 Hi, I'm Zor. Welcome to a new Zor education. I have decided to spend some time to introduce just another notation, which is used in physics of electromagnetic field and some other places. You see, when in the course of scientific development people came with some new notion, new approach, etc. They usually invent new symbolics. Newton, for instance, and Leibniz have invented the notation for differentiation. Now, whatever I'm going to introduce right now is just basically a continuation of that path. It's basically a differentiation on one next step. It's just a notation, but notation is very useful in a sense that it short-terms the formula. It makes it probably easier to remember if somebody remembers these formulas. Also, it will help us to express the equations of Maxwell for electromagnetic field in a concise and short form. Again, it's mostly notation. At the same time, I will use this particular notation to introduce a certain concept and you will see how useful it is in this particular case. Obviously, we can do this out. However, again, it's a convenience, and why not? Now, this lecture is part of the course called Physics for Teens, presented on Unisor.com. I do suggest you to watch this lecture from the website because this lecture is part of the course, which means there is a menu, there are some parts of the course, there are topics, there are lectures, there are exams. Every lecture has textual supplement, which basically is like a textbook. Everything is totally free. There are no advertisements. There is pure knowledge, no strings attached. You don't even have to sign in if you don't want to, basically, to take the course and even to take exams, although exams will not be remembered maybe for the next time you would like to take them if you don't sign in, obviously. In any case, let's just go in. Now, this lecture is called Gradient. Gradient is something which I will end this lecture with. So, I will start introducing certain notions, certain notation, certain symbolics, and then I will use it to introduce a concept called Gradient. Okay, so, first of all, what exactly is I'm going to introduce? Only one particular symbol. Well, it looks like Greek letter delta upside down, capital letter. So, the real capital letter is a triangle. That's a delta, capital delta in Greek, and this is just upside down. Now, usually it's called Nabla. I don't know why. I don't know who invented this particular symbol, but what it means is Nabla is basically a triplet of operators' differentiation. So, what is this? First of all, this is partial derivative by argument x. This is partial derivative by y. This is partial derivative by z. If you do not remember what partial derivative is, then you shouldn't really watch this lecture, go back to calculus. In the same website, by the way, Unisor.com has math 14 scores, and there is a very big chapter called calculus, and there are partial derivatives introduced there. So, basically it's four functions which depend on, in this particular case, three arguments. And we will actually use today only functions of three arguments. Why three arguments? Because our world is, well, in geometrical sense, three-dimensional. Let's not talk about zero relativity and time as a fourth dimension. So, our geometrical world is considered to be Euclidean and Cartesian coordinates in it and coordinates are x, y, z. So, that's why we are considering that there is some kind of a frame of reference and there is a function of three arguments. And relative to this function, we do this partial derivative. Partial derivative of this function by x, by y, and by z. So, these are three operators. It's not functions yet. It's three operators. But if we apply this to this particular function, and notation is this, then what it means actually, it means df of x, y, z, by dx partial derivative by x of this function, df of this function by y, and df of this function by z. Triplets of different functions which are three partial derivatives. That's what this actually means. Now, another important concept is a vector. Now, what do we usually consider to be a vector? Well, a triplet of numbers. Usually, for example, if you have again three-dimensional world and you can have a vector from the beginning of coordinates to a particular point p with coordinates x, y, and z, and this would be a vector. So, this vector would be actually x, y, z, right? Coordinates of this point. Now, so again, we used to think about the vector as a triplet of numbers. Well, in this particular case, it's triplet of real numbers. Coordinates are real numbers. Positive or negative, whatever. Obviously, it's a stretch, but why don't we call a set of three of anything a vector of three-dimension? In this particular case, it's three operators. In this particular case, it's three functions. For each specific x, y, and z, these three functions will be three numbers, right? But generally speaking, if I just write it like this, it's three functions. Why don't we call this a vector? It's a vector of functions. This is a vector of operators. And then there are vectors of real numbers. I mean, they're all kind of vectors. Well, not exactly, obviously, because, for example, you can add two vectors of numbers and get another vector of numbers. Can you add two vectors of functions and get a vector of function? Well, maybe if you have something like one would be f1 function, f2 function, and f3 function, it's a vector of functions. And you have another vector of functions, which is g1, g2, and g3. Maybe you can add them together and have a new vector of functions, f1 plus g1, f2 plus g2, f3 plus g3. Yeah, maybe, why not? So, I mean, can you multiply this vector of function by number, by k? Yes, you can multiply the vector of functions by k, put k in front of each one of them, and that would be a multiplication. You will have another vector of functions. And linear combinations would work, et cetera. There is nothing wrong with calling these a vector. Sometimes they're called pseudo vectors, just to emphasize that they're different from traditional vector. Or, again, vector of operators, which means a set of, in this case, three operators, vector of functions, a set of three functions. In this case, the word vector and the word set are basically equivalent. Okay, so, we have introduced a new concept called a vector of operators. These three operators of differentiation are combined together, and they're called by a new symbol. It's just a symbol, nothing else. I mean, I can use this one, but this is shorter and much nicer, obviously, and easier to remember, right? So, basically, so far, I've just introduced a new symbol. Now, what's next? Okay, next is we are going to a specific example of how this thing can be used to get some real physical sense in it. Okay, so let's consider that you have this function. Now, these x, y, and z would be three dimensions, three coordinates. So, x, y, and z define a point in space, in some coordinates, in some quotation coordinates. Okay, so we have x, y, z as a point in space. What can be a value of the function which is defined in this and many other points in space? Well, for example, air pressure. Okay, that's some number which is associated with every point. Air pressure here is one number, air pressure there is another number, and here we have a function of three arguments. Arguments are coordinates of the point, and f is a pressure at that particular point. What else? Well, level of radiation at that particular point, or electric potential in that particular point. So, you know that the space can be basically the electromagnetic field, and if you have, let's say, a charge at one particular point, then at every other point in space around it, you can introduce a concept of electric potential, which is basically a work required to bring a unit positive electric charge from infinity to that particular point, or from this particular point to infinity, depending on the sign, et cetera. So, there are different things which can be actually modeled as a function of a location, function of three arguments, and these functions are exactly what we are considering right now. Now, we also consider these functions to be smooth enough so the differentiation works. I mean, there are some functions which are not differentiable at all, but we are considering nice case when, let's say, the pressure is changing from here to there gradually. So, there is something like a speed of change along any direction. Or what's the direction of the maximum change, et cetera. Okay, so we are assuming that this is a physical function, physical characteristic of a point at space. So, let's say we have a point in space. Now, what we are interested in is, if this function represents, well, let's talk about pressure, air pressure. Now, if from this function I go to this direction, air pressure is changing, obviously somehow. If I go from here to there, it's also changing, but differently, and I go to there, it's still different. Now, my task is to find a direction of the greatest change of the pressure, because why is it important? Well, most likely, if this is a direction of the greatest change, then most likely the masses of air would be moved to this or this direction, depending on which one is greater, which one is smaller, but it's along the greatest change that's where most of the air will go. Because a change from here to here is greater than from here to there, then the air will be more likely going into the direction of the greater change, because the force will be greater. What's the pressure? The pressure is basically a force on molecules. So if the pressure is greater, then the molecules will move that way. So it's important for, let's say, weather forecasting, just an example. There are many other examples. So we are interested in the direction of the greatest change, greatest growth or greatest drop, whatever. Now, what is a change? Change is a derivative, basically, right? Okay, so let's just do it one thing at a time. So what we are talking about is, we would like to find a vector r, which is infinitesimal increments from this location to the next location. These are very small, well, infinitesimal, obviously, in a limit case. So basically what I am assuming is that from point p we go to point q with location x plus delta x, y plus delta y, z plus delta z. So I move from here to here, and I will check what's the difference of my function when I move from here to here. And then I will try to find such delta x, delta y, and delta z to maximize that difference. Generally speaking, right? That will be the direction of the greatest change, right? So I will find delta x, delta y, and delta z in such a way that to maximize the change of value of the function from p to q. So basically I have to find f of q minus f of p, and find the maximum. So this is basically f of x plus delta x, y plus delta y, z plus delta z, minus f of x, y, and z. That's my difference in the value of the function as I move from p to q. I don't need this anymore. So I have to maximize this. I have to find delta x, delta y, and delta z in such a way that this would be maximum. Assuming that they are really infinitesimal because I would like to know the direction from the point. My increment should be as small as possible. All right? Okay, fine. So let's just think about how to do it. Now this is just plain arithmetic. Now this would be equal to... Now instead of incrementing all three arguments together, I will increment them one by one. First I will increment the x. So I will put f of x plus delta x, y, z. Don't change y and z. Only the x. Minus... f of x, y, z. Plus. Now this I will increment by y. x plus delta x, y plus delta y, z. Don't touch the minus f of this. x plus delta x, y, z. Plus. Now I will increment the third component. f of x plus delta x, y plus delta y, z plus delta z. Minus f of x plus delta x, y plus delta y, z. Now you see that in this particular case this would be cancelling this, this would be cancelling with this, and I will still have all three minus initial, right? So that's equality. Fine. Why did I do that? For a very simple reason. If you remember with functions of one argument, let's say you have a function of x, okay? And it's somehow changing, okay? Now if you would like to know f of b minus f of a, if this is a and this is b, I would like to know this difference. Now remember what derivative is. Now there is a point here such that if I multiply a minus b, no b minus a, sorry, b minus a by derivative at this point c, there is a theorem about this basically in calculus. It's some kind of average growth, right? So there is some kind of a value, some kind of a point c that the value of derivative would actually deliver this. Because what is derivative? It's a tangent, right? So that's basically, and this is the difference in function. This is the difference in argument. So this piece is f of b minus f of a, and this piece is b minus a. So if I multiply this by tangent of this angle, so my question is to find an angle. If these points b and a are very close to each other, then c would be in between, which means also very close to each other. So it's approximately b minus a times derivative at point a, let's say. So considering b and a are infinitesimally close to each other, that would be an exact equality, right? So the difference would be an infinitesimally of a greater order, something like this. So let's not go into the details of this. It's a known fact from calculus. Again, if you've talked about this, go back to the calculus lecture, which there is a lecture about this particular quality. There is a theorem about this. So I will use it, all right? And again, I will use the fact that partial derivative is a derivative by one argument only. So right now my y and z, for instance, are not changing at all. So they are constant. And this is basically a function of one argument, and the difference between two values of function are equal to the difference between their arguments, x plus delta x and x, that's the difference. And derivative by x, and this is the partial derivative, at point one or another doesn't really matter because these points are infinitesimally close to each other, by dx. That's what it is. Now what is this? Similarly, x plus delta x here and here, and z here and here, are constant. And the only difference is in argument y. So that would be my delta y. And again, partial derivative of x, y, z, y in this particular case. And the third argument. Now why are we using x instead of x plus delta x? Because they are very infinitesimally close to each other, so it doesn't really matter. As delta x would go to zero, this function of x plus delta x would be exactly the same as function of x, right? And now the same thing with the third one. Now we have only z, which is changing. And these are constant. So basically it would be change of the argument. Argument is z, not y, times partial derivative by z. And again, I'm not using this partial derivative. It's a function. So in theory, I should have taken the value of this function at point x plus delta x and y plus delta y. But again, as delta x and delta y are presumed infinitesimal variables, they are closing to x and y correspondingly. I can use x and y. Okay. Now look at this. And look at this. And look at this. Vector r is delta x, delta y, and delta z. That's a vector of increment, deviation from the initial point. So basically if my P is a vector, then P plus r is equal to Q, right? P is a vector from the origin of coordinates to point P. r is displacement, infinitesimal displacement into some direction. I don't know which direction. Depending on these deltas. And the Q would be the final point where I'm moving to. And this is the difference between value of this function f at point Q minus value at point P, right? So let me just rephrase it. I will put nabla f of x, y, z point r. Now this is a triplet of functions, right? If I put nabla f, means f is here, here and here. So it's three functions. Well, three functions is not exactly a vector in our traditional understanding. But let's just think about this as a vector. Besides, for every fixed x, y and z function has certain values, three numbers. So basically it can be considered as a vector, as a vector of numbers basically. For each x, y and z, we have this would be at three functions and value of these functions at point x, y and z. At r is also some kind of a vector here. And this is a scalar product of two vectors, that product, right? Because this is exactly a scalar product. This is first component of the first vector, times second component of the first vector, times second component of the second vector and the third component of the first vector times third component of the second vector, right? So that's a classical expression for a scalar product. And that's why it's very convenient. Now, what do we know about this? We know that this is equal to absolute value of delta f, absolute value of vector r, time cosine of angle between them. That's basically the absolute value of this scalar product. Plus or minus, I don't really care right now. We need to change, doesn't matter to change, greater or less. So this is a final expression of absolute value of the change between whenever I move my location from p to q. When is it maximum? Well, obviously when cosine is equal to zero, right? Which means vector r should be collinear with vector delta f. And that's basically the answer to our problem. Remember the problem? Where exactly is the direction I have to basically move or something where the function changes the most? Well, the answer is direction is this vector, this vector. Which is actually df by dx, df by dy, df by dz. So this is a vector, vector of three functions. But for every specific location at any specific point p where x, y and z are fixed, these three functions have certain values. So this is a vector in a real traditional sense. So for every point x, y and z, knowing these three functions, I calculate the value of these three numbers. And that's the vector of my nabla f. And that's exactly the direction I have to change my location if I want to change in a direction of the greatest change of the value of the function. Now, this vector is called gradient of f of x, y, z. So if I take from the function three derivatives, getting new three functions, then this would be, this triplet of these functions would be a gradient because it defines for each location, it defines a direction where the value of the function changes the most. So this is called gradient. There is an abbreviation, grad f of x, y, z, which is this or another f of x, y, z. Now, I would like to point that there is nothing in between. There is no dots. There will be a dot that will be another operator. That will be an extension. So this is a concept we have introduced from the formal standpoint. But now you see there is a very physical, very tangible application of this. So if f describes, let's say for example, a pressure, air pressure, then the gradient of f basically describes the direction of the most change in the pressure and most likely that would be direction of the wind, at least general direction. And at each point it can go in different direction, obviously. We are talking about at each point. The gradient is a vector which is defined for each point. It's a triplet of functions. And for each point we can calculate the value of these three functions for each x, y, z. And that gives me a real numerical vector which shows the direction of the greatest change, greatest growth or greatest drop in value or whatever. The greatest change is the greatest. Well, that's it. That's it for today. Now, let me just give you a little bit preamble for the next things. Now, the nubla is used in electromagnetic equations of Maxwell very significantly. Maxwell himself did not use it. That's the latest development. But later, not latest, later development. But it's convenient. And that's why after I have recorded the lectures about three different Maxwell equations, and that was using nubla, actually, but they didn't go into the details of what specific needs for this. So in this particular case, nubla applied to a scalar function. Now, f is a scalar function, which means for x, y, z, I have only one value, f, like pressure, for instance. But if my function is not a scalar but a vector, for example, for each x, y, z, I need a direction of the wind. That's a three-component vector, right? So that would be not a scalar, it would be a vector field. In this case, f is a scalar field. And we can apply operator nubla to a scalar field to get vector functions. Now, we can apply operator nubla differently to a, let's say, vector and get something else. That's called divergence, and that would be the next lecture. And then something else as well, there will be a curl of the vector field. And that would be also another lecture. But these lectures, I have decided to record as well and put them into the menu of the course prior to the Maxwell's equation. Because I just realized that it's important to talk about these relatively mathematical, less maybe physical concepts that do have a physical application, a real mathematical concept to support my Maxwell equations calculations, derivations, etc. Okay, that's it for today. I do recommend you to read the notes for this particular lecture. So it's physics for teens. You go to part which is called waves. Then the topic would be after that field waves. And the gradient would be one of the lectures in that particular department. That's it. Thank you very much and good luck.