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

Functions Lecture 1| Unacademy JEE | LIVE Daily | IIT JEE Mathematics | Sameer Chincholikar

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Lecture 1 | The Fourier Transforms and its Applications

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Lecture by Professor Brad Osgood for the Electrical Engineering course, The Fourier Transforms and its Applications (EE 261). Professor Osgood provides an overview of the course, then begins lecturing on Fourier series.

The Fourier transform is a tool for solving physical problems. In this course the emphasis is on relating the theoretical principles to solving practical engineering and science problems.

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this presentation is delivered by the Stanford center for professional development we are on the air okay welcome one at all and as I said on the TV when you were walking in but just to make sure everybody knows this is e 261 the Fourier transform and it's applications Fourier transforms at all Fourier and my name is Brad Osgood circulating around are two documents that give you information about the class there is a general description of the class course information how we're going to proceed some basic bookkeeping items I'll tell you a bit more about that in just a second and also a syllabus in a schedule and I'll also say a bit more about that in just a second let me introduce our partners in crime in this course we have three courses fins thomas john thomas one stand up where's Thomas there we go Rajiv Agarwal I smell that right very good Reggie going to stand up is Rajiv and Michael medias okay so far am gonna correct that okay that's like a Metis everybody thank you all right so now and we will be setting up times for the review sessions and so on all right you know so that'll be that will be forthcoming we have a web page for the course some of you may have already visited that but let me give you the and it's the addresses on the one the sheet one of the sheets that's being passed around but let me write that up now so you can be sure to visit it and register for the class because it is on the web page that you will find course handouts course information I will email people via the web page alright so you have to be registered I five to send an announcement to the class post an announcement and send out an email then that'll be done through the web page and you have to be registered on the Paige in order to get those emails I won't be doing it through access all right so it is at like many of the other classes HTTP slash slash however you do those where the colons go or is it here EE class Sanford et you you can find it very easily edu slash e 261 okay go there if you have not already and and register yourself for the class all right now let me say a little bit about the information that you have I want to say a little bit more about the mechanics I'll talk more about the content in just a second the let me say a bit about the syllabus and schedule and the course reader the syllabus is as I said on the on the top an outline of what we're going to be doing I hope a fairly accurate outline of what we're going to be doing but it's not a contract alright so there will be a natural ebb and flow of the course as things go along and when we get to particular material or what we cover in what order this is more or less I say accurate but it is not written in stone what you should use it for however is to plan your reading so I things will be much better for all of us if you read along with with the material as the syllabus has the schedule basically outlines all right because there's there times what I'm going to want to want to skip around a little bit there are times and I'm going to derive things there are times when I'm not going to derive things and you'll get much more out of the lectures our time together if you've read the material thoroughly before you come to class so that's one thing I ask you to do we have two exams scheduled we have a midterm exam and a final exam I'm going to schedule the midterm exam midterm exam is is already actually on here at least tentatively sort of toward the end of October we'll have it outside of class that is it'll be a sit down regular exam but I want to do it for 90 minutes rather than 15 min 15 minutes is just too short at a time for a class like for material like this so it'll be a 90 minute exam and we'll schedule it several sessions outside of class this is the way I've usually done it hasn't been any problem it's worked out alright for everybody so we'll have you know alternate times and so on and the final exam is scheduled by the char's office do not come to me right before the final exam saying oh I scheduled a trip out of town I hope that's not a problem right you know what the dates are ahead of time we'll also have regular problem sets none of these things that I'm saying should be new to you you've been through the drill many times the first problem the problem sets are going to be I had a starling innovation last time I taught the course where I handed out the problem sets on Monday and had them do the following Wednesday so you actually had you know like a week and a half to do the problem set so there was overlap between the two and people thought that was just a brilliant idea so we're going to do that again this year except for the first problem set and I decided it was not such good policy to hand at the very first problem set on the very first day of class so I'll hand that out on Wednesday and I'll post that also or at least I'll post it on that sure I'll hand it out it'll be available on Wednesday and it'll be due the following Wednesday and again these sorts of things are pretty routine for you I'm sure even through many times it will be practice although again not necessarily every time without fail to have MATLAB problems on the homework one or two MATLAB problems on the homework so I'm going into the assumption that people have some experience with using MATLAB you don't have to be terribly advanced and also access to using MATLAB so if you do not have experience using MATLAB and you do not have access to MATLAB get some experience and get some access won't be hard okay now let me say us a little bit about some of the things this is the course reader for the course it's available at the bookstore and also available on the course website all right doesn't have the problems in it but it has the material that were going to be covering in class now this is a basically a stitched together set of lecture notes that I've been using for a number of years in the class and I sort of tinker with it every time I teach the class but because it is a stitched together set of lecture notes they're the organization is sometimes a little bit odd like you have an appendix in the middle of the chapter and what that means is it was used to be an appendix to a set to a particular lecture that went on that particular day and it never got moved to anywhere else all right so the organization can be a little bit funny you can help on this all right that is if you find typos if you find errors if you find things that are less than clear in their in their wording if you want if you if you have some other ideas for you apples or other explanations please tell me I am working on this I have to say that the I'm hope because these these were written as a set of lecture notes these are meant to be a good and I hope helpful companion for the class that is they're meant to be read and they're meant to be used so you can help as generations of students in the past have helped try to refine these and turn them into something that's really a good accompaniment to to the class as we go on okay one other thing that's special this quarter is the class is as always taped and the lecture notes the lectures are going to be available to everybody but this time for the first time the lecture is going to be available to the world all right Stanford is decide on an experimental basis we're sort of competing with MIT here I think to try to make some classes some of the materials for some classes available to the world all right so the Elektra notes are going to be everything's going to be done through the website but instead of needing a Stanford ID to view the tape lectures I think anybody in the world can view these lectures was a little bit daunting I have to watch my language to try to dress well all right so we'll see what we'll see what goes with that I will however issue a warning I will not answer the world's email all right I will answer email from the class but I will not answer and I think I speak for the TAS here the TAS neither will answer the world's email on this all right how we're going to keep the world out of our inboxes I'm not sure exactly whether this is going to be a problem or not but at any rate that's what's happening okay all right any questions about that any questions about the mechanics of the course or what your expectations should be what my expectations of you are okay all right now I always like to take an informal poll actually when we start this class that's what it's a number of times now and it's always been a mixed crowd and I think that's one of the things that's attractive about this class so let me ask who are the E's in this class who are an electrical engineering your undergraduate or graduate all right so that's a pretty strong show of hands but let me also ask who are the non E's in this class all right that's also a pretty strong show of hands the EES are as is typical the majority of the students in the class but there's also a pretty strong group of students in this class who are not elect engineers by training by desire by anything all right and they usually come from all over the place I was looking at the web I was looking at the classes before I got the class and I think there's some people from chemistry somebody from chemistry anybody from chemists I thought there were somebody up see back there all right and other some people from Earth Sciences uh somebody from somebody is talking actually from Earth Sciences this morning somebody from Earth Science okay where else I think there was an Emmy couple of Emmys maybe yeah all right now that's important to know I think the course is very rich in material all right rich in applications rich in content and it appeals to many people for many different reasons okay for the ease and who are taking the class you have probably seen a certain amount of this material I don't want to say most of the material but you probably seen a fair amount of this material scattered over many different classes but it's been my experience that one of the advantages of this class for electrical engineering students either undergraduate or graduate students is to see it all in one piece all right to put it all in your head at one time at least once all right because the subject does have a great amount of coherence it really does hang together beautifully for all the different and varied applications there are core ideas and core methods of the class that it is very helpful to see all at once alright so if you have seen the material before that's fine I mean that is I mean that you can you can draw on that and draw on your experience but don't deny yourself the pleasure of trying to synthesize the ideas as we go along I mean there's nothing so pleasurable as thinking about something you already know trying to think about it from a new perspective try think about it from a new point of view trying to try to fold it into some of the newer things you'll be learning so I have I've heard this from electrical engineering students many times in the past that it's a it's a pleasure for them to see the material all together at once it may seem like a fair amount of review and in some cases it will be but not in all cases and even if it is a review they're often slightly different twists or slightly new takes on things that you may not have seen before I may not have thought of quite in quite that way so so so that is my advice to the electoral engineering students for the students who have not seen this material before they're coming out of from a field and maybe only heard you know secret tales of the Fourier transform and its uses well I hope you enjoy the ride because it's going to be a hell of a ride a heck of a ride as we go along alright now for everyone I sort of feel like I have to issue I don't know if I call this a warning or just sort of a statement a principal or whatever this is a very mathematical class this is one of the sort of Holy Trinity of classes in the Information Systems lab in electrical engineering the electril engineering is a very broad department and split up into a number of laboratories along research lines I am in the Information Systems lab which is sort of the mathematical part of the subject there's a lot of signal processing coding Theory imaging and so on and this course has been for a number of years taught by faculty sort of thought of as a cornerstone in the signal processing although it has a lot of different applications to a lot of different areas the other courses in that Holy Trinity are 263 dynamically near dynamical systems and 270 a statistical signal processing who's taken to say whose likes let me ask you so because this is also very common who's taken to 63 in the class also a strong majority and who's taken to 78 yeah ok so there's a fair a little little bit less but still number of people we will actually see not so much with 270 oh well actually with both classes with 263 in 278 you'll actually see some overlap that I also hope you find interesting the language will be slightly different the perspective will be slightly different but you see this material in this class melding over into the other classes and vice-versa and again I think it's something that you can really draw on and I hope you enjoy all right so it is those those classes and the perspective that we take the faculty your teaching those classes is a pretty mathematical one but it's not a class in theorems and proofs you can breathe a heavy sigh of relief now all right I can do that but I won't all right I will derive things I'll derive a lot of a number of formulas I'll derive it and I'll go through those derivations or I'll hope that you go through the derivations in the book when I hope and I think that they will be helpful all right and when in some case that is there's an important technique or there's an important idea that you'll see not only in the tick Euler instance but over all that you'll see the same sort of derivation the same sort of ideas be applied not only for one formula but for other sorts of formulas and also in some cases to my mind as twisted as that may be I sometimes think of the derivation of a formula almost as identical with a formula I mean to use the formula effectively almost as to know the derivation because it's to know where it applies and to know how it applies and where to expect to use it all right so that's why I will go through those things for the purpose of teaching a certain amount of technique and for the purposes of sort of having those techniques really at your fingertips so that you can apply them again in a situation that may not be quite identical with with what we did but will be similar enough so that the simp so that the ideas may apply in this situation that's that's very important we will also do plenty of different sorts of applications but again because the field the subject is so varied and because the clientele because the students in the class are also varied will try to take applications from different areas will have applications from electrical engineering but will also have applications from physics and from other areas i i've also done in the past and will see if i get to this some applications from Earth Sciences for example and we'll just see how they go so we all have to cut each other a little bit of slack and if an application or particular area is not exactly to your liking well chances are it might be to somebody's liking to your right or left so you say cut everybody should cut each other a little slack and just enjoy the ride I should also say that many of the more specialized applications are found in more specialized courses all right so we will touch on a lot of things and I will use the words that are used in a lot of different courses and a lot of different subjects but we won't always do see an application to its bitter end so to speak or we won't do every pot we certainly won't do every possible application because there are just so many of them so you will find you will not run out of ways of using the Fourier transform and Fourier analysis techniques in any classes here they go it goes on and on and on but we'll only be able to see a certain amount of a certain amount of that all right and actually that leads to a very important point release of the start of the class that is where do we start all right that is this subject which is so rich and so diverse forces you forces me forces all of us to make hard choices in some ways about where what we're going to cover where we're going to start what direction we're going to go and all the different choices are defensible you will find books out there that take very different taps toward the subject they take different starting points they have different emphases they go off in different directions and you can make a good argument for any one of those choices but you have to make a choice so for us we are going to choose I have chosen not we me I have chosen to start the class with a brief discussion of Fourier series and go from there to the Fourier transform all right whereas it is also very common choice to forget about Fourier series and maybe pick them up a little bit along Angier or pick them pick them up a little bit on the edges or assuming that everybody seen Fourier series then go right into the fray transform I don't want to do that because I think that the subject of Fourier series is interesting enough in it we're not going to do very much with it but it's interesting enough in itself again it's something you may have seen in different context but it provides a natural transition to the study of the Fourier transform and it is historically actually the way the subject developed okay so that's how we're going to that's how we're going to do things will start with Fourier series and use them as a transition to Fourier transform now first of all what is this concerned with overall I it may be a little bit too strong a statement but for our purposes I want to identify the idea of Fourier series as almost identified with the study of periodic phenomena alright so for us it's identified most strongly with a mathematical analysis of periodic phenomena now it certainly shouldn't be necessary for me to justify periodic phenomena as an important class of phenomena you have been studying these things for your entire life pretty much ever since the first physics course you ever took where they do the harmonic oscillator and then the second physics course you took where they did the harmonic oscillator and then the third physics course you took rhythm they did the harmonic oscillator you have been studying periodic phenomena alright so that shouldn't be a controversial choice Fourier series goes much beyond that but it is first and foremost for us associated with a study of periodic phenomena the Fourier transform in although again it doesn't maybe doesn't do it's just justice completely is can be viewed as a limiting case of Fourier series it has to do with a study of the mathematical analysis on phenomena so if you want to contrast Fourier series and Fourier transforms then that's not a bad rough-and-ready way of doing it doesn't it say it doesn't capture everything but it captures something so Fourier transform as a limiting case and in a meeting that I'll make more precise later is limiting case of Fourier series Fourier series of free series techniques is identified with or has to do with is concerned with how about that for weaseling way out of it is concerned with the analysis of non periodic phenomena so again it doesn't say everything but it says something and one of the things that I hope you get out of this course especially for those of you who have had some of this material before are these sort of broad categorizations that help you sort organize your knowledge all right it's a very rich subject you've got to organize it somehow otherwise you'll get lost in the details all right you want to have certain markers along the way that tell you how to think about it how to organize it what what what a particular formula what cat it what general category it fits under okay now it's interesting is that the ideas are sometimes similar and sometimes quite different and sometimes it's the situation is simpler for periodic phenomena sometimes the situation is more complicated for periodic phenomena so it's not as though there's sort of a one-to-one correspondence of ideas but that's one of the things that we'll see and one of the reasons why I'm starting with Fourier series is to see how the ideas carry over from one to the other see where they work and see where they don't work alright some ideas carry easily back and forth between the two some phenomena some ideas some techniques some don't and it's interesting to know when they do and when they don't sometimes the things are similar and sometimes they're not now in both cases there are really to kind of inverse problems there's a question of analysis and there's the question of synthesis two words that you've used before but it's worthwhile reminding what they mean in this context the analysis part of Fourier analysis is has to do with breaking a signal or a function I'll use the term signal and function pretty much interchangeably alright I'm a mathematician by training so I tend to think in terms of functions but electrical engineers tend to think in terms of signals and they mean the same thing all right so analysis has to do with taking a signal or a function and breaking it up into its constituent parts and you hope the constituent parts are simpler somehow then the complicated signal that as it comes to you so you want to break up a signal into simpler constituent parts I mean if you don't talk in just in terms of signals here or you don't use exactly that language that's the meaning of the word analysis I think close enough whereas synthesis has to do with reassembling a signal or reassembling a function from its constituent parts a signal from its constituent parts kind of stitch one alright and the two things go together all right you don't want one without the other you don't want to you don't want to break something up into its constituent parts and then just let it sit there all these little parts sitting on the table with nothing to do you want to be able to take those parts maybe modify those parts maybe see which parts are more important than other parts and then you want to put them back together to get that to get either the original signal or a new signal and the process of doing those things are the two aspects of Fourier analysis I use I use the word analysis they're sort of in a more generic sense now the other thing to realize about both of these procedures analysis and synthesis is that they are accomplished by linear operations series and integrals are always involved here both analysis and synthesis free analysis analysis and synthesis are accomplished by linear operations this is one of the reasons why the subject is so I don't know powerful because there is such a body of knowledge on and such a deep and advanced understanding of linear operations linearity will make this a little bit more explicit as I go as we go on further but I wanted to point it out now because I won't always point it out all right because when I say linear operations when I'm thinking of here integrals in series all right eg ie integrals and series both of which are linear operations the integral of a sum is the sum of the integrals the integral of a of a constant times a function is a constant interval the function and so and similarly with sums alright because of this one often says or one often thinks that Fourier analysis is part of the study of linear systems alright in engineering there's there's a there's their courses called linear systems and so on and sometimes Fourier analysis is thought to be a part of that because the operations involved in it are linear I don't think of it that way I mean I think it's somehow important enough on its own not to think of it necessarily as subsumed in a larger subject but nevertheless the fact that the operations are linear does put it in a certain context in some in some ways in some cases more general context that turns out to be important for many ideas alright so often so you see you often hear that Fourier analysis Fourier analysis is a part of the subject of linear systems the study of linear systems so I don't think that really does complete justice to Fourier analysis because of because of the particular special things that are involved in it but nevertheless you will you'll hear that okay now let's get launched alright let's start with with the actual subject of Fourier series and the analysis of periodic phenomenon a periodic phenomena and Fourier series as I said it certainly shouldn't be necessary for me to sell the importance of periodic phenomena as something worth studying you see it everywhere all right the study of periodic phenomena is for us the mathematics and engineering or mathematics and science and engineering of regularly repeating phenomena that's what's always involved there's some pattern that repeats and it repeats regularly right so it's the mathematics and engineering so this is an engineering course I'll put that before science or maybe I won't even mention science mathematics and engineering of regularly repeating patterns I'm relieving a couple of terms here I'm leave all these terms somewhat vague what does it mean to be regular what does it mean to repeating what is a pattern in the first place but you know what you know what I mean you know it when you see it and the fact you can mathematically analyze it is what makes the subject so useful now I think although again it's not ironclad trouble is this subject is so rich that every time I make a statement I feel like I have to qualify it well it's often true but it's not completely true and sometimes it's not really true at all but most of the time it's true that it's helpful but not always helpful but most of the time helpful occasionally helpful to classify periodicity as either periodicity in time or periodicity in space all right you often see periodic phenomena as one type or the other type although they can overlap so you often periodic phenomena often are either periodicity in time a pattern repeats in time over and over again you wait long enough and happens again so for example harmonic motion so eg harmonic motion a pendulum I think bobbing on a string G harmonic motion or periodicity in base periodicity in space the city in space alright now what I mean he is there is often a physical quantity that you are measuring that is living on some object in space one dimension two dimensions whatever that has a certain amount of symmetry alright and the periodicity of the phone on is a consequence of the symmetry of the object so it's often the cow giving example just a second so here you have say some some physical quantity physical not always but often you know physical quantity distributed over a region with symmetry the region itself repeats all right the region itself as a repeating pattern all right so the periodicity of the phenomenon the periodicity of the physical quantity that you're measuring is a consequence of the fact that it's distributed on on over some region that itself has some symmetry so the periodicity arises from the symmetry for periodicity here of the object of the of the physical quantity that you're measuring arises because the periodicity of the are the symmetry of the object where tributed where it lives I'll give you an example there from the symmetry matter of fact I'll give you the example the example that really started the subject and we'll study this is the distribution of heat on a circular ring so eg the distribution of heat on a circular ring alright so the object the the physical quantity that you're interested in is the temperature but it's a temperature associated with a certain region and the region is a ring all right the ring has circular symmetry it's around okay so you're measuring the temperature at points on the ring and that's periodic because if you go once around you're at the same place so the temperature is periodic as a function of the spatial variable that describes where you are on the ring time is not involved here position is involved all right it's periodic in space not periodic in time periodic in a spatial variable that gives you the position and the periodicity arises because the object itself is symmetric because the object repeats that's why this sort of example is why one often sees and this actually turns out to be very far-reaching and quite deep that free analysis is often associated with questions of symmetry in a sort of most mathematical form you often find for a series developed in and in this context and Fourier transform is developed in the context of symmetry so you often see so you see Fourier analysis let me just say free analysis analysis is often associated with problems or just not off with with analysis of questions that have to do with that have some sort of relying symmetry so let me say often associated with problems with symmetry just leave it very general this is the very first of all that for the problem of distribution of heat on a ring we're going to solve that problem that was the problem that Fourier himself considered alright they introduced some of the methods into the into the whole subject let's launch everything all right so again it's not periodicity in time its periodicity in space and for those of you who have had or may have courses in this that the mathematical framework for this very general way of looking for a analysis is group theory because the theory of groups in mathematics is a way of mathematize the ADEA of symmetry and then one extends the ideas for elseís into to take into account of groups that is to say to take into account the symmetry of certain problems that you're saying and it really stays very quite it's quite far-reaching we're not going to do it we'll actually have a few occasions to to go to go into this but but with a light touch all right I'm just telling you I'm just giving you some indication of where the subject goes all right now what are the mathematical descriptors of periodicity well nothing I've said so far I'm sure it is new to you at all you just have to trust me that at some point before you know it some things I say to you will be new I hope but one of the mathematical descriptions of periodicity again that in the two different categories say the numbers the quantities that you associate with either either a phenomena that's periodic and timer function or a phenomenon that's periodic in space for periodic and time for periodicity in time you often use the frequency all right frequency is the word that you hear most often associated with a phenomena that is periodic in time you use frequency the number of repetitions the number of cycles in a second say if a pattern is repeating whatever the pattern is again if I leave that term sort of undefined or sort of vague it's the number of repetitions of the pattern in one second or over time all right that's the most common descriptor mathematical descriptor of a phenomenon is periodic that's periodic in time for a function for a phenomenon is periodic in space you actually use the period that's the only word that's really in use in general for the particulate well one thing a time so for periodicity in space you use the period all right that is sort of the physical measurement of how long the long the pattern is before repeats somehow all right the measurement of how whether its length or some other quantity measurement of how let me just say how big the pattern is that repeats they're not the same all right they have a different feel they rise off from from different sorts of problems that's probably too strong a statement but I think I think it's fair to say that mathematicians tend to think in terms of mostly in periodic they tend to think in terms of the period of a function or the period is the description of periodic behavior whereas engineers and scientists tend to think of systems evolving in time so they tend to think in terms of frequency they tend to think of how often a pattern repeats over a certain period of time all right that's like everything else is that statement has to be qualified but I get tired of qualifying every statement so I'll just leave it at that now of course the two phenomena are not completely separate or not always completely separate they come together periodicity and time and periodicity in space come together in for example wave motion all right that is traveling disturbance a travelling periodic disturbance so the two notions of periodicity come together two notions here periodicity and time periodicity in space come together in EEG wave motion understood very generally here as a periodic as a regularly repeating pattern that changes in time that moves because more jumps up a little bit I think of their skipping so a regular a moving a subset regularly moving disturbance you know a group of freshmen through the quad you know just they're everywhere mostly regular mostly moving all right now there again the two descriptors come in the frequency and the wavelength so again you have frequency and wavelength you have frequency nu and wavelength usually associated usually denoted by this is for periodicity in space and for periodicity and time frequency nu for periodicity in time that's the number of times and repeats in one second this is cycles per second the number of times that the pattern repeats in one second so for example you fix yourself at a fix your position in spate both time and space are involved so you fix yourself at a point in space and the phenomenon washes over you like a water wave all right and you count the number of times you're hit by the wave in a second and that's the frequency that's the number of times that the phenomenon comes to you for periodic for periodicity and time the function the phenomenon comes to you for periodicity in space you come to the phenomenon so to speak all right so I fixed myself at a point in time the wave washes over me at a certain characteristic frequency over and over again regularly repeating it comes to me new times per second the wavelength you fix the time and allow the platen and see what the phenomena looks like to distribute it over space so for periodicity in space fix the time and see how the phenomena is distribute to see the pattern distributed over space distributed my writing is getting worse distributed then the length of one of those a complete to speak is the period or the wavelength length is a term that's associated with the periodicity in space for a traveling traveling phenomena for a wavelet wave for wave motion so the length the length of the disturbance I say one complete disturbance if I can say that one complete pattern is the wavelength now like I say ever since you were a kid you've studied these things and especially don't know the number by lambda but I bring it up here because of the one important relationship between frequency and wavelength which we are going to see in a myriad of forms throughout the quarter that is there's a relate in the case of wave motion there is a relationship between the frequency in the wave length determined by the velocity and there could be two different phenomena all right periodicity in time and periodicity in space may not have anything to do with each other but if you have a wave traveling if you have a regularly repeating pattern over time then they do have something with to do with each other and they're governed by the formula distance equals rate times time which is the only formula that governs motion all right so there's a relationship between frequency and wavelength that is distance equals rate times time I love writing this in a graduate course because it's the up the equation in calculus actually in all of calculus I think this is pretty much the only equation used in very clever ways but the only equation and in our case if the rate is the velocity of the wave then this translate V is the velocity the rate of the wave of the motion and the equation becomes as I'm sure you know many times lambda that's the distance that this this the the wave travels in one cycle it traveling it's traveling at a speed V if it goes nu cycles in one second then it goes one cycle in 1 over nu seconds let me say that it going to make sure I got that right if it goes nu cycles in one second if it just passed you nu times in one second then in 1 over nu seconds it rushes past you once rushing past you once means you've gone through one wavelength so distance equals rate times time the time it takes to go one wave length is 1 over nu seconds so I have lambda equals V times 1 over nu or lambda nu equals V again a formula européenne many times now why did I say this if you've seen it many times because I never have the confidence that I can talk my way through that formula for one thing so I always have to do it secondly it exhibits a reciprocal relationship to quantities all right there's a reciprocal relationship you can see it more clearly over here where the constant of proportionality or inverse proportionality is the velocity all right lambda is proportional to the reciprocal of the frequency or the restore the frequency is proportional to the reciprocal of the wavelength at any rate or the or expressed this way lambda times nu is equal to V so there's a reciprocal relationship between the frequency and the wavelength all right this is the first instance when you talk about periodicity of such reciprocal relationships we are going to see this everywhere all right it's one of the characteristics of the subject hard to state as a general principle but but they're plain to see that in the prop in in in the analysis and the synthesis of signals using methods from Fourier series or Fourier analysis there will be a reciprocal relationship between the two between the quantities involved all right I'm sorry for being so general and but you'll see this play out in case after case after case and it is something you should be attuned to all right all right so you may never have thought about this in these types of simple enough formula you've used millions of times all right you may not have thought about it somehow in those terms but I'm asking you to think about stuff use you once saw in very simple context and how those simple ideas sort of cast shadow into much more involved situations all right the reciprocal relationship between as well as we'll learn to call it the reciprocal relationship between the two domains of Fourier analysis the time domain in the frequency domain or the tie or the store the time domain and the spatial domain or the spatial domain in the frequency domain and so on is something that we will see constantly alright and I will point that out but if I don't point it out you should point out to yourself you should be attuned to it because you will see it and it's one of those things that helps you organize your understanding of the material because sometimes when you're called upon to apply these ideas in some context that you haven't quite seen you have to ask yourself it's at least the good starting place is to ask yourself questions like well should I expect a reciprocal relationship here you might lead you to guess what the formulas should be or guess what the relationship should be so you say well somehow I want to use for a analysis to do this problem so I'm sure I should be looking for some sort of reciprocal relationship the quantities that I'm interested in somehow should be related in some kind of reciprocal way and what that might mean might be more or less involved depending on the particular kind of problem but you'll see it trust me you'll see it okay right now we're almost done for today why does mathematics come into this in the first place I mean periodicity is evidently sort of a very physical type property why is it allow any kind of mathematical description well it does because there are very simple maybe not so simple mathematical functions that exhibit periodic behavior and so can be used to model periodic phenomena so math comes in because there are simple mathematical functions that model that are periodic that repeat and so can be used to model periodic phenomena I am speaking of course of our friends the sine and cosine now you may think again we've only talked about elementary things in very elementary contexts but you know I have a PhD in this subject and I get excited talking about sines and cosines I mean you know and it's not just creeping old age I mean I think there you know there's a lot there's a lot to reflect on here and sometimes the miraculous nature of these things cosine of I'll use I'll use T is the variable cosine of T and sine of T our periodic of period two pi that is cosine of T plus two pi is equal to cosine of T for all values of T and sine of two pi + t + 2 pi is equal to sine of T why dead silence because the sine and cosine are item don't tell me I want to do it because this I'll do it over here because the sine and the cosine are associated with periodicity in space because the sine of the cosine are associated with an object that regular repeats the simplest object that the regularly repeats does circle you didn't meet sine and cosine that way first you met sine and cosine in terms of ratios of psiy lengths of sides in triangles that's fine but that's an incomplete definition the real way of understorey way but the but them but the more sophisticated way the ultimately more far-reaching way of understanding sine and cosine is as associated with the unit circle where the cosine of t is the x-coordinate and the sine of t is the y-coordinate and T is Radian measure I'm not going to go through this in too much detail but the point is that the sine of the cosine are each associated with the phenomenon of periodicity in space they are periodic because if you go once around the circle that is to say T goes from T to T plus 2 pi you're back where you started from all right that's why it's periodicity in space all right that's the definition of sine and cosine that exhibits their their periodic phenomena not the definition in terms of right triangles it's not the definition it's not that the definition in terms of right triangles is wrong it just doesn't go far enough it's incomplete all right it doesn't reveal that fundamental link between the trigonometric functions and periodicity and it is fundamental if not for that mathematics could not be brought to bear on the study of periodic phenomena and furthermore this clear and will quit in just a second that is not just 2pi but any multiple of 2pi positive or negative I can go clockwise or I can go counterclockwise I can say the cosine of t plus 2pi n is the same thing as cosine of 2t and the sine of 2pi t plus 2pi n is the sine of T for n any integer n 0 plus or minus 1 plus or minus 2 and so on and so on the interpretation is that when n is positive I'm going count and it is just an interpretation is just a convention when n is positive I'm going counter clockwise around the circle when n is negative I'm going clockwise around the circle but it's only when you make the connection between periodicity and space and the sign of the cosine that you see this fundamental property all right now all right I think we made it out of junior high today that's that was my goal all right what is what is most amazing and what and what was what we'll see you next time is that such simple functions can be used to model the most complex periodic behavior all right the simple from such simple things some simple acorns mighty oaks grow or whatever you excuse me whatever whatever stuff you learn out there that the simple these simple functions that associated with such a simple phenomena can be used to model the most complex really the most complex periodic phenomena and that is the fundamental discovery of Fourier series all right and is the basis of Fourier analysis and we will pick that up next time thank you very much see you then

Programming Basics: Statements & Functions: Crash Course Computer Science #12

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Today, Carrie Anne is going to start our overview of the fundamental building blocks of programming languages. We’ll start by creating small programs for our very own video game to show how statements and functions work. We aren’t going to code in a specific language, but we’ll show you how conditional statements like IF and ELSE statements, WHILE loops, and FOR loops control the flow of programs in nearly all languages, and then we’ll finish by packaging up these instructions into functions that can be called by our game to perform more and more complex actions.

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مرحبا, انا كاري أن, أهلا بكم في كراش كورس لعلوم الكمبيوتر في اخر حلقة قمنا بشرح كيفية كتابة البرامج باستخدام اوامر لغة الآلة و بسسب وجود تفاصيل كثيرة على مستوى لغة الآلة مما يشكل عائقا أمام كتابة برامج معقدة و لإزالة الكثير من هذه التفاصيل , تطورت لغات البرمجة لكي تسمح للمبرمجين بالتركيز على حل المشاكل الحسابية دون أن ينشغلوا بتفاصيل مكونات الكمبيوتر لذا اليوم سةف نستكمل ذلك النقاش وساقدم لكم بعض الاساسيات في بناء البرامج التي تقدمها تقريبا جميع لغات البرمجة "Translated by : Eng. Ahmed Isam." مثل اللغات الصوتية لغات البرمجة لها جمل وأوامر وهي افكار كاملة مستقلة مثل "أريد الشاي" أو "انها تمطر" باستخدام كلمات مختلفة يمكننا تغيير المعنى مثل : من "أريد الشاي" إلى "أريد وحيد قرن" لكن لا يمكن ان نجعلها "أريد المطر" لان ذلك غير منطقي قواعدياّ الشعور النحوية. مجموعة القواعد التي تبني بها وتكتب بها الجملة البرمجية تدعى بناء الجملة (تركيب الجملة-Syntax-) اللغة الإنجليزية بناء جملة، وهكذا تفعل
جميع لغات البرمجة. "A يساوي 5" هي لغة برمجة
بيان. في هذه الحالة، البيان يقول متغير
اسمه ولديه رقم 5 المخزنة فيه. وهذا ما يسمى عبارة تعيين ل
نحن تعيين القيمة إلى متغير. للتعبير عن أشياء أكثر تعقيدا، ونحن بحاجة إلى
سلسلة من البيانات، مثل "A هو 5، B هو عشرة، C يساوي زائد B " يروي هذا البرنامج الكمبيوتر لتعيين متغير
'A' يساوي 5، المتغير 'B' إلى 10، وأخيرا إلى إضافة 'A' و 'B' معا،
وضعت هذه النتيجة، وهو 15 عاما، إلى – أنت تفكر في ذلك – متغير C. لاحظ أنه يمكن أن نطلق المتغيرات مهما فعلنا
تريد. بدلا من A، B و C، ويمكن أن يكون التفاح،
الكمثرى، والفواكه. لا يهتم الكمبيوتر، طالما المتغيرات
تتم تسمية فريد. ولكن من المحتمل أن يكون أفضل الممارسات لاسم
لهم الأشياء التي تجعل المعنى في حالة شخص آخر هو محاولة لفهم التعليمات البرمجية. برنامج، وهي قائمة من التعليمات،
ويشبه وصفة: غلي الماء، إضافة الشعرية، انتظر 10 دقيقة، واستنزاف والتمتع بها. في نفس الطريق، ويبدأ البرنامج في
يعمل البيان الأول وهبوطا في وقت واحد حتى يضرب النهاية. وحتى الآن، فقد أضفنا رقمين معا. ملل. دعونا نجعل لعبة فيديو بدلا من ذلك! بطبيعة الحال، فإنه من السابق لأوانه التفكير
الترميز لعبة كاملة، لذلك بدلا من ذلك، سوف نقوم استخدام مثالنا لكتابة قصاصات صغيرة من
التعليمات البرمجية التي تغطي بعض أساسيات البرمجة. تخيل أننا بناء ممر المدرسة القديمة
لعبة حيث لديها جريس هوبر للقبض على البق قبل أن ندخل في جامعة هارفارد مارك 1 و
تعطل جهاز الكمبيوتر! على كل مستوى، وعدد من البق يزيد. نعمة أن نقبض عليهم قبل أن تبلى
أي التبديلات في الجهاز. لحسن الحظ، لديها عدد قليل من التبديلات اضافية ل
الإصلاحات. للبدء، سوف تحتاج إلى تتبع
من مجموعة من القيم التي تعتبر مهمة ل اللعب، مثل ما هو مستوى اللاعب في وضع التشغيل،
النتيجة، وعدد من البق الباقين، و كذلك عدد من التبديلات الغيار في غريس
المخزون. لذلك، يجب علينا أن "تهيئة" المتغيرات لدينا،
وهذا هو، تعيين قيمة الأولية: "مستوى يساوي 1، والنتيجة تساوي 0، والبق يساوي 5، ناقلات الغيار يساوي 4، واسم لاعب يساوي "أندريه". لخلق لعبة تفاعلية، نحن بحاجة إلى
السيطرة على تدفق البرنامج إلى أبعد من مجرد يمتد من أعلى إلى أسفل. للقيام بذلك، ونحن نستخدم البيانات التحكم في التدفق. وهناك عدة أنواع، ولكن إذا البيانات
هي الأكثر شيوعا. يمكنك نفكر بها "إذا X صحيح،
ثم القيام Y ". مثال اللغة الإنجليزية هو: "إذا أنا
متعب، ثم الحصول على الشاي " إذا كان الأمر كذلك: "أنا متعب" هو بيان صحيح،
بعد ذلك سوف يذهب للحصول على الشاي إذا كان "أنا متعب" غير صحيح، ثم سأفعل
لا يذهب للحصول على الشاي. بيان IF هو مثل شوكة في الطريق. الطريق الذي كنت تأخذ غير مشروط سواء
التعبير هو صواب أو خطأ – حتى هذه ويطلق تعبير القوائم المشروطة. في معظم لغات البرمجة، تعليمة if
يبدو شيء من هذا القبيل …. واضاف "اذا والتعبير، ثم، بعض التعليمات البرمجية، ثم تنتهي بيان إذا ". على سبيل المثال، إذا "مستوى" 1، ثم نحن
تعيين النتيجة إلى الصفر، لأن لاعب هو مجرد بداية. ونحن أيضا تعيين عدد من الحشرات إلى 1، للحفاظ على
من السهل في الوقت الراهن. لاحظ الأسطر من التعليمات البرمجية التي هي مشروطة
على-بيان إذا المتداخلة بين IF وEND IF. وبطبيعة الحال، لا يمكننا تغيير التعبير الشرطي
إلى كل ما نريد لاختبار، مثل "هو النتيجة أكبر من 10 "5 أو" هو الخلل أقل من 1 ". وإذا، البيانات يمكن الجمع بين بيان ELSE، الذي يعمل بمثابة التقاط كل حالة التعبير غير صحيح. إذا كان مستوى ليس 1، رمز داخل
وسيتم تنفيذ كتلة ELSE بدلا من ذلك، و عدد من الأخطاء التي يجب أن المعركة هي نعمة
تعيين 3 مرات رقم المستوى. هكذا مستوى 2، سيكون من ستة البق، وعلى
مستوى 3 هناك 9، وهلم جرا. لا يتم تعديل النتيجة في كتلة ELSE،
حتى يحصل على نعمة للحفاظ على أي النقاط التي أحرزتها. وإليك بعض الأمثلة من البيانات إذا، ثم، شيء آخر
من بعض لغات البرمجة شعبية – أنت يمكن أن يرى بناء الجملة يختلف قليلا، ولكن
هيكل الأساسي هو تقريبا نفس. إذا، البيانات يتم تنفيذ مرة واحدة، مشروطة
يتم اختيار المسار، والبرنامج ينتقل. لتكرار بعض العبارات مرات عديدة، ونحن بحاجة
لخلق حلقة المشروطة. طريق واحد هو بيان الوقت، وتسمى أيضا
حلقة while. كما كنت قد خمنت، وهذا حلقات قطعة
من التعليمات البرمجية "بينما" هو شرط صحيح. بغض النظر عن لغة البرمجة، فإنها
ننظر بشيء من هذا القبيل: في لعبتنا، دعنا نقول في بعض النقاط،
زميل ودية restocks غريس مع التبديلات! الصيحة! لتحريك له تجديد مخزوننا الظهر
تصل إلى حد أقصى قدره 4، يمكننا استخدام حلقة while. دعونا المشي من خلال هذا الرمز. أولا نحن سوف نفترض أن النعمة لديه 1 فقط
غادر أنبوب عندما يدخل زميلها. عندما ندخل في حلقة في حين، فإن أول شيء
الكمبيوتر لا هو اختبار لها مشروطة … و التبديلات أقل من 4؟ حسنا، التبديلات حاليا 1، لذلك نعم. الآن نحن ندخل في حلقة! ثم، فإننا ضرب سطر من التعليمات البرمجية: "التبديلات متساوين
التبديلات زائد 1 ". هذا هو مربكا بعض الشيء لأن المتغير
تستخدم نفسها في عبارة تعيين، لذلك دعونا فك عليه. كنت دائما تبدأ من خلال معرفة الحق
جانب من توقيع متساوين أولا، لذلك ما تفعله "التبديلات زائد 1" الخروج أن تكون؟ حسنا، التبديلات حاليا القيمة 1، لذلك
1 زائد 1 يساوي 2. ثم، ويحصل على حفظ هذه النتيجة مرة أخرى في
التبديلات متغيرة، الكتابة فوق القيمة القديمة، حتى التبديلات الآن يخزن قيمة 2. لقد ضرب نهاية حلقة في حين، والتي
يقفز برنامج النسخ الاحتياطي. فقط كما كان من قبل، ونحن اختبار الشرطي ل
نرى ما اذا كنا في طريقنا للدخول في الحلقة. هي التبديلات أقل من 4؟ حسنا، نعم، ناقلات يساوي الآن 2، لذلك نحن ندخل
الحلقة مرة أخرى! 2 زائد 1 يساوي 3. حتى يتم حفظ 3 في التتابع. حلقة مرة أخرى. هو 3 أقل من 4؟ نعم إنه كذلك! في حلقة مرة أخرى. 3 زائد 1 يساوي 4. لذلك نحن إنقاذ 4 في التتابع. حلقة مرة أخرى. هو 4 أقل من 4؟ …. لا! وبالتالي فإن الشرط هو الآن خاطئة، وبالتالي نحن
الخروج من حلقة والانتقال إلى أي المتبقية الشفرة. هذه هي الطريقة التي تعمل حلقة في حين! هناك أيضا شيوعا لحلقة. بدلا من أن تكون حلقة التي تسيطر عليها حالة
التي يمكن تكرار إلى الأبد حتى حالة هو زائف، وFOR حلقة يتم التحكم العد.
يكرر عدد محدد من المرات. انها تبدو شيئا من هذا القبيل: الآن، دعونا نضع في بعض القيم الحقيقية. حلقات هذا المثال 10 مرات، لأننا
حدد أن المتغير 'ط' يبدأ في القيمة 1 وترتفع إلى 10. إن أهم ما يميز لحلقة هو أن
في كل مرة يضرب NEXT، فإنه يضيف إلى 'ط'. عندما 'ط' يساوي 10، يعرف الكمبيوتر
انه تم يحلق 10 مرات، وحلقة المخارج. فإننا يمكن أن يحدد العدد إلى ما نريد
– 10، 42، أو مليار – والامر متروك لنا. دعونا نقول أننا نريد أن نعطي لاعب مكافأة
في نهاية كل مستوى لعدد التبديلات فراغ لديهم خلفها. كما يحصل اللعبة أصعب، فإنه يأخذ المزيد من المهارات
أن يكون التتابع غير المستخدمة، لذلك نحن نريد مكافأة لترتفع القائمة على أضعافا مضاعفة على المستوى. نحن بحاجة لكتابة قطعة من التعليمات البرمجية التي تحسب الدعاة – وهذا هو، بضرب عدد في حد ذاته عدد محدد من المرات. حلقة مثالية لهذا! أولا دعنا تهيئة متغير جديد يسمى
"مكافأة" وتعيينه إلى 1. ثم، علينا خلق FOR حلقة ابتداء من الساعة 1،
وحلقات يصل الى رقم المستوى. داخل تلك الحلقة، ضربنا مرات مكافأة
عدد من التبديلات، وحفظ تلك القيمة الجديدة العودة الى مكافأة. على سبيل المثال، دعنا نقول التبديلات يساوي 2،
ومستوى يساوي 3. لذلك لإرادة حلقة حلقة ثلاث مرات، والذي
يعني مكافأة هو الذهاب الى الحصول مضروبا التبديلات … بواسطة التبديلات … بواسطة التبديلات. أو في هذه الحالة، وأحيانا 2 مرات 2 مرات 2،
الذي هو مكافأة من 8! هذا هو 2 إلى قوة 3RD! هذا الرمز الأس هو مفيد، ونحن قد
تريد استخدامها في أجزاء أخرى من التعليمات البرمجية لدينا. انها تريد ان تكون مزعج للنسخ ولصق هذا
في كل مكان، ويجب أن تحديث متغير أسماء في كل مرة. أيضا، إذا وجدنا الخلل، سيكون لدينا لمطاردة
حول وتحديث كل مكان كنا فيه. كما أنه يجعل رمز أكثر مربكة للنظر في. الاقل هو الاكثر! ما نريده هو وسيلة لحزمة حتى الأس لدينا
كود حتى نتمكن من استخدامها، والحصول على نتيجة، و ليس من الضروري أن ترى كل التعقيد الداخلي. نحن نتجه مرة أخرى تصل إلى مستوى جديد من
التجريد! لتجزئة وإخفاء التعقيد، برمجة
يمكن اللغات حزم قطع من الشفرة في وظائف الكشف عن اسمه، وتسمى أيضا أساليب أو الوظائف الفرعية
في لغات البرمجة المختلفة. ويمكن بعد هذه الوظائف يتم استخدامها من قبل أي دولة أخرى
جزء من هذا البرنامج فقط عن طريق الاتصال اسمها. لننتقل كود الأس لدينا في وظيفة! أولا، يجب علينا أن تسمية. يمكن أن نطلق عليه أي شيء نريد، مثل HappyUnicorn،
ولكن منذ رمز لنا بحساب الأس، دعونا نسميها الأس. أيضا، بدلا من استخدام أسماء المتغيرات محددة،
مثل "التتابع" و "المستويات"، نحدده أسماء المتغيرات العامة، مثل قاعدة والتصدير
قيمها الأولي ستكون "مرت" في وظيفة لدينا من أي جزء آخر من
البرنامج. بقية من التعليمات البرمجية لدينا هو نفسه كما كان من قبل،
مدسوس الآن إلى وظيفتنا ومع الجديد أسماء المتغيرات. وأخيرا، نحن في حاجة الى ارسال نتيجة لدينا
كود الأس إلى جزء من البرنامج التي طلبت ذلك. لهذا، ونحن نستخدم بيان عودة، وتحديد
أن يتم إرجاع القيمة في 'نتيجة'. لذلك لدينا كود ظيفة الكامل يبدو مثل هذا: الآن يمكننا استخدام هذه الوظيفة في أي مكان في منطقتنا
البرنامج، ببساطة عن طريق الاتصال اسمها ويمر في رقمين. على سبيل المثال، إذا كنا نريد لحساب 2
قوة ال44، يمكننا أن مجرد دعوة "الأس 2 فاصلة 44. " ومثل 18 تريليون يعود. وراء الكواليس و 2 و 44 الحصول على حفظها في
المتغيرات قاعدة والتصدير داخل وظيفة، انها تفعل كل الحلقات لها عند الضرورة، ومن ثم
ترجع الدالة بالنتيجة. دعونا نستخدم وظيفة المسكوكة حديثا لحساب
مكافأة النتيجة. أولا، نحن تهيئة مكافأة ل0. ثم نتحقق إذا كان لاعب لديه أي المتبقية
التبديلات مع-بيان إذا. إذا فعلوا ذلك، فإننا ندعو وظيفة الأس لدينا،
يمر في التبديلات ومستوى، والتي تحسب التبديلات لقوة مستوى، والعوائد
ونتيجة لذلك، وهو ما حفظ في المكافأة. قد يكون هذا الرمز وحساب مكافأة مفيدا في وقت لاحق، لذلك دعونا الانتهاء من ذلك بوصفها وظيفة للغاية! نعم، باستدعاء وظيفة أن وظيفة! وبعد ذلك، انتظر …. يمكننا استخدام هذه
وظيفة في وظيفة أكثر تعقيدا. دعنا نكتب واحد أن يحصل على استدعاء كل مرة
لاعب انتهاء المستوى. ونحن سوف يطلق عليه "levelFinished" – فإنه يحتاج
لمعرفة عدد من التبديلات اليسار، أي مستوى كان، والنتيجة الحالية. تلك القيم
يجب أن تحصل مرت في. داخل وظيفة دينا، ونحن سوف حساب
مكافأة، وذلك باستخدام وظيفة calcBonus لدينا، وإضافة ذلك إلى درجة التوالي. أيضا، إذا كانت النتيجة الحالية أعلى من
النتيجة اللعبة عالية، ونحن حفظ مستوى جديد تسجيل واسم اللاعبين. وأخيرا، نعود النتيجة الحالية. الآن نحن نحصل على يتوهم جدا. وظائف يدعون ظائف يدعون
المهام! عندما ندعو سطر واحد من التعليمات البرمجية، مثل هذا
تعقيد مخفيا. ونحن لا نرى جميع الحلقات الداخلية و
المتغيرات، ونحن فقط نرى نتيجة أعود كما لو كان السحر…. على درجة من مجموع 53. ولكنها ليست سحرية، انها قوة
التجريد! إذا فهم هذا المثال، فإنك فهم قوة من الوظائف، وجوهر كامل البرمجة الحديثة. انها ليست مجدية في الكتابة، على سبيل المثال،
متصفح ويب كقائمة واحدة طويلة gigantically البيانات. وسيكون من الملايين من خطوط طويلة والمستحيل
لفهم! بدلا من ذلك، البرنامج يتكون من آلاف
وظائف صغيرة، كل منها مسؤولة عن ميزات مختلفة. في البرمجة الحديثة، فمن المألوف أن
رؤية وظائف لفترة أطول من حوالي 100 خط من التعليمات البرمجية، لأن في ذلك الحين، هناك على الارجح
وهو الأمر الذي يجب أن انسحبت وقدمت في وظيفتها الخاصة. Modularizing البرامج في وظائف لا
يسمح مبرمج واحد لكتابة كامل التطبيق، ولكنه يسمح أيضا فرق من الناس للعمل
بكفاءة على برامج أكبر. يمكن للمبرمجين مختلفة تعمل على مختلف
وظائف، وإذا كان الجميع يحرص على الرمز يعمل بشكل صحيح، ثم عندما يكون كل شيء
وضعت معا، ينبغي للبرنامج بأكمله العمل أيضا! وفي العالم الحقيقي، والمبرمجين ليسوا
إضاعة الوقت في كتابة أشياء مثل الدعاه. لغات البرمجة الحديثة تأتي مع ضخمة
حزم من وظائف مكتوبة مسبقا، ودعا المكتبات. هذه فقد كتبت بواسطة المبرمجون الخبراء، جعلت كفاءة
وخضعت لاختبارات صارمة، ومن ثم تعطى للجميع. هناك مكتبات في كل شيء تقريبا،
بما في ذلك الشبكات، والرسومات، والصوت – المواضيع سنناقش في الحلقات المقبلة. ولكن قبل أن نصل إلى تلك، ونحن بحاجة الى التحدث
حول الخوارزميات. مفتون؟ يجب ان تكون. سوف اراك الاسبوع المقبل.

Learn About RBI Functions In Hindi – Complete Course on Economics By Aman Srivastava

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RBI Functions – In this lesson Aman has discussed in detail about the RBI and has explained various functions related to RBI. The Preamble of the Reserve Bank of India describes the basic functions of the Reserve Bank as: “…to regulate the issue of Bank Notes and keeping of reserves with a view to securing monetary stability in India and generally to operate the currency and credit system of the country to its advantage.” Learn more about its functions in this lesson which is a part of a complete course on Economics.
You can find the entire course here:

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C Programming Tutorial | Learn C programming | C language

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C Programming Language is the most popular computer language and most used programming language till now. It is very simple and elegant language.
1) This is by far the most comprehensive C Programming course you’ll find here, or anywhere else.
2) This C Programming tutorial Series starts from the very basics and covers advanced concepts as we progress. This course breaks even the most complex applications down into simplistic steps.
3) It is aimed at complete beginners, and assumes that you have no programming experience whatsoever.
4) This C Programming tutorial Series uses Visual training method, offering users increased retention and accelerated learning.

Every programmer should and must have learnt C whether it is a Java or C# expert, Because all these languages are derived from C. In this tutorial you will learn all the basic concept of C programming language. Every section in this tutorial is downloadable for offline learning. Topics will be added additional to the tutorial every week or the other which cover more topics and with advanced topics.
This is we will Learn Data Types, Arithmetic, If, Switch, Ternary Operator, Arrays, For Loop, While Loop, Do While Loop, User Input, Strings, Functions, Recursion, File I/O, Exceptions, Pointers, Reference Operator , memory management, pre-processors and more.

#Ctutorialforbeginners #Ctutorial #Cprogramming #Cprogrammingtutorial #Cbasicsforbeginners
c tutorial for beginners. C programming tutorials for beginners. C Programming Language Tutorials

Time: 00:12:35 – Lesson 2 – C programming introduction and first ‘hello world’ program
Time: 00:25:45 – Lesson 3 – simple input & output ( printf, scanf, placeholder )
Time: 00:41:07 – Lesson 4: Comments
Time: 00:44:32 – Lesson 5 – Variables and basic data types
Time: 00:52:41 – Lesson 6 – simple math & operators
Time: 1:00:00 – lesson 7 – if statements
Time: 1:09:00 – lesson 8 – if else & nested if else
Time: 1:20:00 – lesson 9 – the ternary (conditional) operator in C
Time: 1:28:56 – Lesson 10 – Switch Statement in C
Time: 1:43:35 – Lesson 11 – while loop
Time: 1:52:24 – Lesson 12 – do while loop
Time: 2:01:14 – Lesson 13 – for loop
Time: 2:11:25 – Lesson 14 – functions in C
Time: 2:22:54 – Lesson 15: Passing parameters and arguments in C
Time: 2:31:40 – Lesson 16: Return values in functions
Time: 2:41:33 – Lesson 17: scope rules in C
Time: 2:51:08 – Lesson 18: Arrays in C
Time: 3:02:28 – Lesson 19: Multidimentional arrays in C
Time: 3:12:33 – Lesson 20: Passing Arrays as function arguments in C
Time: 3:24:54 – Lesson 21: Pointers in C
Time: 3:35:36 – Lesson 22: Array of pointers
Time: 3:43:38 – Lesson 23: Passing pointers as function arguments
Time: 3:57:44 – Lesson 24: Strings in C
Time: 4:12:17 – Lesson 25: (struct) structures in C
Time: 4:27:10 – Lesson 26: Unions in C

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Lecture 0 – Introduction to Computer Science I

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This is first lecture from the series of course “Introduction to Computer Science I”, Harvard OpenCourseWare with Instructor David J. Malan.
The Instructor is just awesome and this course is most taken, most awaited. Surely it will make you better understand Computers and Computer science.
The topics covered in this lecture are: Intro to Binary. ASCII. Algorithms. Pseudocode. Source code. Compiler. Object code. Scratch. Statements. Boolean expressions. Conditions. Loops. Variables. Functions. Arrays. Threads. Events.
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