 Hi, I'm Zor. Welcome to a new Zor education, excuse me. Today we will continue studying the characteristics of fields. We actually started this in a couple of previous lectures. The characteristics which we have already learned about was gradient. Gradient is basically the magnitude and direction of the largest change of the field. So if we will use, for example, the air pressure or density of air molecules, if you wish, as a scalar field, it's a scalar because it's one particular value at each point in a three-dimensional world. Then the gradient is which direction and by how much this density is growing. Density obviously relates to the pressure. So the gradient of the air density is basically the direction of the wind where the wind goes and it's a vector. So gradient basically converts a scalar field which is density of the air molecules into a vector field. At any point in space we know the direction of the wind, so that was a gradient. I will use the same comparison, the same example, if you wish, of the field. Air pressure, density of air molecules and wind for yet another characteristic of the field which is called divergence. Now, again, back to the previous lecture where we were using gradient. Let me just remind you that the gradient of the scalar field is a scalar which is defined for every point in the three-dimensional Cartesian coordinates. So the gradient of this grad is basically a vector, as I was saying, it's the vector of the wind, where the most significant change of the air molecule density is. So that's partial derivative by x, that's the first coordinate, second coordinate and the third coordinate. So a vector with these three coordinates, as it was basically explained in the previous lecture, defines the direction of the greatest change, the maximum rate of change, if you wish. So for a scalar we are converging into the vector. Now, at any point, XYZ is just any point in the three-dimensional world. Now today's diversion characteristic is basically a reverse in some sense. If gradient allows from air molecule density to go to the wind, this is density or pressure and this is the wind direction, the vector. Diversion is actually a reverse operation. So let's assume we know the wind, how knowing the wind, the direction and magnitude of the largest rate of change, how can we convert it into the original air pressure, so where there was more molecules than average and that's the higher pressure. And where there was the lower density of the molecules, the low pressure. And that's why the wind goes basically from the higher to the low. So again, for every characteristic which we have given to us, the wind from the wind, we can actually reverse back and find where exactly the density was greater. Well, that's where the wind is blowing from, right? So that's the kind of explanation of what we would like actually to have about diversion. So diversion from the vector gives me a scalar. Now let's explain it in more technical details. Now by the way, before I start, if you remember this thing was actually represented as nabla f of x, y, z, where there is nothing in between here. Where nabla was basically our completely artificial construction, which looks like a vector, but it's actually not the triplet of numbers as any vector is, but it's a triplet of operators of differentiation, partial differentiation by every coordinate. And if you are not comfortable with differentiation or partial differentiation, please stop right now, because this lecture will not be really very helpful for you. Go back to calculus. And now this particular website Unisor.com contains Math 14 scores, and there is a calculus part, so you can go there and refresh that. So I assume that we don't have any problems with partial differentiation. Okay, so this is nabla, then this thing, this thing actually looks like a product of a vector, well pseudo vector or triplet of operators, or sometimes we can just say simply vector by a scalar, by constant or a scalar actually. So the differentiation would be for each particular component, and it looks like we multiply our vector by a scalar. At any point x, y, z, it's a fixed number constant, that's why it's a scalar field. Okay, so it's just the abbreviation, a shorthand, etc., so it makes writing a little bit shorter and easier maybe. Now, we will use the same nabla for divergence, but it will look different, obviously. It will not be just a multiplication, because now our source of information is vector, vector where something went, for example. Okay, so we know that, we scratch this, and we don't need gradient anymore, I was just refreshing what was in the previous lecture. So, again, divergence is basically a characteristic of the field, now a characteristic of which field, what kind of a field. Now, for a gradient, we were talking about scalar field, and converted using the gradient converted into vector field, vector field, because at every point in a three-dimensional world, there was a vector defined. So, we had three components. Now, right now, we are assuming that vector field is given. So, the wind, for example, at every point, at every point in space, we have the wind, which means we have a vector, which has magnitude and direction. Now, we have to use it to find out where exactly the air pressure was greater, where density of the air molecules was greater. Okay, so how can we do it? All right, well, here is how. For example, this is our three-dimensional world, and at every point, we have a vector. So, let's say this is a point Pxyz, and there is some kind of a vector here, which is Vxyz. So, at every point, we have a vector. That's our wind, if you wish. It's always convenient to have some kind of physical representation of this. So, we have a wind at every point. What I would like to do is, I would like to know what is the density of the molecules, air molecules, at this particular point. How can I do it? Well, here is how. Let's draw a little parallel pipette here. So, around point P, we have drawn a very tiny parallel pipette with dimensions which eventually will be, with dimensions delta x, delta y and delta z, which will be, as you understand, infinitesimal. Now, I would like to know what's the density if I know the wind. Okay, fine. If I know the wind, I know this vector V. I know all these three components of this vector V. It's a Vx component, Vy component and Vz component. So, it's a vector. It has three components. It's projections on every axis. So, what does Vx component do? Well, the X component do is the speed with which the wind goes from the left to the right. Vy is, how is it? From front to back, or from back to front, doesn't matter. And Vz is from bottom to up. So, these three components each make their own contribution. And instead of talking about one particular vector V, which is going in some direction, I can actually analyze situation by coordinates. Okay, so let me see what kind of contribution Vx does. Okay, there are a certain number of molecules going in and certain molecules going out from this parallel pipette. And I would like to count them somehow. Okay, how can I do it? So, let's put some letters here. A, B, C, D, E, F, G, H. So, let's talk about the component Vx. Now, the coordinates of A is if my point in the middle, P, has x, y, z, and my A, B is equal to delta x, the component of A would be x minus one-half of delta x, y minus one-half of delta y, and z minus one-half of delta z. Right? Because the x, y, z is right in the middle and this is to the left, to the down, and to the front. So, every particular... I think I should really put this kind of easier to understand. This would be x, this would be y, and this would be z. I think that's easier. Okay, so now we are to the left of x, to the front, to the negative direction of y, and negative direction of along the z. Okay, so this is coordinate at this point. Okay, now I'm talking about contribution only of the Vx. Now, what is the speed of molecules going from the left to the right? So, all these molecules going from A, E, H, D to B, C, G, F, going through this parallelogram. Parallelipipid, sorry, parallelipipid. Well, I know the area A, E, H, D, same as, by the way, B, F, G, H, and the area is equal to delta y times delta z. Right? So, x goes this way. So, this way is delta y and this way is delta z. So, this is delta x, this is delta y, and this is delta z. Okay, now the speed. Now, I'm talking about only the x component, right? So, I have a speed Vx, but I think that considering my delta x, delta y, and delta z are ultimately infinitesimal, I can always consider that the speed within the whole square, A, D, H, E, is the same as the one which is in the middle. Again, when I'm squeezing this in, that would be obvious. And again, if the difference exists, it will be the difference of a higher order. It's a typical with differentiation kind of assumption. So, the speed would be my point x, coordinate x would be x minus one-half delta x. Now, it's in the middle of this square, right? Of the left square. So, y and z would be exactly the same as the center. So, it would be y and z. So, that would be my speed. Okay, now this would be my area. So, the number of molecules would be proportional to the product of the speed and the area. Speed is basically how much they move within unit of time, within second. And if I multiply it by the area, it would be the number proportional to the number of molecules. Obviously, it depends on some properties of air, but we're not talking about this. Properties of air would be, everywhere would be the same. So, if I will multiply by delta y and delta z, that would be a number, well, proportional to a number of molecules coming into the parallelepiped during a unit of time. Now, what comes out? So, this is the plus. What comes out? It comes out through this line, b, c, g, f. The area of this line is the same, but the speed would be speed on this side, which is plus one-half of delta x, y, c. And times delta y times delta z. So, this will be my income, this will be my expenses. This is coming proportional to coming air molecules, this is proportional to leaving this parallelepiped. So, how much is left? Well, obviously, we have to subtract one from another, right? So, what happens if we subtract one from another? Let me subtract from this, I will subtract this because this is going out, this is going in. So, if I would like to know what's the contribution to the output world, I have to subtract whatever comes out from the comes in. So, I need the contribution to the output world. So, it would be vx of x plus one-half delta x, y, z minus vx x minus one-half delta x, y, z. Multiplied by delta y, delta z. Now, again back to calculus, you remember that we had a very simple property that whenever, if you have a function, function of one argument. So, this is let's say x, this is x plus delta x. Now, the f of x plus delta x minus f of x divided by delta x is derivative. That's basically a definition of derivative. So, we can say that approximately equal to this, obviously, right? And the precision is as delta x goes to zero, this goes to basically equality. Limit of this is equal to this. Which means this approximation can be used in this particular case because what this is, y and z, they are not participating because right now we are talking about change of one argument x. So, my partial derivative of, my partial derivative by x, dvx of x, y, z is basically, well we can say approximately, but it's equal to vx of x plus one-half delta x, y, z. Minus vx of x minus one-half delta x, y, z divided by delta x, right? So, this difference I can replace with this times this. And that's what I'm going to do. So, this is approximately. And again, the approximation is greater and greater as I'm squeezing my parallelepiped dvx of x, y, z by dx, partial derivative by x. The difference between them is delta x. So, that's why I multiply it by delta x, delta y and delta z, which is actually delta volume, right? So, this is a contribution of my wind from the left. Well, the x component of the wind, that's what it is. It's my contribution to the output world. Okay, great. Now, what's my contribution, let's say, along the y-axis? Well, absolutely the same as you understand. So, my whole contributions would be equal to in a unit of time. This plus by y and by z, z by z, x by y, by x. This is a total contribution of all of them. Now, my volume goes to zero, obviously, as I'm squeezing this. It's actually not really important. What is important is some of these, which is dvx by dx plus dvy by dy plus dv z by dc. That's what's important. Well, obviously I should put arguments, x, y, z, for each one of those. So, the sum of these would be the contribution of the wind at this particular point, p at x, y and z. Okay. So, this expression actually is called divergence of the vector v. And again, I assume that v has components x, vx, dy and vz. So, v is actually a function of three argument. But it's not just a function, it's a triplet of functions. Which are by x, projection by x, projection by y and projection by z. It's a vector. So, v is a vector. It has three components. And this particular expression is called its divergence. And obviously it's positive if my wind contributes to the world more than it consumes from the world. Because I've just summarized by all three components the contribution and consumption basically along each component. And that's why I have this sum. Now, I can express this thing slightly differently if I'm using the familiar symbol nabla. Again, a pseudo vector if you wish or triplet or triad. I mean different people can use different things. And if you use it as a kind of a vector because it's also like three components although not numerical component but operators of differentiation. So, how can I express this with this? Well, look, this is a vector. This is a vector. So, it looks like scalar product, that product. Remember, if I'm multiplying a, b, c times, I should really use curly brackets as a vector. Vector a, b, c. These are projections of x, y, and z by vector d, e, f. So, what's this scalar product? It's a, d plus v, e plus c, d. It's a scalar. So, look what we have done. Divergence of the field I'll put this, which is a vector. It's nabla, scalar product with vector. That's what basically this expression is, right? So, again, this is, divergence is a net contribution of the wind at point x, y, z to the world. If it's positive, it means that we are going greater out than consumed in, summarized in all three directions, because maybe you, maybe you have contributing positively along x, but negatively against y or z. But in any case, when we are summing, summarizing them together, you have a net contribution. So, if my divergence of my vector field, which is, let's say, speeds of the molecules, vectors, velocities of the molecules. So, if this is my model, then obviously the wind blows from the point x, y, z somewhere else. So, it corresponds to a greater density of the molecules. Because if you remember again, we had this multiplier, delta, delta, delta volume. So, if without that multiplier, it's a density, right? So, our quantity was quantity of molecules, which we are getting into the parallel pipette was divergence times delta volume, which is the volume of the parallel pipette. So, without that volume, it's a density. If you divide it by volume, it will be density. So, this is basically for molecules of air is the variable, a scalar, which basically represents the density. Now, if you have another model, for example, you have electric field. Now, what electric field basically does, instead of molecules, you have charges, electrons, right? So, exactly the same thing, the density of the electrons in any particular point of space where electric field exists is the divergence of the field. But instead of the wind, we should use the intensity of the electric field. We will talk about this when we will go back to electromagnetic fields. And the application of all these field properties to electromagnetic field, we will definitely talk about this particular thing. Basically, the first Maxwell's equation is about exactly this, that the divergence of the electric field corresponds to, with some multiplier, corresponds to the density of electric charge in that particular point. Now, when we are talking about air pressure or density of molecules of air, or we are talking about density of electric charge, we are not talking about absolute density, we are talking about density above and below some kind of average. Now, the average in case of electricity is when you have the same number of positive and negative charges. Let's say electrons and protons. Now, if we are talking about the air, well, there is something which we can call average air pressure or air density. And if everything is average, there is no wind. The wind exists only because there is some kind of above average or below average density in certain places. And that's why the wind goes from the higher to the lower. Okay, so that's basically what I wanted to say about divergence of the field. Again, let me remind that if gradient of the scalar field converts it to vector field, let's say from air density to the wind, direction and strength of the wind, the divergence goes basically the other way around. Gaging the wind, so gaging the vector field, we are trying to find out where are those areas where the air density and pressure are greater from which the wind actually goes. So it's kind of, logically, it's reverse operation. Okay, now I do suggest you to read the notes for this lecture. So you go to unizord.com to Physics 14's course. It's the part which is related to waves. And in the waves, there is a topic called field waves. And in that topic, I'm introducing all these concepts. Gradient, divergence, the next will be curl, two-dimensional curl and three-dimensional curl. And only then I will go to wave equations of Maxwell, which basically use these concepts to express the properties of electromagnetic field, the wave properties of electromagnetic field. Because equations are actually using these concepts, divergence, gradients, the curl, to express what's the relationship between magnetic and electric components of electromagnetic field is. So that's it for today. Thank you very much and good luck.