 Hi, I'm Zor. Welcome to Unizor Education. Today we will talk about differential along a curve. Well, presumably we know what differential actually is a function. So this is basically the build-up from that concept. This lecture is part of the advanced mathematics course for teenagers and high school students presented on unizor.com. So all the lectures are accessible from that website as well as very detailed notes for each lecture and for registered students. There are some exams which they can take and I do recommend actually to watch the lecture from from this website from unizor.com and I do recommend to sign in to participate in the educational process which includes like exams results which your supervisor can take a look at etc. And the site is free by the way. All right, so we will talk about differential along a curve. First of all, let's just recall that we will use the definition of a curve as basically a parametric pair of functions. Basically, t is a parameter x and y are coordinates of a point and as t is changing the value within certain interval from from a to b then x of y and t and y of y no sorry x of t and y of t are smooth functions which result if you have some kind of a coordinate system in some kind of a curve which can be anything actually. So it's not like a function where for each x you should have only one particular value of y. Now the curve given parametrically is basically it can loop, it can intersect itself etc. So it's much more broader kind of a definition of a curve. Well, obviously the purpose is to analyze movements on the plane which will be at the end of this lecture very very detailed addressed. Okay, so we will talk about parametric curve. Now differential along the curve. Well, basically the purpose of this is to talk about the lengths of the curve. Well, let me just make the picture much simpler and let's talk about the lengths of the curve. So let's consider this is your curve. Now, as t is changing from a to b or whatever else interval it's defined, the x and y coordinates are changing from this point to this point and now we are interested in the lengths of the curve. Well, obviously, we will do the same thing as we did many times before. We will just divide into segments and each piece of curve we will replace with a segment, a straight segment, right? So let's say this is my t0, actually x of t0, this is x of t1, this is x of t2, etc. This is x of tn. There are some points in between and this is correspondingly y of tn and let's say this is x of tn plus first and y of tn plus first plus one. Okay, so this is a segment which we would like to use as approximation of this curve. Now, you remember from the concept of differential of a function, we were talking about if you have some kind of a tangential line, then this is, this is the value of the function and this is the value of basically dx times derivative, this piece actually. So we're talking about the difference between the value of function at this point and the value of function at this point. So this is the difference, this is delta y and this is a differential which is dy, which is equal to derivative times dx, right? And we were talking about that this little piece, which is the difference between real increment of the function and its differential is infinitesimal variable when this point goes to this one of a higher order than the increment differential dy itself. And since it's a higher order, then in the limit we can basically replace one with with another. Well, same thing here. The difference in length between this curve and this segment, whenever we are getting these two points closer and closer, is an infinitesimal variable of a higher order than the length itself. And basically this is the foundation for something which in the future we will learn as integration. So whenever we are summing up all these little segments and compare it with the lengths of the curve, the difference will be of an infinitesimal of a higher order. And that's very important. So as it is customary, as we have already done many times, what we will do is we will approximate the lengths of the this small piece of a curve with this particular lengths along the along the straight line and we will go to a limit when these two points are getting closer and closer. Now, what actually have we done here? We have divided this interval where our parameter is defined into small pieces, t0, t1, etc., etc., up to t something like N, capital N. And what we are actually talking about right now is we are getting this division more and more dense, which means that the number of points is going to infinity and the difference between the points, well, let's say maximum difference between the points is going to zero. So they are all getting closer and closer. How can we do it? Well, we cannot actually like divide in half, then each half divide in half, etc., etc. I mean, there are many different processes when we will achieve this type of more and more fine division of this interval between these points. And as it actually going on, my corresponding xN, or x of tN and x of tN plus 1 are getting closer and closer. Same thing with y. We are obviously assuming our functions are smooth enough. We are talking about smooth curve, no gimmicks, no crazy stuff. So the smooth behavior dictates that whenever my delta tN goes to zero, my delta x of tN also goes to zero, which is the difference between this and this. And obviously the same thing for y. For any index N, because all of them are getting smaller and smaller, all the intervals between these t. And therefore, all the intervals between x and all the intervals between y. Now, why do I need these two? Well, very simple. Because if these are two points, then this is delta xN and this is delta yN, right? Because between y of tN plus 1 and y of tN, that's the delta yN. This one. Delta yN is basically delta y of tN. That's what I actually meant, obviously. And if that's the case, we can obviously calculate the lengths of the segment, right? So what's the lengths of the segment? By Pythagorean theorem, delta s of tN is equal to square root of delta x of tN, square plus delta square y of tN. So this is the lengths of the segment, this one. And we are actually saying, again, same thing as we were saying before, that this length of segment approximates the lengths of the curve. And it's a good approximation. Good in terms of, if I will make this division of these points denser and denser, which means if my delta tN goes to zero, now we are talking about infinitesimal variables because each of these is infinitesimal. And what's most important that, as far as infinitesimals are concerned, the difference between this length and this length is infinitesimal of a higher order than this one. So we are actually replacing this process, using the letter D instead of Greek letter delta. It assumes that we are talking about infinitesimal variables. So we are talking about ds of t. I will omit the tN because it's basically for any t, right? We can say that this is dx of t square plus dy of t square, right? And this is called differential along the curve, basically. Whenever these are differentials along the x-axis and along the y-axis, this is differential along the curve. And now you can say this is by definition, if you wish, because all whatever I was just explaining about is just the base for something like this and some justification for basically saying that's what it is, right? All right. Now, we understand that whenever we are talking about this d instead of delta, we are not talking about some kind of fixed increment. We are talking about a process when each of these is infinitesimal variable and it's all based on the delta tN to be an infinitesimal variable which we usually denote as dt, right? Now, let's go a little bit further back. Now, what is differential of the function? This is just a plain function, right? Well, obviously, differential of the function, as we know, is equal to, let me put it here, dx of t is equal to derivative of t times dt, right? This is another definition we actually came up with before, right? That's just a plain function. So, if I was substituted here, what do I have? I will have square root of x of t square plus y of t derivative here, derivative there, two derivatives, and dt can go outside of the square root. So, I will have here. So, this is basically another representation of the differential along the curve for a function, for a curve given as a parametric pair of some parameter t, okay? Fine. Basically, this is it about differential along the curve. This is my final kind of a formula. It's obvious because, you know, the theorem, Pythagorean theorem is kind of a simple thing, and this is just a plain consequence from the definition of the differentials, right? Now, what I would like actually to say is, and now we are going to application of all these mathematics, let's actually consider some physics behind it, all right? And I understand this is the course of math, but, you know, math actually exists for some purpose. Sometimes it's just an exercise of the mind, and sometimes it has practical applications, and this is definitely has practical applications. So, let me just go with this. So, differential along the curve can be considered as an infinitesimal piece of a curve, length of the piece of the curve. Whenever the edges are getting closer and closer to each other, right? So, if this is the curve, so this piece, this piece has certain lengths, and this length is obviously some kind of a function of its edges and these functions, but whenever these two edges are getting closer and closer, then the length gets closer and closer to this particular expression, where dt is the increment of the argument from here to here. Okay, now we're talking about physics. Well, it's basically about velocity and speed. If you have a point moving along this curve, this trajectory, usually S of t is lengths covered from t equals to zero to some kind of a time t, right? So, if this is the beginning of the movement, then S of zero obviously would be equal to zero. This is S of t, and this is S of t plus dt, and I'm using now dt as infinitesimal increment of the argument. So, this length is S of t, this length is S of t plus dt, and the difference between them, which is dS of t, is the difference between the lengths of, we have covered during the time t plus dt minus lengths we have covered to the point t. So, this minus this, and as we know, this is this, right? I don't want to write it again. Now, the concept of velocity and speed, two different concepts, by the way. Speed is absolute value of speed, and velocity is a vector. So, the vector of velocity has two characteristics. It's absolute value and direction. So, the absolute value of the velocity is the speed, and direction, what is direction of vector of velocity? That's direction of the movement, right? Which is always tangential line to the trajectory, right? So, it's tangential to trajectory. That's the direction of vector, and its absolute value is the speed. Now, how do we calculate the speed? Well, if it's a movement along the straight line and the speed is constant, then you have the distance, and you divide by time, right? Generally speaking. Well, if it's a curve, and the speed is different, and we don't really know what it is, but we do know how much we have covered at the point t. We know the function s of t. Well, we don't know exactly the s of t, but we can actually have the coordinate of the point at any point. So, what is the approach in this case, if everything is non-constant? Well, we have a small increment of time from t to t plus delta, t plus dt, and we calculate the average speed on that particular period of time, right? So, the average speed would be equal to the distance between these two points divided by increment of the time, right? And again, I'm using d here because I assume that we are dealing with two infinitesimal variables. I'm interested with the speed at point t. So, I step forward from t forward to t plus dt, calculate the average speed. But now, since dt is infinitesimal, it goes closer and closer. Whatever the limit is, is basically the speed at that particular point. That's a reasonable definition of the speed at point, if the speed is changing all the time. All right. Now, you know this expression for differential of the distance, right? So, we can say that this is, right, if I divide it by t. So, we have the expression of the absolute value of the speed. Now, let's think about velocity. You know that we can always represent vector as some of other vectors. There is a parallelogram rule, right? So, what I will do in this particular case is I will represent my vector, which goes along the tangential line as two components. One is parallel to x, and I call it vx of t. And another is component projection on the line parallel to y-axis, and we will call it y, not x, but t. Now, let's think about what is this vector? Well, obviously, this vector is along the x-axis, right? This is the same direction, because that's how I made it. And what is actually its absolute value? Well, this is an increment of x from here to the next point, right? So, if I divide it by the same rule, if this is my t plus dt, then this is my dx, and I divide it by time. So, it would be actually the speed in the direction of the x-axis, right? So, x of t is actually this vector. It's absolute value, if you wish. And correspondingly, y of t is the value of that other vector, the speed along the y-axis. And again, the Pythagorean theorem actually here is exactly the same, right? So, we can say that v is a sum of two vectors, this and this. This vector is horizontally directed, and its value is equal to derivative of the x-coordinate by time. And this vector is vertically directed along the y-axis, and its value is equal to y-coordinate, derivative of y by time. So, this is something which is used in physics a lot. Whenever you solve the problems, you, in many cases, it's very helpful to basically represent your speed as a sum of two components. One goes vertically, and there are some functions y of t which direct this movement, and another is horizontal, which is another component, the x of t, whatever, dictates the horizontal movement. For instance, vertical movement can be related to the force of gravity, and horizontal movement can be basically some kind of a direction forward, which is defined by initial force, initial speed, or whatever. So, it's very important to understand that this differential along the curve actually leads to some very important physical concepts like speed and velocity. And I'm not talking about the average speed now, average speed like everybody knows what it is, and it's not interesting because it's average. What's more interesting is a speed at any particular point. So, if you have a curve defined by this, you can actually having this type of differential and derivatives, you can actually calculate the speed at any particular moment t, not the average speed. So, for this, you need derivatives. So, basically, from the Newtonian time, physics always uses these concepts of derivative and differential and integral as well, and Newton and some other people were actually in the beginning of creating calculus from the need to address these issues like what is our speed at any particular time when it's changing. So, just a little view maybe into the course of physics, which I'm just thinking about right now, where I would actually to approach rigorously such concepts as speed using the differentials and calculus in general, because that's exactly how it's supposed to be done. That's exactly the laws which govern the Newtonian mechanics. Okay. So, differential along the curve, that's it. That's all I wanted to talk about today. I do recommend you to go to the website Unisor.com and read the notes to this lecture. They are quite detailed. Basically, it's like a textbook if you wish. And I do encourage you to sign in as a student with some supervision or maybe self-supervision and take exams, for instance, wherever it's available. It's not available for all lectures or all topics which I'm addressing, but it is available in many algebra and geometry, for instance, topics. So, do that. Other than that, that's it. Thanks very much and good luck.