suggested video was Peter Schiff was right and I clicked on that video and it turned out it was about a guy named Peter Schiff who was going on CNBC and Fox Business and Fox News and all these other TV shows predicting the economic crash in 2006 and 2007 and early 2008 while everyone else was laughing at him and this was October 2008 while the crash was happening and everyone else before it was showing this time making these predictions that there was going to be this huge crash and everyone was laughing at him saying this will never happen and I thought that was really interesting especially because in October 2008 at the time everyone on TV was saying no one could have seen this coming where I was watching this guy predicting this stuff very accurately one two three years prior yeah 2008 crash was in a way a blessing in disguise at least for the Austrian school yes that's a good point I was I was actually an engineering student at the time I knew nothing about economics I decided to go into chapters a bookstore and get some economics books that picked up a couple one of them was economics in one lesson I tried reading the other economics books I got first I could not can I make they had three tails out of what they were trying to say is found the writing boring and confusing but when I opened economics in one lesson and I read about the broken window fallacy I was swept away and I did some research and I found out that both the author of economics in one lesson Henry Hazlitt and Peter Schiff were both part of the Austrian school and a little while later I switched my major into economics and I'm here now so the reason I asked you on was because I read an article you wrote for Mises Canada and it was responding to a recent article on Lew Rockwell calm where Walter Bloch recounted two cases of eminent economists casually referring to the Austrian school as a cult one of them was blocks thesis adviser Gary Becker the Nobel laureate and he said to quote him by a cult I mean a small number of dedicated followers who speak mainly to each other and interact little with let us call the mainstream economists so does the Austrian school fit Becker's definition of a cult in your opinion in my opinion if it does fit backwards definition of a cult so does every other school of economics and that was the point I was trying to make actually that article from Rocco was a few years old I wrote that article in response to a blog post Paul Krugman wrote very recently just about a week ago now where he said that international econ people who study international economics there's about 30 or 40 of them he describes it as a floating craps game as the same people going around conference conference around the world just mainly talking to each other and reading what everyone else is reading and I when I read this I was reminded of how Gary Becker described a call to walter block words both denote and they Connaught so they they have their explicit meaning and then they have all sorts of suggestions and Gary Becker's definition is sort of it it may denote a cult but of course when people hear the word cult they think of they think of people all drinking poisoned kool-aid or they think as I suggested before about animal sacrifice and it's really sort of a slam to to use that word even even if a school of thought is like a cult in some superficial way right so they're probably actually the the defining characteristic of a cult though is blind adherence to its leaders but anyone who follows the Austrian school at more than a superficial level will realize that there's lots of infighting within the Austrian movement even prominent economists of the school today still pick fights with its founders with people find major disagreements with Ludwig von Mises and Rothbart and Hayek and that I don't think is is characteristics of many other cults where you're expelled if you question the leader like for instance Iran's inner sanctum I've heard that Rothbard himself was was excommunicated from it for questioning her yes that's that's definitely the story with Iran there many other death cults where people who have questioned the leader they were either pushed out of the cults or they were executing themselves no one is X has been executed from the Austrian school the Austrian school is a academic orientation so anyone can call themselves an Austrian and no Austrian can really do anything about it and in fact a lot of people who call themselves Austrian I personally wouldn't agree that their Austrians but they they still classify themselves as Austrian but I can't do anything about it if they want to if they feel comfortable calling themselves Austrian Who am I to say that they are arms and the contention that Austrians don't engage with the the mainstream is also problematic because Austrians do in fact engage with the mainstream quite often but it's it's often very one-sided you don't always get a fair response and you always get it's hard to get your views in the mainstream journals especially when you come from such a such a sort of different approach well it depends on how you define mainstream there are lots of Austrians publishing in mainstream journals a lot of them happen to be based on the history of thought or history of economics or more applied journals or usually lower ranked journals if you're not professional Austrian economists an academic Austrian economist you have to publisher you're gonna be fired from your job you're not gonna get hired anymore so you have to find outlets to publish your work academic outlets to publish your work and there are only two or maybe three libertarian Austrian publications out there and if a hiring committee sees you've only been publishing in two or three journals that that aren't famous within the within the profession then you're not gonna get a job so any professional economists you see who calls himself an Austrian you have to know that they've published at least some of the work in mainstream journals perhaps of medium caliber definitely but maybe some of the older guys they'll have definitely some publishing more prestigious journals as well I suppose a difference between Krugman's 30 or 40 people in the what he calls old boy networks is that you know if he was in the the international trade circuit and maybe the people in the international trade circuit and the people in the say labour economics circuit maybe they all sort of leave each other alone but Austrian economics tries to look at the whole economy and many different parts of it and has an alternative view in many areas so in a way it rubs up against all these all these different groups and all these different mainstream schools of thought in a way that they don't rub up against each other well with the other mainstream schools one distinction is that it's true that the 30 or 40 people in each little circuit do mostly interacts with themselves but because they also hang and large and mainstream environments at their home university they no other they at least no other people in their departments who are part of other circles so they are aware of their work they're aware of the work of other mainstream economists but most Austrian economists work at MIT's here ranked schools or lower so they don't even have that advantage of being in the same department as as prominent mainstream economists so they are really at a large disadvantage for being heard of let alone being read and discussed they're at a disadvantage in being heard of within the academic established was being recognized as individuals within the academic establishment let alone getting mainstream publications let alone rubbing elbows with the top guys well you heard about the Austrian school from Peter Schiff going out on on new shows and arguing with people I think relative to the number of Austrians we're very well represented in popular media in popular books we do a good job and even people at sort of doing very original Austrian research take the time to go and engage with a broader audience and even the general public yes so I think that's probably the distinctive feature of the Austrian school with respect to other mainstream economists it's that Austrians tend to be a lot more passionate about their work and a lot more dedicated to engaging a broad audience with their research when as opposed to mainstream economists there they seem to be rather comfortable staying in their academic niches and not and not trying to make digestible representations of the work for educated layman and it makes sense in a way because in order to find out about the Austrian school at all in the first place you need to be doing more than just showing up to class and majoring in economics you have to be going out of your way to read popular books and to go to the blogosphere and you have to be that student who just can't get enough economics and goes out of his way to find more and is their time and that's really how people discover the Austrian school yeah it's the Austrian school is in there but it's really hidden away you got a you got to work for it it's not something that you just stumble upon and now I'll take us to our second topic which is another essay you wrote about mathematical economics versus literary economics which literary being of course a bit of a smear what we mean when we say literary is of course economic theory that uses verbal logic to work through its reasoning so calling the Austrian school economists as literary economists is an old division that's been thrown at us since at least the time of Mises because in human action in one of the chapters on mathematical economics versus Cadillac –tx he says that the style of economics that he does has been often derided as literary economics or as he's been derided personally as a literary economist so I was I was a little bit taken aback when I saw John Cochrane the University of Chicago economists use use that term to describe other kinds of non mathematical economics on his blog I'm not sure if he's aware of the history of methodology and the divisions people have thrown around for a few decades now but it was it was a nice nice little echo of the old days but again literary economics as you said is a slurs it's intended to be an insult because when you read when you think of literature you think of Airy prose and it's something that's not well founded in facts or or reality or it's not very logically rigorous but the Austrian method is is very logically rigorous in fact it's based on the science of logic as opposed to the mainstream economy economics where they have a sound out of thin air where they assume Homo economicus where they have rational expectations that strict form which is unrealistic and then they use mathematics which is a type of logic I guess but they use it in a way that's disconnected from reality right so a a mainstream economist might say well math is a very rigorous type of logic and it brings it all out into the open mm-hm but we might contend that yes it's very rigorous and yes when you when you do the math right it's clearly correct but there's that extra step at the start where you convert your understanding of what's going on into mathematical language and you have to do a lot of simplifying and a lot of assumptions for instance the common assumption that people are making choices in in continuous terms so that they cannot take derivatives which of course we humans don't really do in fact if you you know when you first learn calculus it's kind of a head-scratcher which sort of goes to show that you're not doing it on a daily basis to make everyday decisions right so yeah that's a common charge I guess or common defense of mathematical economics that it's more rigorous than words but it doesn't matter how rigorous something is if your foundations are nothing if you have a very rigorous castle but it's made out of sand it's the first little wash of water is gonna tear it all away mathematical economics is based on poor assumptions and the details that it goes through our unrealistic so if the purpose of economics is to understand reality mathematical economics by abstracting away from reality entirely it is a poor substitute as opposed to the Austrian method of starting with realistic foundations of human of the human condition which is there's only one a priori understanding of the human condition and that is humans acts that is they make choices consciously and then from there you can understand human action in general you can interpret what people are doing now or that if they're making a conscious decision that means that they are using they're trying to achieve some sort of end using means and that opens up a an entire field of inquiry and we call that field of inquiry economics john maynard keynes had a bit of a different critique of math he was a non mathematical economist although all of his followers later became mathematical economists but he said something along the lines of when you don't use math then you can sort of set something to the back of your mind and continue with your line of reasoning and then reintroduce it later whereas if it's hidden in the bottom of Appendix thirteen in a mathematical assumption that you made you can't very well go back and and say well now now let's consider if we if we relaxed that but that's that's a very different sort of critique because it's not about keeping the logic clearance and simple and true well I'm I take a couple issues with what you said there I think first of all Keynes was a mathematician and less so an economist in fact it was trained as a mathematician who's never trained as an economist he in his later life he started writing about economic subjects but it was definitely trained as a mathematician his father was a mathematician and in his and in his work it was very mathematical and he insisted he really liked the mathematics as well now the claim that with logical economics with verbal economics you can use you can have an idea and keep it in the back of your mind and then introduce it later I think that's probably true but I think in practice it works out the opposite in that most logical verbally economist when they make assumptions and they make an economic when the trying to expound an economic theory they're very repetitive about what's supposed to be happening and what what the relations are – to reality and what we're assuming what we're not assuming as opposed to the mathematical economists who who I think they hide a lot of shenanigans in their math so you say oh let's start with these so-called realistic assumptions of this and that or whatever and then you take a derivative that has no relation to reality or you end up saying L when consumers end up our consumers are maximizing their utility by the following you know twice differentiable function which I think that's that's tricky mathematics that's those are mathematical shenanigans one thing I'm very impressed with with Austrian economists is the level of humility an Austrian will frequently say that proxy ology can tell us this it can tell us given that person's their knowledge and their their goals how they will respond to information how they will try it attempt to attain their means but it can't tell us things like numerical relations so an Austrian economist would never for instance even attempt to find a numerical relationship between inflation and unemployment that they expected to be consistent through the ages or through the decades like the Phillips curve right and I think that's because Austrians start out with methodology and they understand intrinsically what economics is supposed to accomplish and what it can accomplish and epistemology is a very major part of the training that an Austrian has to go through yeah one of the central lessons of the first 200 pages of human action is that there are no and it's a recurring theme throughout the book that there are no constants in human relations in human action and when there are no constants you can't have any math because math intrinsically depends on constants if you say X plus y Z you even if you say x and y and z are variables the strict relation X plus y is a constant relation right and if there are no constant relations it calls into question how well we can use econometrics and Mesa says this again and again that if you do a study of and you gather some data and you look at it you are only a state you are doing economic history you are looking at the particular you know the prices or the unemployment rate or whatever you're looking at at a particular time in a particular place in the past that alone does not tell you necessary truths about what's going to happen in the future or in a different time and a different place right so when you find out what the price elasticity of potatoes in the Ukraine in 1995 was that tells you only what the price elasticity of potatoes in the Ukraine in 1995 was it'll tell you nothing about what price elasticity of potatoes in 1996 was in the Ukraine or well what it was in Slovenia or Russia or Afghanistan it's it's very econometrics is based on space and time and economics as a science according to the Austrians is a it's supposed to be a general field of inquiry whereas econometrics insists upon it's not rather than insist upon it is in fact an exact and precise science but it can't be a predictive science and it can't be a science where you get general theories it's it's a historical science so you can only ascertain what happened in the past with with econometrics at a definite time in a and at a definite place but you're you said before your your introduction and what first impressed you with the Austrian school was Peter Schiff making a prediction so that prediction was not part of his economic theory is what you're saying but it was a prediction that he was able to make using his judgment and his understanding of theory Peter Schiff is is an entrepreneur and entrepreneurs by definition have have some sort of knowledge about or have some sort of expectation about the future and their expectations can be formed in menu informed by many things and Peter shifts expectations about the future were informed by his understanding of Austrian theory but just because you understand Austrian Theory does not mean that you can be a good entrepreneur because it takes more than a good theory to to be a good entrepreneur there's more you have to be aware of in fact there were many Austrian well I wouldn't say many there were definitely several Austrian economists who as latest 2007 some even 2008 were denying that there was a housing bubble and that wasn't because they had a different theory it was because they had a different interpretation of the history of the what was what was going to be relevant at the time the the relative importance of interest rates versus versus the legislation that was in place versus how entrepreneurs might react this has brought up a lot against the Austrians in that after the crisis a lot of Russians were predicting massive inflation and there's some debate about what that meant but most make most mainstream people and even a lot of Austrians will admit now that they lost that they miss predicted what was going to happen with inflation but that's only not a problem for the theory that was a problem with what what were the actual facts on the ground what ended up being relevant for for price inflation and what other counter veiling forces were at play right so the Austrian entrepreneurs and the prognosticators weren't acting as theorists when they were predicting inflation they were acting as entrepreneurs or pundits or any anything other than a theorist as a pure theorists predicting the future is is hard for everyone it turns out there also that famous graph from Christina Romer that showed what they expected unemployment to be without stimulus and with stimulus and then they got stimulus and real unemployment was much higher than even their baseline yes and their response was that oh we set the base right baseline wrong it turns out things must have been worse right so even though Christina Romer is a Keynesian and at least probably calls herself a positivist that is someone whose theory lives and dies by by the data and the evidence and is tested against statistics really was not willing to throw out their theory when they made a wrong prediction right and Walter Bloch brings this up in his Lew Rockwell piece in that the best economists even though they talk a big empirical game in their heart of hearts they know that economic theory isn't really determined by data and so-called empirical evidence economic theory is is logical and the best economist recognized this so there so that's why when was a block did some econometric analysis and found out that rent controls actually increased the availability of housing his adviser didn't praise to him for his groundbreaking he says he said he was an idiot and that he should do the calculations again right because he knew for a priori reasons or for theoretical reasons that that result could not be true and also that statistics is a noisy and uncertain thing especially when you don't have a sort of like laboratory where you're I don't know shooting particles at each other in a totally controlled environment there's a lot of noise and you're trying to tease values out of the noise but sometimes you get an absurd answer and what that means hopefully is just that the noise took over that you-you-you got a lot of noise and your answer is just wrong yeah so this goes back to the constant relations and economics business there are no constant relations in economics we can't know what is the true cause of anything just by observing it in reality so that's why we need to abstract away from reality using theoretical tools and the Austrians say the best theoretical tools start with the understanding that all humans act that is they make conscious decisions trying to use means to get certain ends and we understand in our daily lives that other people behave that way I understand if somebody you know walks out the door to the hotdog stand across the street that he doesn't you didn't do it because a strong breeze blew him out the door he did it because he was hungry and he wanted a hotdog right and just because that's a very mundane sort of obvious observation doesn't mean that it can't inform a theory of economics or a theory of human action you're right I think a lot of people have problems accepting that such a basic insight can have such resounding implications for science in a way I think people they really want to be able to to know things to that that in that they can't know they want to be able to predict the future they want to know exactly be able to fine-tune some economic policy so it's just perfect and there's not an expert out there who has that sort of prescient knowledge to to make those sort of determinations and to consider all the relevant factors in the real world what we can do is turn to our theory and with strong use of the cater asparagus qualifier say how will a effect be given that all else is held constant yeah I mean that's that's really the only way we we can learn about anything in the real world even even physicists they they have to make simplifying assumptions to understand their theories they always assume that there's no friction or no gravity or something to understand how a ball works but in the real world we don't need to have a PhD in theoretical physics to know how to throw a baseball so that another person catches it same with we don't need a full knowledge of economics to become an entrepreneur or a good stock analyst but if you do have a strong understanding of theory you can make better judgments about the future but not perfect judgments because again there are all sorts of random things in the world that could affect anything about anything whether it be a rock falling to the ground or whether our stock is gonna blow up or crash and in in some ways the sign of a good theory or Austrian theory at least is that it can be helpful to entrepreneurs it it does illustrate or at least it demonstrates possibilities to entrepreneurs and identifies trade-offs that they might need to make where whereas a neoclassical world where all the every business returns a profit of zero unless it has unless it's a monopoly is really not helpful for someone in the real world trying to run its business and earn a profit right there's there's that and also that I think it's the more correct theory of reality so if you have a better understanding of reality then you can make better judgments about reality and make better decisions based on those judgments so if you if you follow a theory that says there's no way that there's going to be a an economic crash then you're not going to prepare for an economic crash and when an economic crash does occur you're gonna be caught off guard you're gonna lose a lot of money but if you were someone like Peter Schiff and you had a theory that could help inform you about the possibility of crashes what causes crashes how to identify crashes then you can prepare yourself and be ready for when it comes and this is all a rather heretical view within the economic mainstream do you see it becoming more accepted in the future I think it's already starting to become more accepted there was a paper by the IMF economists I forget his name Raj Raj something a couple of years ago that was that didn't mention it by name but but it did espouse a theory of the business cycle that was essentially Austrian at its core and if I'm any indication popular Austrians like Peter Schiff like Ron Paul like Tom woods and others have a very profound influence on a younger generation so in the next five to ten years as we see more young people get their PhDs and into the academia you're going to see a larger influx of Austrian or Austrian influenced academic literature getting published by mainstream journals so the the John Cochran's and the Paul Krugman's of the world they're not going to suddenly become Austrians but they're gonna read older and young people who are just now hearing about the Austrian school and reading about it they're going to get PhDs and eventually hopefully get tenured positions and 20 30 years down the road maybe we'll have made some serious ground we can only hope I agree with you and goes back to the old adage that science progresses one funeral at a time I think what what's happening right now you're gonna see the real fruits of it bear in about 20 or 30 years as you said where can my listeners find you online best places the mrs. dot see a blog page I'm gonna be updating there about a couple times a week now so missus dossier and then there's a little tab up there for blocks Thank You ash Thank You Garrett if you enjoyed this episode of economics detective radio you can head over to economics detective comm for additional content and links the music for this podcast was created by Cassandra McLeod who you can find at soundcloud.com under the stage name minaret that's mi na space ret

Probability Part 1 || Basic Definitions || GATE Lectures for Engineering mathematics ||



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otherwise this lecture of engineering mathematics on topic basic definition probability part one is brought to you by lecture social calm so guys let's start the lecture and first of all let's see what are the topics that we will cover in this lecture we will see some basic definition about the sample space then we will see definition of a random experiment we will see about event events and the types of events and we will also see definition of probability we will see the definition of conditional probability so let's start is sample space sample space it is that color sum of all possible outcomes of an experiment okay let's suppose our sample statement so all possible outcome of an experiment let's suppose an example like like if our experiment is like talking of tossing a coin ticking for example is like tossing a coin one time one time let's apply our experiment is this so our sample space will be what all possible outcome so if you are causing at point sorry if you are tossing a coin then our all possible outcome will be I just to be head or it will be n okay similarly clear our if you are experimented like a rolling die taken one time then then our temperatures will be what all possible outcomes means if you are rolling your dice then our outcome will be one it may be one it may be 2 3 4 5 & 6 okay so these are the sample sketches four digits pin linked company is 1 2 3 4 5 6 is the sample space for rolling a dice one time experiment and this head and 10 is the sample size for tossing a coin 1a okay next embellishment represents the all possible outcome of any experiment okay let's see the random experiment random experiments are that Perriman whose outcomes are unpredictable okay those outcomes up until unpredictable is known as random experiment like a topping and unbiased called coin is random experiment it could be anything it could be head out 10 okay we don't know so it takes unpredictable so we will call it as a random experiment and the draw card from a pack of 20 52 cards okay so it is all equals out the random experiment now let's see the definition of even even the the outcome of any experiment is known as sealing or mathematically we can say that event is the subset of of sample sizes like here this is sample space for tossing a coin in if you are tossing a coin when sample if aliens had intent means it could be head it could be 10 so this is the outcome these outcomes are known as even this outcomes are known as even lighter we can say getting head while tossing a coin is a event okay similarly in rolling a die case we can say that getting getting any number like getting one getting to getting sick is a like a event okay and we can set up getting even number while we are rolling a dice it is also a event getting odd number while toss when rolling a dice it is also available basically event if there are some outcomes someone comes or one outcome the shelf Olympus okay it is a subset of sampling okay so what event is there like getting head while tossing a coin one time okay or you can say getting even number while rolling dice getting odd number ending so now let us see the types of events what let's first of all let's see what is complimentary built if you are telling le any when see any winter in event e then we can call it complimentary of this event like a PC or you can write it like T bar okay so complementary of any event e will be known as EC or a bottom and it will contain all the outcomes all the outcomes of sample space that is not in E pockets it contains all the outcome of sample space that is not in P means let's suppose our sample space for rolling a die let's let's suppose our experiment is like a rolling work type so our temperatures will be 1 2 3 4 5 6 ok let us suppose our event is like getting a odd number okay so we can write our event is like getting an odd number one we can write it like a 1 3 & 5 okay this will be all event it is a subset of this our sample space this is our event getting odd number so or complimentary wait will be easy or we can create T bar so it will contain all the all the outcomes of them sample space that is not in this our event so here in this event we have 1 3 type so let's remove this event these outcomes and I write the rest of the outcomes of sample space so we can write is here 2 4 & 6 so complimentary event of getting odd number will be getting even number okay this will be the complimentary event of this pivot okay this it contains all the outcomes of sample space that is not in our many 8 okay now honestly the equally likely events to even e and F are said to be equally likely if their probabilities are equal instability of E is equal to probability of n so probability of F then we can call it the equally likely even okay we haven't seen the cement definition of probability we will see it later let's start for for now we can understand this if their probability will be equal then it is known as equally likely event okay so let's move further mutually X is equal which will be exclusive and this is important so any two events E and F are said to be mutually exclusive if attacking initially excusing if there is no common outcome between that no common outcome in E and F okay it means we can write it as the e intersection F of F is different the intersection type means they each represent the common factor in common or contain T and F we can write it e intersection F is equal to 5 years I represent the null set it will bring the null set and e intersection F is equal to nothing when it's null set okay mutually excuse me there is no common outcome in PL death okay knowledge to the connective exhaustive even – even e and f are said to be correct – at the vacuum if this both even this both even gives complete self Alice's complete sample space it means e Union F it represents a Union F it means all terms of D as well as outcomes of F both combined together then it will give us sample if it s we are representing sample space is F in later okay so collective exhaustive infant this gift is both even combined together then it will give complete sample itself then this event will be no less collect to exhaustive events okay now let's see the independent even this is not very important to events E and F are said to be independent if if probability of T doesn't affect doesn't affect F has happened or not if s happened or not and vice versa okay means the probability of happening of e does not affect the probability of happening of F and probability of happening of F doesn't affect the probability of happening of e then this two events are known as independent even okay image very explicit events we can say that generally people get confused with mutually exclusive and the independent events okay so let me tell you one main difference is mutually exclusive will be like it will be on same sample space okay but in independent events there will be temple essence will be different okay we will see it by Venn diagram then you will get more idea okay we will see it by Venn diagram let's do some good diagram then we will get more a deal let's let's quickly make a check this okay let's suppose this list empty this residence F okay so in this represent this rectangle is representing our whole temple say this rectangle is representing whole sample space s this circle is representing the event T in this semicircle little painting event if so this reason chicken this reason will be our e Union F okay now let's see again look about this rectangle is representing us and perspective this representing e the circle in this circle is let us suppose representing event F so this reason is your e intersection F okay let's suppose this is our temperature s this representing over this entire district anything about e between side reason is representing even T so this outside reason will be our season T bar or we can say eg use complementary complementary of E and the let's suppose this is over self-punishment and there are two events like E and F okay there are two events like T and F and this is like this means there is no there is no any common outcome okay here we can see this towards digitally E and this is totally F and there is no commonly there is no any common area between these so this type of event we call it mutually experience which will virtually explain you mutually exclusive email okay sample space is same but there is no common outcome in E and F okay so this type of event is known as mutually actually queue and let's suppose about if we have like this this is one example is that this one in the year it is B this is another temple is he – here term belief is is – and it is f so we have temple x-rays are different so and the probability of happening of e does not affecting the probability of happening of F okay so this type of events is known as independent event in the pendant even okay so independence difference is an example of this is like a if you are calling if you're tossing a coin two times so our first time result will not affect our second time today okay so like a tossing a coin two times is like they're both two times are like independent from each other so it is like that independent events okay now let's see the definition of probability the probability of an event is defined as the ratio of the number of grades in event can happen to the number of ways sample space can happen okay I think it's clear like left of our symbol space like one two three four five six okay the number of times temperature else can happen like number of ns me like here click OK this is sample space for rolling a die so probability is nothing but the ratio of number of events any vent can happen divided by the number of sample it can happen okay so like our sample stage is like one two three four five six we have if you are rolling advice then number of elements we have we can say number of outcome is like six years okay one two three four five six six let's suppose our event is like getting a odd number getting odd number is nice even so we will get or something like 1 3 5 okay so number of outcomes will be a number of ways it can happen a physical number of ways and even to calc Apple so Roman number of which means 1 3 5 so there are three ways that this event will happen okay we could give 1 3 5 then the tunes can happen to number of ways that even cannot happen like 3 and the symbolic that can happen like a six ok so here probability will be three by six or we can say 1 by 2 okay so this is basic definition of probability now let us see the conditional probability so conditional probability of any event e ok let's suppose we have any event e well such that its probability is some greater than 0 let's suppose our event is like that okay then let's suppose there is another event F that suppose there is another event F then probability of happening f when he is already happened probability of happening F when it already happens this is known as the conditional probability of F and it will be equal to your probability of F intersection e divided by probability of e okay this is the conditional probability of F when E has already happened he has already happened okay we can write it like a like here the probability of e F intersection T will be like number of times E and F happening together / simple offices and here also number of days he can happen / and policy with a numerator and denominator we both have divided by number of sample of the space number of number of this sample space can happen so that can be divided so we can write it like number of outcomes a in the X intercepts and t / number of outcomes in E okay we can write it like this also so this represent the conditional probability of F when t has already happened okay then thank you like like like my video clip and the subscribe in our channel and if you have any doubt you can ask me in my comment box or email effect lecture for Chrome and there you can download my handwritten note from my website lecture for silicon thank you

Math 2B. Calculus. Lecture 01.



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good afternoon and welcome to the fall quarter this is nice to be calculus and my name is Natalia comer OVA little professor here at the math department and I'll be teaching you this subject first I want to go over the basics of this class and tell you a little bit about how that's been organized first of all I have created a website with all the information already today I'm still spell everything out for you so first of all the textbook so this is probably the most confusing part of the whole thing so it's called calculus early transcendentals so it's 7th edition of calculus by Stewart so the most important part is their early transcendentals there is a 7th edition Edition that does not contain this words look at your book if it doesn't say that that's the wrong book now there's various shapes and forms of this book for instance what I have only contains single variable calculus so that good for 2a and 2b and so that'll be fine for this lab there is also a bigger book that has you know parts that contain two other classes but if you're planning to take to D or to e you should get the full book the big one there is also an electronic book okay and so there are many questions that people ask me over email is it okay to use a second hand copy and the answer is yes it's okay if you bought your book from a friend if it's used and your electronic stuff that we work if you cannot create an account that's okay all you need from this book is the chapter material and the homework problems okay so I do not require you to buy a new copy of course a secondhand copy is okay any questions about the book okay so now exams we're going to have two midterms on October 18th and November 8th and we're going to have a common final exam question no absolutely not so common final we just helped on Saturday December 7 1 3 so I'll tell you a little bit about the common final but first I'd note that according to the policy of the mathematics department you're not allowed to use books notes or calculators during any tests so it's a closed book test no cell phone no calculators are allowed now what's the common final if you've taken to a you know what that is it means that you come here on Saturday okay and it's that it's a common file of a final it's held across all the sections of this class so there are unique requirements for everybody you have to produce your valid UCI ID card for the midterms and the final you have to make sure that you have an idea and you also have to make sure that you are recognizable on that picture so very often students produce something that looks like this it's just the circle instead of a face so make sure that you get your ID replaced such as your pictures so another thing about the common final is that if you could not make it you should let us know early on and you don't let let me know you let the Secretary's at the math department know you have to fill out the form that this contained online you have to follow the link from the website there is a special form that the standard form that we will fill out and there should be no problems that they will arrange for franca but of course you have to have a valid reason questions about the final listen so the book no nothing like this I will provide a paper copy of the exam and all you need is a pencil and an eraser yes no calculator are we allowed to bring like a simple no no but we always make sure that you can do all the math it will open your head there will be nothing norms for better will a calculator be needed for the class no no you can use this will you do Homer the force but in class no okay so now some other silence that will have apart from the midterm to the final we will have homework okay so the way it works in this class is homework is optional which means that it's not graded nonetheless this is proved the most essential part of this class because everything that you're tested on is based on this option home okay so the own list of homework problems is provided on the website of the class and it goes by section number so for instance section six point one and it gives you a list of four word problems so you sit down and do as many as you can after we have covered the material if you're fine with all the horrible for each section you need to get an ID laughs okay the if you're fine with most of them you'll get a name and so so this is your way to study do the home report no but you will test them directly but we will have quizzes quizzes are held once a week at the discussion session a session on Thursdays and the quizzes are completely based on the homework assignments for the previously ok so if you've done your homework you'll know how to do your whose problem it's either just over problem taken from the list or something that is very very close to so in order to prepare for the quiz you have to do the homework and the quizzes are graded now another part of assignments is live work when you raise your hand if you know whatever work is do you know what it is oh I see so some you know together resist so this is an online homework assignment so on the website for the task I created a link that takes you to the web works at home page this page is not actively as of now they'll activate it in about two weeks when the first assignment is posted so the first assignment will be posted on the 10th of October okay so until then we don't have to worry about it so you go there you log in and you do your problems there will be eight web work assignments during the whole corner there will be every week posted on Thursday and do the next Friday the first homework assignment is the beginning of the class in their curative such that later on where were problems could test your knowledge drug you know quite long ago so everything that you have studied up to that date can be tested each assignment has a varying number of problems and I posted the full schedule of all the web work assignments that is already known now so you will know the due dates and the dates when these things are posted you have an extension for Thanksgiving okay for Thanksgiving week you have a little bit more time to finish so there are a few things that you need to know about web work so among the class files I posted a PDF file that tells you about web work so one you think that they told us about this year is that somehow it doesn't work very well from so it works very well from campus when you're on campus if you're away from campus you have to use VPN to connect and the instructions of how to set it up are given on the website of the class in their various scores associated as well good works because sometimes you may type something in the wrong pot and the things that you made a mistake all these things please read what I posted it gives you a lot of information and so if you experience technical problems with refworks please do not email me because I will not be able to help you I can only confuse you if you ask me technical questions about you know the website setup and stuff email them there is a email address provided on the website of web work and they'll be able to resolve your technical issues if you have a problem with your mathematics then I'm your guy okay email me and or come to the office hours and able to help you with any math folks okay but not with technical problems and website or business today questions okay great consists of 40% final 20% of each of the two midterms and percent and 10 percent quizzes so the lowest quiz and the lowest WebWork are dropped because of this policy there is no possibility to make make up for losses okay there is one seat here if you need to sit down and there is also one over here there are also two seats over there in the middle okay so if you miss the quiz or if you miss a WebWork don't worry about it because you have a chance to miss my question so it's the web where it's like in online coverage or is it like an online site it's like an online homework assignment I think you have an infinite number of attempts oh yeah you'll be doing it at home but not during quizzes so quizzes are like exams for questions okay now I think the last thing is this so early transcendentals so what does it mean it means that if you took calculus through a prior to fall 2012 you will have some catching up to do so the mathematics department has changed the syllabus we used to teach things like logarithm for park sign for e to the X we used to teach that into B now these things are taught in 2a and you're supposed to know them you're supposed to know these things and you're supposed to know their derivatives and their calculus okay you have to know what they look like you should be able to put them so how many of you have not seen those before okay very good sort that's good however if you want to refresh your memory about these complicated functions there is some online video material you can watch some videos and refresh your memory about these things such that you are well prepared for what we have here instead of the same will be studying sequences and series which are much more fun listen No so there is no current standard great so so the finals are graded across the whole you know all the sections then if they think that it was too hard or too easy they're going to uniformly you know add some pointers multiplied by a factor so everybody gets the same treatment and then after that there's no curving also there is no extra credit there's nothing you can do so everything is very straight forward in this class all the information is in the homework if you have questions if you don't understand something come to my office hours which I will announce next week I don't know yet when my office hours are and then we'll be fine averages the final so if they think that the common final was too hard they're going to kind of boost that's great just for the final yeah you don't go to this class no and they're not supposed to what's the website I think you can find it under booking there's a link also you can go to my home page and very live from there but it's easier to go from the triple each month just to make sure is that the web word is doing for the following correctly yes it's it's the folks at on this Thursday it's a newer the Friday next week so it's totally week yes so you have a week on one day to finish it just okay and you can always ask me any questions during the class just raise your hand okay so now we do some review of some essential stuff that you learn into a people don't see something please let me know so I avoid some parts of the whiteboard okay so today we cover section 4.9 anti-derivative okay it's ready so let's suppose that the derivative of function capital f is used by lowercase F to hold values of X then F capital is called the antiderivative okay so for example let's suppose that F is equal to x squared what's the antiderivative function there are two opposite operations you could take is riveted and you can go back and say the antiderivative question yes so this happens to be one of the anti derivatives of this function in fact how many are there infinite limits let's say so by adding your constant C we create an infinite family of functions each of which is a 90 derivative of this how can you check so this is a new derivative to check we say what is f prime of X we have to differentiate so I have 3x squared 4 3 0 is X square check because it coincides with my original function question you cannot see this is very bad so are you feeling that can you see above this line yes okay so we have a theorem if out yes an antiderivative then F plus C is the most general antiderivative and here C is the constant yeah so let's practice and do some examples so let's suppose that f of X cosine yeah fine yeah okay so basically we're looking for a function whose derivative is equal to cosine another example is given to find X to the n not equal to minus 1 so may I ask you please do not talk during this class if you have questions ask nope so we have a rule to evaluate the antiderivative of this it's called the power rule so it tells up tells us that the antiderivative it's given by this function and in fact this is the rule that we use here to calculate the antiderivative in our very first example so I have a question why is it that we have to require that M is not equal to maximum what happens to this for global and equal zero yes we divide by zero so this formula is obvious not applicable something else goes on goes on there so that's my next example this very special case when n is equal to minus 1 is 1 over X what's the antiderivative natural log of X in fact it's like this natural log of the absolute value of X plus C and we have to say that this formula after the holds on any interval that does not contain zero so if I give you if I give you an interval from 1 to quiet first on this interval any such function with the constant C will serve as an antiderivative of the law if I want to write down the most general antiderivative on the whole real line it's something there are slightly more complicated so f of X is f of X let's see for positive values of X in a slow rhythm minus X I'm sorry I know the c1 okay plus c2 when F is negative in the NT derivative is not defined for x equals 0 because the original function is not defined for x equals 0 no they can be equal but in general they don't have to be so this is the most general form of the antiderivative of the function 1 over X so this function can have a discontinuity it goes like log X log minus X but here when you meet when you hit the x equals 0 we can have a different constant so it's in general and discontinuous functions very good we'll also have something like this for negative values of and so so let's go he is X to the minus 4 all right so now we have a very similar situation so are using the power rule okay my power is minus 4 so I have to X – 4 + 1 + 1 – 4 + 1 X – 3 / – 3 so the most general antiderivative is given by the following discontinuous function so the anti-derivative again is not defined for x equals zero because the function itself is not defined there and it may experience a discontinuity as you cross 0 because we are allowed to take different constants for negative and positive values of x so what we need to know for this class is the list of anti derivatives it's best to know those back hard so we have a table in the textbook that goes like this we have a function and we have a particular so what are the most common functions that you you should know by heart [Applause] rule if you have a function f any function f x constant C so the anti derivative also gets multiplied similarly with the summation the antiderivative of a sum is the sum of two anti derivatives now particular examples of functions here I do this for completeness so continue to stable here let's do cosine s this time this is sine X negative cosine secant square is me tangent secant tangent is secant these two follow from the definition of the derivative of secant and tangent we know the derivative of this is equal to this then the antiderivative of this business this table works both ways to go from here to here you have to take a derivative to go from here to here you have to take the antiderivative what else we have something that is associated with arcsine the impact engines so this is the best one why is it the best on in singles itself right the easiest hunter ever they follow this is equal to its own derivative posted equal to its own engineer first so let's practice so a simple common problem that we encounter is a look at this function we go to the table of course we don't find that function in the table however if we simplify this function will find its components in the table right so the first thing here is to simplify and then we'll be able to use the table so I'm going to say that this is 5 must cross X and here I divide through by X sine of 3x plus X minus 1/2 okay now each of the components here and be found in the table and I'm going to use the table soul-g problem Sergey so the derivative of G is this therefore I have to find the derivative antiderivative so it's five plus nine x how do they get the first term I we will learn how to integrate in this class but we have to find an antiderivative of a constant so where can you find answers for example here if N equals zero that's a constant 1 so n equals 0 gives me F to the power 1 divided by 1 so that's F so the antiderivative of 1 is X and here this tells me that multiplication by constant just carries through so this number 5 appears in front of the antiderivative ok sign is the anti derivative cosine just grows close right from the table this one is easy again that's got the power function so it's 3x squared over 2 plus this one is also easy because we use the same rule power rule only now we don't have an integer power the power is equal to minus 1/2 so it has X 1/2 divided by 1/2 and then don't forget plus 6 this usually cost one point then you test ok question sir do we have to simplify the 1/2 to radically um well it's easy like this probably I wouldn't take a point off for this but different graders are different okay then next question is a little more sophisticated we can talk about differential equation in the cab with the huge topic and their whole courses thought on this but I'll just show you this so the problems leg is fine F if F prime is equal to X to the 6 then F of 1 is equal to 3 so I need to find the function f given this information two pieces of information the first piece of information pertains to its derivative right and the second one tells me what the value of the function f is at one point x equals to 1 so that's what I need to find so from this equation I can find F by looking at the most general antiderivative so general antiderivative I take the antiderivative of X to the sixth which is X to the 7 over 7 plus C so I've found a whole bunch of function f they all differ by this constant and that is why I am given the second condition this condition will help eliminate most of these and zero in on the relevant use F of 1 equals 3 how do you know but what is it exactly so I go f 1 is 1/2 the same / 7 Nazi and that's supposed to be equal to 3 so I can say that 1 7 plus C equals 3/4 C equals 3 minus 1 which is 20 therefore my function f not the most general one but the actual function f that solves both of these ok that's given right X to the seventh over 7 now out of all of these functions identified the one that satisfies not only the first equation but the second one to question okay so now will refresh your memory putting cards to graphing antiderivative and we'll talk about the notion of velocity let's suppose that the function f is given graphically something like this so it starts up here goes negative this and leave this okay something like this let us sketch the graph of the antiderivative so no formulas are given and I want to draw F capital so how do I do this in principle this is the derivative of this function now remember what is an integer what is the derivative the derivative is this rate of change it's the rate of change it's the rate at which the function changes if we think of the independent variable as time if the derivative is how quickly that function changes okay it tells me the slope or the rate of change and the rate of change can be interpreted as a velocity so let's suppose that this is a velocity of motion and as you can see as time goes by changes sometimes will go faster then will slow down okay at this point the velocity is equal to zero and here it becomes negative which means that will go backward right then after this point we again turn we start going forward so by using this information I'm going to draw the position given the velocity okay I have to recreate the position given the velocity I start with some arbitrary point okay let's suppose that we know that we start at one and now solo the velocity here is positive which means that I'm going forward I go forward means that my position the coordinate of my position increases so for a while between time equals zero and time it was one I go forward slower and slower and slower but I move forward so I move forward this is my positive direction my accordion increased and this point has stopped at this point my derivative is equal to zero which means that I'm going to have a maximum here and now my velocity becomes negative at this point I start going backward that's exactly what I'm drawing here I start going backward faster and faster faster at point two i if my diversity is the fastest negative and then it become slower and slower slower so somehow at point two we get something like this and I stop at point three because my blocks taken is zero after three I continue to go forward so my corner increases and then the velocity decreases so somehow I level off I start doing slower slower slower than eventually almost go wait stop questions so you should be able to take the graph of a function and draw its antiderivative but I want you to think about velocity okay I want you to think if this is positive this increases if this is negative this decreases if this is zero it means that I experience either a maximum or a minimum I don't change at the country first very good so now in the last problem I think it's the hardest of all we will talk about not form the velocity but also acceleration because they're both connected to derivatives anti derivative so the problem is like this suppose that the acceleration of a particle is given by this function so here is my vocabulary and yes disposition okay this is the communications and you know that the velocity is the derivative of the position information is the derivative of the bellows do you know this yes okay so what is given is the acceleration and also some information about the position at the beginning the position is – in the velocity and the beginning is equal to minus one find the position as a function of time given this information so I have to do it step by step I start by saying that acceleration is the derivative of the velocity so V prime okay is T plus one if I know the derivative of the velocity I can find the velocity by taking the antiderivative we of T is found by calculating the antiderivative on this function which is really d squared over 2 plus T plus C so given the acceleration this is a a is the same as V Prime and it's given by T plus 1 so if I know V prime I know V given by this unfortunately I have this unknown constant here but I can fix that by saying that look V of 0 is equal to minus 1 that tells me find the appropriate C so we use V of 0 is minus 1 so what is near 0 0 over 2 plus 0 plus C and it's supposed to be equal to minus 1 therefore C is minus 1 therefore is G squared over 2 plus C minus 1 so I completed the first step using the information about the acceleration I found the most and I also use this condition about the initial velocity that helped me identify the particular constant C okay so that's step number one step number two I know the velocity now but I need to know the position so step number 2 I go here the velocity is the derivative of the position so B is the same as s Prime and that's given by this formula that we just derived just weird or 2 plus t minus 1 from this I can find the position by taking the antiderivative so if I know the derivative of s I can find the answer again I have an unknown concept called a so what's this constant that's the last step I'm going to use this piece of information the initial position s of 0 is given by 0 6 is to post be equal to 2 which means that a equals 2 so I can write down the answer s of T is given by G cubed over 6 plus T squared over 2 minus T plus 2 so this is a formula that defines the position of the particle as a function of time question okay thank you very much

Understand Calculus in 10 Minutes



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TabletClass Math learn the basics of calculus quickly. This video is designed to introduce calculus concepts for all math students and make the topic easy to understand.

okay understanding calculus in ten minutes okay so that's the topic of this video now notice the video is insane or my topic isn't our title is learn calculus in 10 minutes or master calculus or totally you know understand it completely just basically understanding calculus ten minutes like understanding what it is you know that's what the whole point of this and really my kind of main goal here is to kind of maybe demystify what calculus is and lessen the intimidation factor there's a lot of students out there that's all I could never do calculus it's too complicated etc you know there's a lot to learn in this topic but hopefully you'll kind of see the big picture here but let me just before we get into a real fast let's just talk about where you would take calculus as a math student so generally in high school okay so in high school for most students you start off with Algebra one then you go to geometry you go to algebra two and then you go to precalculus in your last year of high school now some students accelerate this and they'll actually end up in calculus in their last year now when you go to college I would say maybe about 70% of the majors but a degrees in college right so at the university level many of them may be well definitely over half we're going to require you to at least have one semester of calculus so we don't take it in high school you'll you'll see a little bit of it in college count if you're some sort of science or technical major or finance whatnot you'll probably take even more of it so it's probably going to be and a lot of your futures if you're going going the college route okay so that's this basically where where it lies in the spectrum of learning all right calculus in ten minutes let's get right to it so calculus basically helps us figure out kind of two problems to kind of things in the math I'm going to start with one that's kind of easy that would be like the area and volume problem okay so I'm just going to sketch out here real quick so this is like a rectangle right so if I said find the area of this rectangle you we have a formula for this if you recall so it would be like length times width right so the area of this rectangle is going to be the length times width no problem okay so we know this because we are given a formula let's do a circle all right what if we have a circle and this is on a perfect circle obviously but if I said find the area of this circle hopefully some of you out there or maybe most of you remember there's a particular formula for this okay it's PI R squared R being the radius which is like from the center out this distance M pi is a number of 3.14 approximately so I can use this formula to figure out the area of this circle and for this rectangle I can use this formula now let's just do one more example what if I had a triangle okay so again if I'm learning to know the area this would be one-half base times height right so this would be the base of the triangle this would be the height okay so formulas help us find the area and by the way this goes for volume as well so these formulas come in handy with these kind of basic figures like triangles and you know circles and etc so let's take a look at word calculus helps us out okay where it's the tremendous value and by the way I can't overstate the importance of calculus and mathematics engineering science I mean it's it's huge really is as powerful as you you know may think it is probably more so all right so what have we had some crazy figure like this for example I'm just trying to draw something okay so what if I said find the area of this particular object or this figure well probably you're going to be looking for a formula right you're going to be okay yeah I could find the area of this just give me the formula well guess what there is no formula for something like this this is very challenging okay so calculus helps us figure out the area of crazy-looking figures like this okay and volume they don't have to be as you know you know abstract as this but but this is what calculus can do for us this is the power of it because without calculus we would have to go through and just kind of like maybe try to estimate the area of this particular thing but calculus can give us the exact area okay now let's suppose that I took this this figure and it's let's kind of put the SEC on a on axis okay and I circled it around spun it around so maybe it's kind of doing this business now I don't know what it would exactly look like but it might look like a an object maybe something like this kind of maybe sketching this out all right and then it looked like it would have some sort of hole in the middle right maybe like like this to have to deal very difficult to kind of maybe a magic but you can see if you did have this hmm it's kind of filled this in a little bit right and then you just kind of rotated this around you would create some sort of weird looking figure like this so this figure you would be looking for the volume right like how much water would be able to fill this particular you know let's say it's oh maybe it's a vase or something you could you could put in okay once again calculus will tell us exactly how to do this and you may think wow this is it once you must have to go through all kinds of crazy advanced calculations in order to do that and surprisingly it's really not as as you think okay once you understand the steps you can do it but you can see the power of being able to figure this this stuff out let's go ahead and take a quick look real basic example on how we find area and volume in calculus alright so maybe some of you out there already taking algebra and whatnot I'm just making a real quick XY axis alright let's kind of maybe a little figure like so okay alright so let's suppose I wanted to find the area from here to here underneath this curve alright let's save this area right here all right let's say this part underneath this curve I'm wanting to find area so that that shape if you look at it there it kind of curves on top right it goes down so you would be asked to find the area of this object okay once again you're looking for formula there is no formula that exists now in calculus or in mathematics in general okay what we have when we have curves and things we have little things called functions all right then basically just describe the curve so this curve could be described by this x squared okay you don't really need to know that right now because that's what the purpose of this video but this this little rule right here just described this curve this is the rule that tells us that this is the shape of this is described at this particular rule okay we call these things functions so as long as you have this this function or the description of what of this curve okay then we can use calculus to find the area so let's say on our little graph this is it's a 1 this is 2 let's say this is 3 4 & 5 ok so what would the way we would do this is we would go ahead and and have this little crazy symbol okay might seem as good for it's called the elongated s but it's basically means the Sun okay and I want to get too far into it because that's not the purpose of this video but we're going to say is a hey we want to find the area underneath this particular curve and and I want you to start from two and go to five all right so that's how we write that okay then we put a little tiny little DX here that will be kind of like in my part to this video here I'm gonna get to that in a second okay but but this is the set up all right to find the area underneath this curve okay you're saying well yeah that looks pretty complicated that's fine but really in the steps to actually find the area are not complex at all what we do in calculus once we write this out this long thing you can kind of think of this long symbol this and elongated s is fine find the area like find me the area underneath this particular curve all right that's all it means this means this is from the left from the left side of the curve and this is to the right side not difficult right this is the description of the curve once again if we're given it not difficult okay so what we do now okay is is a little rule that says take this number this exponent in this power we just add one to it okay so we're going to keep that same variable add one so two plus one is what three pretty simple then whatever that result is we're going to divide by that number okay so all we're going to do is take this right here and we're going to use this to find the area the way we do that is the following okay we're going to subtract we're going to use this thing all right and we're going to subtract the following okay we're going to plug in five over here you can kind of start from the bigger side so we'll plug in five cubed right from our right side and I'm not going to get too far into it and we're going to plug in from the left side like so and when we do this little calculation okay we're going to actually get the area so it's really not difficult in terms of the you know mathematics involved I mean if you take in basic maybe pre-algebra you know middle school math you could figure this out right I would just say I'm telling you the steps now what gets calculus what makes calculus more complicated for people is that these curved descriptions these little curves than these functions they you can get more complicated okay so when these get more complicated you have to learn more rules but it's really matter just learning the rules to to get to this part okay so that's the big deal this is like part one of calculus okay now calculus like I said there's like two big proms it solves for us so that's the first problem area in volume then we erase this and then we'll get into the second cool problem all right right so the second thing calculus does for us and let me go ahead and draw a little XY plot here is calculus helps us determine how steep okay something is what its slope is alright and what do I mean by this um let's suppose from here to here I wanted to know generally how steep this line is going from these two points so I could say well it's kind of going this direction now I'll get into why this is important here in a second we want to know that the direction of curves okay we want to be able to get their steepness because they're always changing if you study this curve this curve is always changing this could be for example just imagine this can be anything you want it to be it could be population growth right over time all right let's think of it that way this could be a stock that's going up in time this can be maybe the the rate of maybe cancer in a particular city that's going up a time so when these when we have what we call like rates of change and things are constantly kind of maybe growing or they're fluctuating we want to be able to estimate between between time periods or between blocks of the curve it's the steepness okay because this gives us an indication of where things are going you can tell here the steepness is different here versus where let's say at this point in this particular graph okay so the steepness is always changing alright we call this we call this steepness I kind of like to use the word as I'm using step but it's really a steepness okay there my word is but actually the technical word is slope all right that's probably better who are free anyways so it's the slope of the curve like hey now where is it kind of so what's its actual slope now you can figure out the slope if you have two points that are on the curve it's pretty easy okay because effectively what you're going to do is just determine like this these two points I can just figure out the rise and that's just this amount right here over the run okay so the slope is defined by the rise over the run so if I can get those two measurements it's no big deal I can just figure it out at the end there there I go but here's here's where calculus really becomes powerful okay what happens when we want to know the slope of a curve and I mean just I'm going to draw a new curve here to make this a bit more pronounced okay maybe something like this okay so we can see that the slope here is kind of going it's kind of doing this and then it's kind of like going this way then it's kind of going this way and then it's going down like so right so the slope of this curve is constantly changing now what if I want to know the exact slope of this curve at this point right here okay that point well the by definition the slope we need the rise and run we actually need two points like here I can kind of estimate it right if I take two points it's kind of going in this direction okay kind of something like that but if you notice I put another point right here it's kind of going like this so you know I'm getting different estimates but I don't want to estimate I want to know the exact slope right on one point of a curve okay and you can think of this as the exact rate of change in one precise moment all right so let's think of this as time okay and maybe this is population growth I want to know the exact rate of population at this exact time maybe it's a May eighth of what every year okay what a particular time you want to know exactly you don't want to know an estimate kind of goes back to our area blind palm you can get estimates but not the exact answer well calculus helps us determine the exact exact answer because in calculus we can determine the slope now once again what we mean is the function okay let's say and this for those you're out there this is not an actual the actual function to this curve I'm just using this for for simplicity stakes but let's say I had this function described by this rule okay now remember we use these function descriptions as when we're trying to find an area and volume in calculus but here we have something called a derivative all right and the derivative basically is it's a rule okay that allows us to find the slope at any point if any exact point along the curve and that's that symbol looks like this it's also looks like so all right this is another symbol in calculus DX over dy and there's even another one F prime etcetera but these crazy symbols are the derivatives so let's take a look how I would find the derivative of a function of 2x squared plus 2x plus 1 alright so far what we call we call this actually the first derivative so all we do is we multiply this exponent 2 times this coefficient this number so 2 times 2 is what for we keep the X and then we subtract 1 from this 2 so that's just X to the first or just X now we go to the next guy we do the same thing so that's X here there's actually a little one we don't write it but there's a 1 up there so 1 times 2 is 2 and then X here to the zero so that's going to be just X to the 0 power is just 1 so this is our first derivative ok of this particular function this curve description and this is a rule to tell us the slope of the curve anywhere along the curve ok but you can see here the what I just did the mechanics of finding this are really quite easy once again it's a bunch of rules so I can use this I'm not going to get into this now but I can use this here this particular rule to easily find the precise slope at this moment of time ok and this helps us solve tremendous problems in mathematics ok here let me um let's let's kind of maybe kind of first of all to be kind of clean this graph up let's say you wanted to know that's how we're doing some sort of scientific study actually we do the better curve and let's let's suppose this is maybe some sort of medicine or pharmaceutical drug or you're testing okay and you want to know like hey wow this is where it's decreasing you know the symptoms of you know let's say cancer you know or reducing you know blab bad blood cells in your body or whatnot you know here you know you're testing you're testing your test and you want to know wow this curve is telling me right in this area that this particular say dosage okay or or combination of what you're using is what works the best and minimizes the issue so you want to know exactly what this point okay so you would want by finding the slope of the curve right here this is what allows us to answer these things precisely okay I call like maximum and minimum problems so anyway I'm sure this video went over ten minutes but hopefully you got something out of it right this is the big deal of calculus it is a big deal it's a huge deal okay but it's not it's not impossible to learn even if you don't have a math background where you're like super strong in math and you struggle math you can get through calculus all right you can definitely get through calculus but it does require you to study a lot of rules and you know the best way to approach it though is to understand the value of calculus so remember the two big things that we studied calculus for is to help us with these area and falling in problems it's a huge okay we use the integral for that all right and then the other thing is to is define the slope okay call this the derivative so we use this symbol okay all right or this symbol Y prime or this symbol there's multiple symbol doing the same thing okay all right so hopefully you got something out of this video for those who are out there taking calculus know a lot about this and you're saying oh well this is not technically correct or this Anette well listen trust me I die I'm a math back I'm a degree in math so that's not the point I'm talking to people out there who you know don't have a clue about calculus or interests in it but anyways thanks for watching and if you like this video please subscribe to my channel have a great day

Einstein's General Theory of Relativity | Lecture 1



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Lecture 1 of Leonard Susskind’s Modern Physics concentrating on General Relativity. Recorded September 22, 2008 at Stanford University.

This Stanford Continuing Studies course is the fourth of a six-quarter sequence of classes exploring the essential theoretical foundations of modern physics. The topics covered in this course focus on classical mechanics. Leonard Susskind is the Felix Bloch Professor of Physics at Stanford University.

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this program is brought to you by Stanford University please visit us at stanford.edu gravity gravity is a rather special force it's unusual it has different than electrical forces magnetic forces and it's connected in some way with geometric properties of space space and time but before and that connection is of course the general theory of relativity before we start tonight for the most part we will not be dealing with the general theory of relativity we will be dealing with gravity in its oldest and simplest mathematical form well perhaps not the oldest and simplest but Newtonian gravity and going a little beyond what Newton certainly nothing that Newton would not have recognized or couldn't have grasped Newton could grasp anything but some ways of thinking about it which will not be found in Newton's actual work but still lutonium gravity the Toney and gravity set up in a way that that is useful for going on to the general theory ok let's begin with Newton's equations the first equation of course is F equals MA force is equal to mass times acceleration let's assume that we have a frame of reference a frame of reference that it means a set of coordinates and as a collection of clocks and those frame and that frame of reference is what is called an inertial frame of reference an inertial frame of reference simply means 1 which if there are no objects around to exert forces on a particular let's call it a test object a test object is just some object a small particle or anything else that we use to test out the various fields force fields that might be acting on it the inertial frame is one which when there are no objects around to exert forces that object will move with you for motion with no acceleration that's the idea of an inertial frame of reference and so if you're an inertial frame of reference and you have a pen and you just let it go it stays there it doesn't move if you give it a push it will move off with uniform velocity that's the idea of an inertial frame of reference and in an inertial frame of reference the basic Newtonian equation number one I always forget which law is which there's Newton's first law second law and third law I never can remember which is which but they're all pretty much summarized by f equals mass times acceleration this is a vector equation I expect people know what a vector is a three vector equation will come later to four vectors where when space and time are united into space time but for the moment space is space and time is time and a vector means a thing which is like a pointer in a direction in space as a magnitude and that has components so component by component the X component of the force is equal to the mass of the object times the X component of acceleration Y component Z component and so forth in order to indicate that something is a vector equation I'll try to remember to put an arrow over vectors the mass is not a vector the mass is simply a number every particle has a mass every object has a mass and in Newtonian physics the mass is conserved that does not change now of course the mass of this cup of coffee here can change it's lighter now but it only changes because mass has been transported from one place to another so you can change the mass of an object by whacking off a piece of it and but if you don't change the number of particles change the number of molecules and so forth then the mass is a conserved unchanging quantity so that's first equation now let me write that in another form the other form we imagine we have a coordinate system an X a Y and a Z I don't have enough directions on the blackboard to draw Z I won't bother there's x y and z sometimes we just call them x1 x2 and x3 I guess I can draw it in x3 is over here someplace XY and Z and a particle has a position which means it has a set of three coordinates sometimes we will summarize the collection of the three coordinates x1 x2 and x3 incidentally x1 and x2 and x3 are components of a vector the components they are components of the position vector of the particle position vector of the particle I will often call either small R or large are depending on on the particular context R stands for radius but the radius simply means the distance between a point and the origin for example we're really talking now about a thing with three components XY and Z and it's the radial vector the radial vector this is the same thing as the components of the vector R alright the acceleration is a vector that's made up out of the time derivatives of XY and Z or X 1 X 2 and X 3 so for each component the compose for each component one two or three the acceleration which let me indicate well let's just call it a the acceleration is just equal the components of it are equal to the second derivatives of the coordinates with respect to time that's what acceleration is the first derivative of position is called velocity we can take this to be component by component x1 x2 and x3 the first derivatives velocity the second derivative is acceleration we can write this in vector notation I won't bother but we all know what we mean I hope we all know we mean buddies by acceleration and velocity and so Newton's equations are then summarized and summarized but rewritten as the force on an object whatever it is component by component is equal to the mass times the second derivative of the component of position so that's the summary of I think it's Newton's first and second law I can never remember which they are Newton's first law of course is simply the statement that if there are no forces then there's no acceleration that's Newton's first law equal and opposite right so this summarizes both the first and second law I never understood why there was a first and second law it seems to me there was just one F equals MA all right now let's begin even even previous to Newton with Galilean gravity gravity as Galileo understood it actually I'm not sure how much of this mathematics Galileo did or didn't understand he certainly knew what acceleration was he measured it I don't know that he had thee but he certainly didn't have calculus but he knew what acceleration was so what Galileo studied was the motion of objects in the gravitational field of the earth in the approximation that the earth is flat now Galileo knew the earth wasn't flat but he studied gravity in the approximation where you never moved very far from the surface of the earth and if you don't move very far from the surface of the earth you might as well take the surface of the earth to be flat and the significance of that is to twofold first of all the direction of gravitational forces is the same everywhere as this is not true of course if the earth is curved then gravity will point toward the center but in the flat space approximation gravity points down down everywhere is always in the same direction and second of all perhaps a little bit less obvious but nevertheless true then the approximation where the earth is Infinite and flat goes on and on forever infinite and flat the gravitational force doesn't depend on how high you are same gravitational force here as here the implication of that is that the acceleration of gravity since force apart from the mass of an object the acceleration on an object is independent the way you put it and so Galileo either did or didn't realize well he again I don't know exactly what Galileo did or didn't know but what he said was equivalent to saying that the force on an object in the flat space approximation is very simple its first of all has only one component pointing downward if we take the upward sense of things to be positive then we would say that the force is let's just say the component of the force in the X 2 direction the vertical Direction is equal to minus the minus simply means that the force is downward and it's proportional to the mass of the object times a constant called the gravitational acceleration now the fact that it's constant Everywhere's in other words mass times G doesn't vary from place to place that's this fact that gravity doesn't depend on where you are in the flat space approximation but the fact that the force is proportional to the mass of an object that is not obvious in fact for most forces it's not true for electric forces the force is proportional to the electric charge not to the mass and so gravitational forces are rather special the strength of the gravitational force on an object is proportional to its mass that characterizes gravity almost completely that's the special thing about gravity the force is proportional itself to the mass well if we combine F equals MA with the force law this is the law force then what we find is that mass times acceleration the second X now this is the vertical component by DT squared is equal to minus that's the minus M G period that's it now the interesting thing that happens in gravity is that the mass cancels out from both sides that is what's special about gravity the mass cancels out from both sides and the consequence of that is that the motion of an object its acceleration doesn't depend on the mass it doesn't depend on anything about the particle a particle object I'll use the word particle I don't necessarily mean the point the small particle or baseb as a particle an eraser is a particle a piece of chalk is a particle that the motion of the object doesn't depend on the mass of the object or anything else the result of that is that if you take two objects of quite different mass and you drop them they fall exactly the same way our Galileo did that experiment I don't know if they're whether he really threw something off the Leaning Tower of Pisa or not it's not important he yeah he did balls down an inclined plane I don't know whether he actually did or didn't I know the the the myth is that he didn't die I find it very difficult to believe that he didn't I've been in Pisa last week I was in Pisa and I took a look at the Leaning Tower of Pisa galileo was born and lived in Pisa he was interested in gravity how it would be possible that he wouldn't think of dropping something off the Leaning power tower is beyond my comprehension you look at that tower and you say that I was good for one thing dropping things off now I don't know maybe the Doge or whoever they call the guy at the time said no no Galileo you can't drop things from the tower you'll kill somebody so maybe he didn't but he must have surely thought of it all right so the result had he done it and had he not had to worry about such spurious effects as air resistance would be that a cannonball and a feather would fall in exactly the same way independent of the mass and the equation would just say the acceleration would first of all be downward that's the minus sign and equal to this constant G excuse me that mean yes now G is a number it's 10 meters per second per second at the surface of the earth at the surface of the Moon it's something smaller and the surface of Jupiter it's something larger so it does depend on the mass of the planet but the acceleration doesn't depend on the mass of the object you're dropping it depends on the mass of the object you're dropping it onto but not the mass of the object that's dropping that fact that gravitational motion is completely independent the mass is called or it's the simplest version of something that's called the equivalence principle why it's called the equivalence principle we'll come to later what's equivalent to what at this stage we could just say gravity is equivalent between all different objects independent of their mass but that is not exact were the equivalents an equivalence principle was about that has a consequence an interesting consequence supposing they take some object which is made up out of something which is very unwritten just a collection of point masses maybe maybe let's even say that not even they're not even exerting any forces on each other it's a cloud a varied a few diffuse cloud of particles and we watch it fall let's suppose we start each particle from rest not all at the same height and we let them all fall some particles are heavy some particles are light some of them may be big some of them may be small how does the whole thing fall the answer is all of the particles fall at exactly the same rate the consequence of it is that the shape of this object doesn't deform as it falls it stays absolutely unchanged the relationship between the neighboring parts are unchanged there are no stresses or strains which tend to deform the object so even if the object were held together by some sort of struts or whatever there would be no forces on those struts because everything falls together the consequence of that is the falling in a gravitational field is undetectable you can't tell that you're falling in a gravitational field by you when I say you can't tell certainly you can tell the difference between freefall and standing on the earth that's not the point the point is that you can't tell by looking at your neighbors or anything else that there's a force being exerted on you and that that force that's being exerted on you is pulling you down word you might as well for all practical purposes be infinitely far from the earth with no gravity at all and just sitting there because as far as you can tell there's no tendency for the gravitational field to deform this object or anything else you cannot tell the difference between being in free space infinitely far from anything with no forces and falling freely in a gravitational field that's another statement of the equivalence principle for example these particles could be equipped with lasers lasers and optical detectives of some sort what's that oh you could certainly tell if you was standing on the floor here you could tell that something was falling toward you but the question is from within this object by itself without looking at the floor without knowing the floor was it well you can't tell whether you're falling and it's yeah yeah if there was something that was calm that was not falling it would only be because there was some other force on it like a beam or a tower of some sort of holding it up why because this object if there are no other forces on and only the gravitational forces it will fall at the same rate as this all right so that's another expression of the equivalence principle that you cannot tell the difference between being in free space far from any gravitating object versus being in a gravitational field that we're going to modify this this is of course it's not quite true in a real gravitational field but in this flat space approximation where everything moves together you cannot tell that there's a gravitational field or at least you cannot tell the difference I will not without seeing the floor in any case the self-contained object here does not experience anything different than it would experience far from any gravitating hating object standing still or uniform in uniform motion no you're accelerating if you go up to the top of a high building and you close your eyes and you step off and go into freefall you will feel exactly the same you feel weird I mean that's not the way you usually feel because your stomach will come up and you know do some funny things you know you might you might lose it but uh but the point is you would feel exactly the same discomfort in outer space far from any gravitating object just standing still you feel exactly the same peculiar feelings one of those peculiar feelings due to they're not due to falling they do to not fall well they do to the fact that when you stand on the earth here there are forces on the bottoms of your feet which keep you from falling and if the earth suddenly disappeared from under my feet sure enough my feet would feel funny because they used to having those forces exerted on their bottoms you get it I hope so the fact that you feel funny in freefall is because you're not used to freefall and it doesn't matter whether you're infinitely far from any gravitating object standing still or freely falling in the presence of a gravitational field now as I said this will have to be modified in a little bit there are such things as tidal forces those tidal forces are due to the fact that the earth is curved and that the gravitational field is not the same in every same direction in every point and that it varies with height that's due to the finiteness of the earth but in the flat space surprise and the Flat Earth approximation where the earth is infinitely big pulling uniformly there is no other effective gravity that is any different than being in free space okay again that's known as the equivalence principle now let's go on beyond the flat space or the Flat Earth approximation and move on to Newton's theory of gravity Newton's theory of gravity says every object in the universe exerts a gravitational force on every other object in the universe let's start with just two of them equal and opposite attractive attractive means that the direction of the force on one object is toward the other one equal and opposite forces and the magnitude of the force the magnitude of the force of one object on another let's let's characterize them by a mass let's call this one little m think of it as a lighter mass and this one which we can imagine as a heavier object will call it begin all right Newton's law of force is that the force is proportional to the product of the masses making either mass heavier will increase the force or the product of the masses begin tons of little m inversely proportional to the square of the distance between them let's call that R squared let's call the distance between them are and there's a numerical constant this for this law by itself could not possibly be right it's not dimensionally consistent the if you work out the dimensions of force mass mass and R will not dimensionally consistent there has to be a numeric constant in there and that numerical constant is called capital G Newton's constant and it's very small it's a very small constant I'll write down what it is G it is equal to six or six point seven roughly times 10 to the minus 11th which is a small number so in the face of it it seems that gravity is a very weak force you might not think that gravity is such a weak force but to convince yourself it's a weak force there's a simple experiment that you can do week week by comparison with other forces I've done this for car classes and you can do it yourself just take an object hanging by a string and two experiments the first experiment take a little object here and electrically charge it electrically charge it by rubbing it on your sweater that doesn't put very much electric charge on it but it charges it up enough to feel some electrostatic force and then take another object of exactly the same kind rub it on your shirt and put it over here what happens they repel and the fact that they repel means that this string will shift and you'll see a shift take another example take your little ball there to be iron and put a magnet next to it again you'll see quite an easily detectable deflection of the of the string holding it next take a 10,000 pound weight and put it over here guess what happens undetectable you cannot see anything happen the gravitational force is much much weaker than most other kinds of forces and that's due to the or not due to but the not due to that the fact that it's so weak is encapsulated in this small number here another way to say it is if you take two masses each of one kilometer not one kilometer one kilogram kilogram is a good healthy mass right nice chunk of iron mm and you separate them by one meter then the force between them is just G and it's six point seven times ten to the minus eleven the you know the units being Newtons so it's very very weak force but weak as it is we feel that rather strenuously we feel it strongly because the earth is so darn heavy so the heaviness of the earth makes up for the smallness of G and so we wake up in the morning feeling like we don't want to get out of bed because gravity is holding us down Oh Oh the equal and opposite equal and opposite that's the that's the rule that's Newton's third law the forces are equal and opposite so the force on the large one due to the small one is the same as the force of the small one on the large one and but it is proportional to the product of the masses so the meaning of that is I'm not heavier than I like to be but but I'm not very heavy I'm certainly not heavy enough to deflect the hanging weight significantly but I do exert a force on the earth which is exactly equal and opposite to the force that they're very heavy earth exerts on me why does the earth excel if I dropped from a certain height I accelerate down the earth hardly accelerates at all even though the forces are equal why is it that the earth if the forces are equal my force on the earth and the Earth's force on me of equal why is it that the earth accelerates so little and I accelerate so much yeah because the acceleration involves two things it involves the force and the mass the bigger the mass the less the acceleration for a given force so the earth doesn't accelerate quickly I think it was largely a guess but there was certain was an educated guess and what was the key ah no no it was from Kepler's it was from Kepler's laws it was from Kepler's laws he worked out roughly speaking I don't know exactly what he did he was rather secretive and he didn't really tell people what he did but the piece of knowledge that he had was Kepler's laws of motion planetary motion and my guess is that he just wrote down a general force realized that he would get Kepler's laws of motion for the inverse-square law I don't believe he had any underlying theoretical reason to believe in the inverse-square law that's correct he asked a question for inverse square laws no no it wasn't the ellipse which was the the the orbits might have been circular it was the fact that the period varies is the three halves power of the radius all right the period of motion is circular motion has an acceleration toward the center any motion in the circle is accelerated to the center if you know the period in the radius then you know the acceleration toward the center okay or we could write let's let's do it anybody know what if I know the angular frequency the angular frequency of going around in an orbit that's called Omega you know a–they and it's basically just the inverse period okay Omega is roughly the inverse period number of cycles per second what's the what is the acceleration of a thing moving in a circular orbit anybody remember Omega squared R Omega squared R that's the acceleration now supposing he sets that equal to some unknown force law f of r and then divides by r then he finds Omega as a function of the radius of the orbit okay well let's do it for the real case for the real case inverse square law f of r is 1 over r squared so this would be 1 over r cubed and in that form it is Kepler's second law remember which one it is it's the law that says that the frequency or the period the square of the period is proportional to the cube of the radius that was the law of Kepler so from Kepler's laws he easily could have that that one law he could easily deduce that the force was proportional to 1 over R squared I think that's probably historically what what he did then on top of that he realized if you didn't have a perfectly circular orbit then the inverse square law was the unique law which would give which would give elliptical orbits so who's to say well then of course there are the forces on them for two objects are actually touching each other there are all sorts of forces between them that I'm not just gravitational electrostatic forces atomic forces nuclear forces so you'll have to my breaks down yeah then it breaks down when they get so close that other important forces come into play the other important forces for example are the forces that are holding this object and preventing it from falling these we usually call them contact forces but in fact what they really are is various kinds of electrostatic for electrostatic forces between the atoms and molecules in the table in the atoms and molecules in here so other kinds of forces all right incidentally let me just point out if we're talking about other kinds of force laws for example electrostatic force laws then the force we still have F equals MA but the force law the force law will not be that the force is somehow proportional to the mass times something else but it could be the electric charge if it's the electric charge then electrically uncharged objects will have no forces on them and they won't accelerate electrically charged objects will accelerate in an electric field so electrical forces don't have this Universal property that everything falls or everything moves in the same way uncharged particles move differently than charged particles with respect to electrostatic forces they move the same way with respect to gravitational forces and as a repulsion and attraction whereas gravitational forces are always attractive where where's my gravitational force I lost it yeah here is all right so that's that's Newtonian gravity between two objects for simplicity let's just put one of them the heavy one at the origin of coordinates and study the motion of the light one then Oh incidentally one usually puts let me let me refine this a little bit as I've written it here I haven't really expressed it as a vector equation this is the magnitude of the force between two objects thought of as a vector equation we have to provide a direction for the force vectors have directions what direction is the force on this particle well the answer is its along the radial direction itself so let's call the radial distance R or the radial vector R then the force on little m here is along the direction R but it's also opposite to the direction of R the radial vector relative to the origin over here points this way on the other hand the force points in the opposite direction if we want to make a real vector equation out of this we first of all have to put a minus sign that indicates that the force is opposite to the direction of the radial distance here but we have to also put something in which tells us what direction the force is in it's along the radial direction but wait a minute if I multiply it by r up here I had better divide it by another factor of R downstairs to keep the magnitude unchanged the magnitude of the force is 1 over R squared if I were to just randomly come and multiply it by r that would make the magnitude bigger by a factor of our so I have to divide it by the magnitude of our this is Newton's force law expressed in vector form now let's imagine that we have a whole assembly of particles a whole bunch of them they're all exerting forces on one another in pairs they exert exactly the force that Newton wrote down but what's the total force on a particle let's label these particles this is the first one the second one the third one the fourth one that I thought that thought this is the ithe one over here so I is running index which labels which particle we're talking about the force on the eigth article let's call F sub I and let's remember that it's a vector it's equal to the sum now this is not an obvious fact that when you have two objects exerting a force on the third that the force is necessarily equal to the sum of the two forces of the two are of the two objects you know what I mean but it is a fact anyway obvious on how obvious it is a fact that gravity does work that way at least in the Newtonian approximation with Einstein it breaks down a little bit but in Newtonian physics the force is the sum and so it's a sum over all the other particles let's write that J not equal to I that means it's a sum over all not equal to I so the force on the first particle doesn't come from the first particle it comes from the second particle third particle fourth particle and so forth each individual force involves M sub I the force of the ice particle times the four times the mass of the Jade particle product of the masses divided by the square of the distance between them let's call that R IJ squared the distance between the eigth article his I and J the distance between the earth particle and the J particle is RI J but then just as we did before we have to give it a direction but a minus sign here that indicates that it's attractive another R IJ upstairs but that's a vector R IJ and make this cube downstairs alright so that says that the force on the I've particle is the sum of all the forces due to all the other ones of the product of their masses inverse square in the denominator and the direction of each individual force on this particle is toward the other all right this is a vector sum yeah hmm the minus indicates that it's attractive excellent but you've got the vector going from like a J oh let's see that's a vector going from the J yes there is a question of the sine of this vector over here so yeah you know absolutely let's yeah I actually think it's yeah you're right you're absolutely right the way I've written that there should not be a minus sign here all right but if I put our ji there then there would be a minus sign right so you're right but in any case every one every one of the forces is attractive and what we have to do is to add them up we have to add them up as vectors and so there's some resulting vector some resultant vector which doesn't point toward any one of them in particular but points in some direction which is determined by the vector sum of all the others all right but the interesting fact is if we combine this this is the force on the earth particle if we combine it with Newton's equations let's combine it with Newton's equipped with Newton's F equals MA equations then this is F this on the ice particle this is equal to the mass of the I particle times the acceleration of the ice particle again vector equations now the sum here is over all the other particles we're focusing on number I I the mass of the ice particle will cancel out of this equation I don't want to throw it away but let's just circle it and now put it over on the side we notice that the acceleration of the ice particle does not depend on its mass again once again because the mass occurs in both sides of the equation it can be cancelled out and the motion of the ayth particle does not depend on the mass of the earth particle it depends on the masses of all the other ones all the other ones come in but the mass of the iPart achill cancels out of the equation so what that means is if we had a whole bunch of particles here and we added one more over here its motion would not depend on the mass of that particle it depends on the mass of all the other ones but it doesn't depend on the mass of the i particle here okay that's again the equivalence principle that the motion of a particle doesn't depend on its mass and again if we had a whole bunch of particles here if they were close enough together they were all moving the same way before before i discuss lumo mathematics let's just discuss tidal forces what tidal forces are once you set this whole thing into motion dynamic young we have all different masses and each part what's going to be affected by each one is every particle in there is going to experience a uniform acceleration oh no no no no no acceleration is not uniform the acceleration will get larger when it gets closer to one of the particles it won't be uniform anymore it won't be uniform now because the force is not independent of where you are now the force depends on where you are relative to the objects that are exerting the force it was only in the Flat Earth approximation where the force didn't depend on where you were okay now the force varies so it's larger when you're far away it's sorry it's smaller when you're far away it's larger when you're in close it changes in a vector form with each individual particles each one of them is changing position yeah and and so is the dynamics that every one of them is going towards the center of gravity of the fire not necessarily I mean they could be flying apart from each other but they will be accelerating toward each other okay if I throw this eraser into the air with greater than the escape velocity it's not going to turn around and fall back changing with what with respect to what time oh it changes with respect to time because the object moves moves further and further away it's not uniformly the radius is changing and it's yeah let's take the earth here's the earth and we drop a small mass from far away as that mass moves in its acceleration increases why does its acceleration increase the deceleration increases because the radial distance gets smaller so in that sense it's not the alright now once the gravitational force depends on distance then it's not really quite true that you don't feel anything in a gravitational field you feel something which is to some extent it different than you would feel in free space without any gravitational field the reason is more or less obvious here you are his is the earth now you're you or me or whoever it is happens to be extremely tall a couple of thousand miles tall well this person's feet are being pulled by the gravitational field more than his head or another way of saying the same thing is if let's imagine that the person is very loosely held together he's just more or less a gas of we are pretty loosely held together at least I am right all right the acceleration on the lower portions of his body are larger than the accelerations on the upper portions of his body so it's quite clear what happens to her he gets stretched he doesn't get a sense of falling as such he gets a sense of stretching being stretched feet being pulled away from his head at the same time let's uh let's all right so let's change the shape a little bit I just spend the week two weeks in Italy and my shape changes whenever I go to Italy and it tends to get more horizontal my head is here my feet are here and now I'm this way still loosely put together right now what well not only does the force depend on the distance but it also depends on the direction the force arm my left end over here is this way the force on my right end over here is this way the force on the top of my head is down but it's weaker than the force on my feet so there are two effects one effect is to stretch me vertically it's because my head is not being pulled as hard as my feet but the other effect is to be squished horizontally by the fact that the forces on the left end of me are pointing slightly to the right and the forces on the right end of me are pointing slightly to the left so a loosely knit person like this falling in freefall near a real planet or real gravitational object which has a real Newtonian gravitational field around it will experience a distortion will experience a degree of distortion and a degree of being stretched vertically being compressed horizontally but if the object is small enough or small enough mean let's suppose the object that's falling is small enough if it's small enough then the gradient of the gravitational field across the size of the object will be negligible and so all parts of it will experience the same gravitational acceleration all right so tidal for these are tidal forces these forces which tend to tear things apart vertically and squish them this way tidal forces tidal forces are forces which are real you feel them I mean certainly new the car the cause of the tides yeah I don't know to what extent he calculated what do you mean calculated the well I doubt that he was capable I'm not sure whether he estimated the height of the of the deformation of the oceans or not but I think you did understand this much about tides okay so that's the that's what's called tidal force and then under the tidal force has this effect of stretching and in particular if we take the earth just to tell you just to tell you why it's called tidal forces of course it's because it has to do with tides I'm sure you all know the story but if this is the moon down here then the moon exerting forces on the earth exerts tidal forces on the earth which means to some extent that tends to stretch it this way and squash it this way well the earth is pretty rigid so it doesn't it doesn't deform very much due to due to these two the moon but what's not rigid is the layer of water around it and so the layer of water tends to get stretched and squeezed and so it gets deformed into the a the form shell of water with a bump on this side and the bump on that side alright I'm not gonna go any more deeply into that that I'm sure you've all seen okay but let's define now what we mean by the gravitational field the gravitational field is abstracted from this formula we have a bunch of particles don't you have need some some coordinate geometry so that would you have the four kind of middle is being pulled by all the other guys on the side I'm not explaining it right it's always negative is that what you're saying doesn't know I'm saying so she's attractive all right so you have but what about the other guys that are pulling upon him a different direction here and we're talking about the force on this person over here obviously there's one force pressing this pushing this way and another force pushing that way okay no the cone no they're all opposite to the direction of the object which is pulling on that's what this – sorry instead well you kind of retracted the minus sign at the front and reverse the ji yeah so it's the trend we can get rid of a – like a RI j and our ji are opposite to each other one of them is the vector between I and J I and J and the other one is the vector from J to I so they're equal and opposite to each other the minus sign there look as far as the minus sign goes all it means is that every one of these particles is pulling on this particle toward it as opposed to pushing away from it it's just a convention which keeps track of attraction instead of repulsion yeah for the for the ice master that's my word you want to make sense but if you can look at it as a kind of an in Samba wasn't about a linear conic component to it because the ice guy affects the Jade guy and then put you compute the Jade guy when you take it yeah now what this what this formula is for is supposing you know the positions or all the others you know that then what is the force on the one additional one but you're perfectly right once you let the system evolve then each one will cause a change in motion and the other one and so it becomes a complicated as you say nonlinear mess but this formula is a formula for if you knew the position and location of every particle this would be the force something you need to solve some equations to know how the particles move but if you know where they are then this is the force on the particle alright let's come to the idea of the gravitational field the gravitational field is in some ways similar to the electric field of our of an electric charge it's the combined effect of all the masses Everywhere's and the way you define it is as follows you imagine an one more particle one more particle amount you can take it to be a very light particle so it doesn't influence the motion of the others and one more particle in your imagination you don't really have to add it in your imagination and ask what the force on it is the force is the sum of the forces due to all the others it is proportional each term is proportional to the mass of the sec strip article this extra particle which may be imaginary is called a test particle it's the thing that you're imagining testing out the gravitational field with you take a light little particle and you put it here and you see how it accelerates knowing how it accelerates tells you how much force is on it in fact it just tells you how it accelerates and you can go around and imagine putting it in different places and mapping out the force field that's on that particle or the acceleration field since we already know that the force is proportional to the mass then we can just concentrate on the acceleration the acceleration all particles will have the same acceleration independent of the mass so we don't even have to know what the mass of the particle is we put something over there a little bit of dust and we see how it accelerates acceleration is a vector and so we map out in space the acceleration of a particle at every point in space either imaginary or real particle and that gives us a vector field at every point in space every point in space there is a gravitational field of acceleration it can be thought of as the acceleration you don't have to think of it as force acceleration the acceleration of a point mass located at that position it's a vector it has a direction it has a magnitude and it's a function of position so we just give it a name the acceleration due to all the gravitating objects it's a vector and it depends on position here X means location it means all of the position components of position XY and Z and it depends on all the other masters in the problem that is what's called the gravitational field it's very similar to the electric field except the electric field and the electric field is force per unit charge it's the force on an object divided by the charge on the object the gravitational field is the force of their on the object divided by the mass on the object since the force is proportional to the mass the the acceleration field doesn't depend on which kind of particle we're talking about all right so that's the idea of a gravitational field it's a vector field and it varies from place to place and of course if the particles are moving it also varies in time if everything is in motion the gravitational field will also depend on time we can even work out what it is we know what the force on the earth particle is all right the force on a particle is the mass times the acceleration so if we want to find the acceleration let's take the ayth particle to be the test particle little eye represents the test particle over here let's erase the intermediate step over here and write that this is in AI times AI but let me call it now capital a the acceleration of a particle at position X is given by the right hand side and we can cross out BMI because it cancels from both sides so here's a formula for the gravitational field at an arbitrary point due to a whole bunch of massive objects a whole bunch of massive objects an arbitrary particle put over here will accelerate in some direction that's determined by all the others and that acceleration is the gravitation the definition is the definition of the gravitational field ok let's um let's take a little break we usually take a break in about this time and I recover my breath to go on we need a little bit of fancy mathematics we need a piece of mathematics called Gauss's theorem and Gauss's theorem involves integrals derivatives divergences and we need to spell those things out there a central part of the theory of gravity and much of these things we've done in the context of a lot of electrical forces in particular the concept of divergence divergence of a vector field so I'm not going to spend a lot of time on it if you need to fill in then I suggest you just find any little book on vector calculus and find out what a divergence and a gradient and a curl we don't do curl today what those concepts are and look up Gauss's theorem and they're not terribly hard but we're gonna go through them fairly quickly here since they we've done them several times in the past right imagine that we have a vector field let's call that vector field a it could be the field of acceleration and that's the way I'm gonna use it well for the moment it's just an arbitrary vector field a it depends on position when I say it's a field the implication is that it depends on position now I probably made it completely unreadable a of X varies from point to point and I want to define a concept called the divergence of the field now it's called the divergence because one has to do is the way the field is spreading out away from a point for example a characteristic situation where we would have a strong divergence for a field is if the field was spreading out from a point like that the field is diverging away from the point incidentally if the field is pointing inward then one might say the field has a convergence but we simply say it has a negative divergence all right so divergence can be positive or negative and there's a mathematical expression which represents the degree to which the field is spreading out like that it is called the divergence I'm going to write it down and it's a good thing to get familiar with certainly if you're going to follow this course it's a good thing to get familiar with but are they going to follow any kind of physics course past freshman physics the idea of divergence is very important all right supposing the field a has a set of components the one two and three component but we could call them the x y&z component now I'll use x y&z are X Y & Z which I previously called X 1 X 2 and X 3 it has components X a X a Y and a Z those are the three components of the field well the divergence has to do among other things with the way the field varies in space if the field is the same everywhere as in space what does that mean that would mean the field that has both not only the same magnitude but the same direction everywhere is in space then it just points in the same direction everywhere else with the same magnitude it certainly has no tendency to spread out when does a field have a tendency to spread out when the field varies for example it could be small over here growing bigger growing bigger growing bigger and we might even go in the opposite direction and discover that it's in the opposite direction and getting bigger in that direction then clearly there's a tendency for the field to spread out away from the center here the same thing could be true if it were varying in the vertical direction or who are varying in the other horizontal direction and so the divergence whatever it is has to do with derivatives of the components of the field I'll just tell you exactly what it is it is equal to the divergence of a field is written this way upside down triangle and the meaning of this symbol the meaning of an upside down triangle is always that it has to do with the derivatives the three derivatives derivative whether it's the three partial derivatives derivative with respect to XY and Z and this is by definition the derivative with respect to X of the X component of a plus the derivative with respect to Y of the Y component of a plus the derivative with respect to Z of the Z component of it that's definition what's not a definition is a theorem and it's called Gauss's theorem no that's a scalar quantity that's a scalar quantity yeah it's a scalar quantity so it's let me write it it's the derivative of a sub X with respect to X that's what this means plus the derivative of a sub Y with respect to Y plus the derivative of a sub Z with respect to Z yes so the arrows you were drawn over there those were just a on the other board you drew some arrows on the other board that are now hidden yeah those were just a and a has a divergence when it's spreading out away from a point but that there vergence is itself a scalar quantity oh let me try to give you some idea of what divergence means in a context where you can visualize it imagine that we have a flat lake alright just the water thin a a shallow lake and water is coming up from underneath it's being pumped in from somewheres underneath what happens that the water is being pumped in of course it tends to spread out let's assume that the height let's assume the depth can't change we put a lid over the whole thing so it can't change its depth we pump some water in from underneath and it spreads out okay we suck some water out from underneath and it spreads in it anti spreads it has so the spreading water has a divergence water coming in toward the towards the place where it's being sucked out it has a convergence or a negative divergence now we can be more precise about that we look down at the lake from above and we see all the water is moving of course it's moving if it's being pumped in the world it's moving and there is a velocity vector at every point there is a velocity vector so at every point in this lake there's a velocity vector vector and in particular if there's water being pumped in from the center here right underneath the bottom of the lake there's some water being pumped in the water will spread out away from that point okay and there'll be a divergence where the water is being pumped in okay if the water is being pumped out then exactly the opposite the the arrows point inward and there's a negative divergence the if there's no divergence then for example a simple situation with no divergence that doesn't mean the water is not moving but a simple example with no divergence is the waters all moving together you know the river is simultaneous the lake is all simultaneously moving in the same direction with the same velocity it can do that without any water being pumped in but if you found that the water was moving to the right on this side and the left on that side you'd be pretty sure that somebody is in between water had to be pumped in right if you found the water was spreading out away from a line this way here and this way here then you'd be pretty sure that some water was being pumped in from underneath along this line here well you would see it another way you would discover that the X component of the velocity has a derivative it's different over here than it is over here the X component of the velocity varies along the x direction so the fact that the X component of the velocity is varying along the direction there's an indication that there's some water being pumped in here likewise if you discovered that the water was flowing up over here and down over here you would expect that in here somewhere as some water was being pumped in so derivatives of the velocity are often an indication that the some water being pumped in from underneath that pumping in of the water is the divergence of the velocity vector now the the the the water of course is being pumped in from underneath so there's a direction of flow but it's coming from from underneath there's no sense of direction well okay that's that's what diverges just the diagrams you already have on the other board behind there you take say the rightmost arrow and you draw a circle between the head and tail in between then you can see the in and out the in arrow and the arrow of a circle right in between those two and let's say that's the bigger arrow is created by a steeper slope of the street it's just faster it's going fast it's going okay and because of that there's a divergence there that's basically it's sort of the difference between that's right that's right if we drew a circle around here or we would see that more since the water was moving faster over here than it is over here more water is flowing out over here then it's coming in over here where is it coming from it must be pumped in the fact that there's more water flowing out on one side then it's coming in from the other side must indicate that there's a net inflow from somewheres else and the somewheres else would be from the pump in water from underneath so that's that's the idea of oops could it also be because it's thinning out with that be a crazy example like the late guy young well okay I took all right so let's be very specific now I kept the lake having an absolutely uniform height and let's also suppose that the density of water water is an incompressible fluid it can't be squeezed it can't be stretched then the velocity vector would be the right thing to think about them yeah but you could have no you're right you could have a velocity vector having a divergence because the water is not because water is flowing in but because it's thinning out yeah that's that's also possible okay but let's keep it simple all right and you can have the idea of a divergence makes sense in three dimensions just as well as two dimensions you simply have to imagine that all of space is filled with water and there are some hidden pipes coming in depositing water in different places so that it's spreading out away from points in three-dimensional space in three-dimensional space this is the expression for the divergence if this were the velocity vector at every point you would calculate this quantity and that would tell you how much new water is coming in at each point of space so that's the divergence now there's a theorem which the hint of the theorem was just given by Michael there it's called Gauss's theorem and it says something intuitive very intuitively obvious you take a surface any surface take any surface or any curve in two dimensions and now suppose there's a vector field that the field points now think of it as the flow of water and now let's take the total amount of water that's flowing out of the surface obviously there's some water flowing out over here and of course we want to subtract the water that's flowing in let's calculate the total amount of water that's flowing out of the surface that's an integral over the surface why is it an integral because we have to add up the flows of water outward where the water is coming inward that's just negative negative flow negative outward flow we add up the total outward flow by breaking up the surface into little pieces and asking how much flow is coming out from each little piece yeah how much water is passing out through the surface if the water is incompressible incompressible means density is fixed and furthermore the depth of the water is being kept fixed there's only one way that water can come out of the surface and that's if it's being pumped in if there's a divergence the divergence could be over here could be over here could be over here could be over here in fact any ways where there's a divergence will cause an effect in which water will flow out of this region yeah so there's a connection there's a connection between what's going on on the boundary of this region how much water is flowing through the boundary on the one hand and what the divergence is in the interior the connection between the two and that connection is called Gauss's theorem what it says is that the integral of the divergence in the interior that's the total amount of flow coming in from outside from underneath the bottom of the lake the total integrated and now by integrated I mean in the sense of an integral the integrated amount of flow in that's the integral of the divergence the integral over the interior in the three-dimensional case it would be integral DX dy DZ over the interior of this region of the divergence of a if you like to think of a is the velocity field that's fine is equal to the total amount of flow that's going out through the boundary and how do we write that the total amount of flow that's flowing outward through the boundary we break up let's take the three-dimensional case we break up the boundary into little cells each little cell is a little area let's call each one of those little areas D Sigma these Sigma Sigma stands for surface area Sigma is the Greek letter Sigma it stands for surface area this three-dimensional integral over the interior here is equal to a two-dimensional integral the Sigma over the surface and it is just the component of a perpendicular to the surface let's call a perpendicular to the surface D Sigma a perpendicular to the surface is the amount of flow that's coming out of each one of these little boxes notice incidentally that if there's a flow along the surface it doesn't give rise to any fluid coming out it's only the flow perpendicular to the surface the component of the flow perpendicular to the surface which carries fluid from the inside to the outside so we integrate the perpendicular component of the flow over the surface that's through the Sigma here that gives us the total amount of fluid coming out per unit time for example and that has to be the amount of fluid that's being generated in the interior by the divergence this is Gauss's theorem the relationship between the integral of the divergence on the interior of some region and the integral over the boundary where where it's measuring the flux the amount of stuff that's coming out through the boundary fundamental theorem and let's let's see what it says now any questions about that Gauss's theorem here you'll see how it works I'll show you how it works yeah yeah you could have sure if you had a compressible fluid you could discover that all the fluid out boundary here is all moving inwards in every direction without any new fluid being formed in fact what's happening is just the fluid is getting squeezed but if the fluid can't squeeze if you cannot compress it then the only way that the fluid could be flowing in is if it's being removed somehow from the center if it's being removed by by invisible pipes that are carrying it all so that means the divergence in the case of water would be zero there was no water coming it would be if there was a source of the water divergence is the same as a source source of water is the source of new water coming in from elsewhere is right so in the example with the 2-dimensional lake the source is water flowing in from underneath the sink which is the negative of a source is the water flowing out and in the 2-dimensional example this wouldn't be a 2-dimensional surface integral it would be the integral in here equal to a one dimensional surface and to go coming out okay all right let me show you how you use this let me show you how you use this and what it has to do with what we set up till now about gravity I think hope a lifetime let's imagine that we have a source it could be water but let's take three dimensional case there's a divergence of a vector field let's say a there's a divergence of a vector field del dot a and it's concentrated in some region of space that's a little sphere in some region of space that has spherical symmetry in other words doesn't mean it doesn't mean that the that the divergence is uniform over here but it means that it has the symmetry of a sphere everything is symmetrical with respect to rotations let's suppose that there's a divergence of the fluid okay now let's take and it's restricted completely to be within here it does it could be strong near the center and weak near the outside or it could be weakened near the center and strong near the outside but a certain total amount of fluid or certain total divergence and integrated there vergence is occurring with nice Oracle shape okay let's see if we can use that to figure out what the field what the a field is there's a Dell dot a in here and now let's see can we figure out what the field is elsewhere outside of here so what we do is we draw a surface around there we draw a surface around there and now we're going to use Gauss's theorem first of all let's look at the left side the left side has the integral of the divergence of the vector field all right the vector field or the divergence is completely restricted to some finite sphere in here what is incidentally for the flow case for the fluid flow case what would be the integral of the divergence does anybody know if it really was a flue or a flow of a fluid it'll be the total amount of fluid that was flowing in per unit time it would be the flow per unit time that's coming through the system but whatever it is this integral doesn't depend on the radius of the sphere as long as the sphere this outer sphere here is bigger than this region why because the integral over that there vergence of a is entirely concentrated in this region here and there's zero divergence on the outside so first of all the left hand side is independent of the radius of this outer sphere as long as the radius of the outer sphere is bigger than this concentration of divergence iya so it's a number altogether it's a number let's call that number M I'm not Evan let's just Q Q that's the left hand side and it doesn't depend on the radius on the other hand what is the right hand side well there's a flow going out and if everything is nice and spherically symmetric then the flow is going to go radially outward it's going to be a pure radially outward directed flow if the flow is spherically symmetric radially outward direct directed flow means that the flow is perpendicular to the surface of the sphere so the perpendicular component of a is just a magnitude of AE that's it it's just a magnitude of AE and it's the same everywhere is on the sphere why is it the same because everything has spherical symmetry a spherical symmetry the a that appears here is constant over this whole sphere so this integral is nothing but the magnitude of a times the area of the total sphere if I take an integral over a surface a spherical surface like this of something which doesn't depend on where I am on the sphere then it's just I can take this on the outside the magnitude of the the magnitude of the field and the integral D Sigma is just the total surface area of the sphere what's the total surface area of the sphere just 4 PI 4 PI R squared oh yeah 4 PI R squared times the magnitude of the field is equal to Q so look what we have we have that the magnitude of the field is equal to the total integrated divergence divided by 4 pi the 4 pi is the number times R squared does that look familiar it's a vector field it's pointed radially outward well it's point the radially outward if the divergence is positive if the divergence is positive its pointed radially outward and it's magnitude is one over R squared it's exactly the gravitational field of a point particle at the center here that's why we have to put a direction in here you know this R hat this art will this R over R is it's a unit vector pointing in the radial direction it's a vector of unit length pointing in the radial direction right so it's quite clear from the picture that the a field is pointing radially outward that's what this says over here in any case the magnitude of the field that points radially outward it has magnitude Q and it falls off like 1 over R squared exactly like the Newtonian field of a point mass so a point mass can be thought of as a concentrated divergence of the gravitational field right at the center point mass the literal point mass can be thought of as a concentrated concentrated divergence of the gravitational field concentrated in some very very small little volume think of it if you like you can think of the gravitational field as the flow field or the velocity field of a fluid that's spreading out Oh incidentally of course I've got the sign wrong here the real gravitational acceleration points inward which is an indication that this divergence is negative the divergence is more like a convergence sucking fluid in so the Newtonian gravitational field is isomorphic is mathematically equivalent or mathematically similar to a flow field to a flow of water or whatever other fluid where it's all being sucked out from a single point and as you can see the velocity field itself or in this case the the field the gravitational field but the velocity field would go like one over R squared that's a useful analogy that is not to say that space is a flow of anything it's a mathematical analogy that's useful to understand the one over R squared force law that it is mathematically similar to a field of velocity flow from a flow that's being generated right at the center at a point okay that's that's a useful observation but notice something else supposing now instead of having the flow concentrated at the center here supposing the flow was concentrated over a sphere which was bigger but the same total amount of flow it would not change the answer as long as the total amount of flow is fixed the way that it flows out through here is also fixed this is Newton's theorem Newton's theorem in the gravitational context says that the gravitational field of an object outside the object is independent of whether the object is a point mass at the center or whether it's a spread out mass or there it's a spread out mass this big as long as you're outside the object and as long as the object is spherically symmetric in other words as long as the object is shaped like a sphere and you're outside of it on the outside of it outside of where the mass distribution is then the gravitational field of it doesn't depend on whether it's a point it's a spread out object whether it's denser at the center and less dense at the outside less dense in the inside more dense on the outside all it depends on is the total amount of mass the total amount of mass is like the total amount of flow through coming into the that theorem is very fundamental and important to thinking about gravity for example supposing we are interested in the motion of an object near the surface of the earth but not so near that we can make the flat space approximation let's say at a distance two or three or one and a half times the radius of the earth well that object is attracted by this point that's attracted by this point that's attracted by that point it's close to this point that's far from this point that sounds like a hellish problem to figure out what the gravitational effect on this point is but know this tells you the gravitational field is exactly the same as if the same total mass was concentrated right at the center okay that's Newton's theorem then it's marvelous theorem it's a great piece of luck for him because without it he couldn't have couldn't have solved his equations he knew he meant but it may have been essentially this argument I'm not sure exactly what argument he made but he knew that with the 1 over R squared force law and only the one over R squared force law wouldn't have been truth was one of our cubes 1 over R to the fourth 1 over R to the 7th with the 1 over R squared force law a spherical distribution of mass behaves exactly as if all the mass was concentrated right at the center as long as you're outside the mass so that's what made it possible for Newton to to easily solve his own equations that every object as long as it's spherical shape behaves as if it were appoint appointments so if you're down in a mine shaft that doesn't hold that's right but that doesn't mean you can't figure out what's going on you can't figure out what's going on I don't think we'll do it tonight it's a little too late but yes we can work out what would happen in the mine shaft but that's right it doesn't hold it a mine shaft for example supposing you dig a mine shaft right down through the center of the earth okay and now you get very close to the center of the earth how much force do you expect that we have pulling you toward the center not much certainly much less than if you were than if all the mass will concentrate a right at the center you got the it's not even obvious which way the force is but it is toward the center but it's very small you displace away from the centre of the earth a little bit there's a tiny tiny little force much much less than as if all the mass was squashed toward the centre so right you it doesn't work for that case another interesting case is supposing you have a shell of material to have a shell of material think about a shell of source fluid flowing in fluid is flowing in from the outside onto this blackboard and all the little pipes are arranged on a circle like this what does the fluid flow look like in different places well the answer is on the outside it looks exactly the same as if everything were concentrated on the point but what about in the interior what would you guess nothing nothing everything is just flowing out away from here and there's no flow in here at all how could there be which direction would it be in so there's no flow in here so the distance argument like if you're closer to the surface of the inner shell yeah wouldn't that be more force towards that no you see you use Gauss's theorem let's do count system Gauss's theorem says okay let's take a shell the field the integrated field coming out of that shell is equal to the integrated divergence in here but there is no divergence in here so the net integrated field coming out of zero no field on the interior of the shell field on the exterior of the show so the consequence is that if you made a spherical shell of material like that the interior would be absolutely identical to what it what it would be if there was no gravitating material there at all on the other hand on the outside you would have a field which would be absolutely identical to what happens at the center now there is an analogue of this in the general theory of relativity we'll get to it basically what it says is the field of anything as long as it's fairly symmetric on the outside looks identical to the field of a black hole I think we're finished for tonight go over divergence and all those Gauss's theorem Gauss's theorem is central there would be no gravity without Gauss's theorem the preceding program is copyrighted by Stanford University please visit us at stanford.edu