Multivariable Calculus Divergence & Curl of a Vector Field

Hello and welcome back to educator.com and multi variable calculus.0000

Today's topic is going to be the divergence and curl of a vector field.0004

Basically, what that means, the divergence and curl are types of derivatives for vector fields.0008

Let us just jump into some definitions and see what we can do.0014

So, we will let capital F(x,y) be a vector field, I am actually going to write everything out explicitly, so it is going to be F1(x,y), which is going to be the first coordinate function, and F2(x,y), which is the second coordinate function.0021

I probably should have done this a little bit sooner... we had mentioned this i, j, k, notation which tends to be very popular in physics and engineering, so all this means, this is just another way of writing F1i + F2j, that is it.0045

It is just a -- you know -- unit vector notation vs. standard vector notation, that is all that is going on here.0063

Okay. So let us go ahead and write down some definitions. Definitions.0071

The divergence of f, so the divergence of f which is also written as div(F), sometimes they put parentheses, sometimes they do not. It equals the following, I will do capital D notation and also use this standard partial derivative notation.0078

So, d1(f1) + d2(f2), which is the same as the partial derivative of the first one with respect to the first variable which usually is x, plus the partial derivative of the second coordinate function with respect to y. That is it.0099

For right now, before we actually talk about what this means, it is just that. Just symbolically.0116

If you are given a vector field, 2 coordinate functions. If you take the derivative with respect to x of the first coordinate function and add the derivative with respect to y of the second coordinate function, it gives you some number called the divergence when you evaluate it at different points.0121

That is it. Okay. We are also going to define the curl.0135

Curl of f, that is equal to d1(f2) - d2(f1), or df2 dx - df1 dy.0140

Now, you should know that curl is also called the rotation of f, so curl is also called... let me put it over here... is also called the rotation, so you will often see -- well not often, but sometimes in certain books, you will see rot(f).0162

Now, note that the curl, this df2 dx - df2 dy, it is exactly what you see under the double integral in Green's theorem, that we learned previously.0190

Note the curl f is exactly what is under the double integral of Green's theorem.0200

Okay. So, let us just do an example of finding the divergence and curl and actually evaluating it.0224

Example 1. We will let f(x,y) = the cos(x,y), that is our first coordinate function, and our second coordinate function is going to be the sin(x²y).0231

So, this is F1, and this is F2. Okay. So, let us go ahead and move to the next page here.0254

Let me go ahead and rite the vector field one more time, so, F = cos(xy), and the sin(x²y).0264

Now, let us go ahead and calculate the divergence. The divergence of F, we said it equals dF1 dx + dF2 dy.0278

When I take the partial derivative with respect to x of this, I end up with -y × the sin(xy), and I hope you confirm this with me... partial derivatives, you just have to go nice and slow... it is very, very easy to make a mistake because you have a lot of x's and y's floating around.0291

Then... + dF2 dy, so that is going to be + x² × cos(x²y). That is the divergence. That is the function. It is a type of derivative is what it is... for a vector field.0311

Now, let us go ahead and calculate the curl. The curl of F is equal to, we said it was dF2 dx - dF1 dy. Let me make my 2 a little bit clearer here.0327

There we go. That is going to equal 2xycos(x²y) - (-x × sin(xy)). This becomes a plus, so I will just write this as 2xy × cos(x²y) + x × sin(xy), that is it.0346

You have your divergence here, and you have your curl here. They are not the same. Clearly they are not the same.0381

Now let us go ahead and evaluate these at a specific point, so, evaluate the divergence and curl, evaluate dif(F) and curl(F)... let us do pi/2 and 1.0390

So, when x = (pi/2,1), this is defined everywhere... this particular vector field. We want to know what the divergence and curl are at a given point.0410

Let us go ahead and do the divergence first. So, the divergence of F, evaluated at pi/2 and 1, is equal to... well, it is going to equal... I just put it into this expression for the divergence, so it is going to be -1 × sin(pi/2) + pi/2² × cos(pi/2)/4.0419

When you go ahead and simplify this out, you end up with a number -- which is all it is -- ... -2.927, so the divergence evaluated at that point is -2.927.0453

We will talk about what this number means, what the negative sign means, in just a little bit. now let us go ahead and evaluate the curl.0466

So, the curl of F at this point (pi/2, 1), again I put this into this expression here, the curl.0475

It is going to be 2 × pi/2 × 1 × cos(pi²/4), right? yes. plus pi/2 × sin(xy), which is pi/2 × 1... and this is × 1 too.0486

When I evaluate this, I go ahead and get the number -0.879. That is it. Divergence, curl, you go ahead and take care of it symbolically, you evaluate it at a certain point, that is what you are doing, you are getting a number.0519

Now, what do these numbers mean, and what is divergence and what is curl, you know, why are we using these terms divergence and curl?0535

What do these mean? What do these numbers mean, and what is div() and curl().0546

Let us go ahead and move to another page here. So, divergence, I will go ahead and give you the definition here, well not the definition, the idea... what this really means.0566

The divergence, it is a measure of the extent to which the vector field is moving away from that point.0575

When you look at the x, y, plane, a vector field is just pick any point at random, there is just going to be some arrow that is going to be emanating from that point. That is what a vector field is... you know this direction, this direction, this direction, all across the plane.0613

Well, the divergence when I calculate it, it is a measure of the extent to which the vector field is actually moving away from that point.0629

That is a positive divergence. If I have a negative divergence, it is the extent to which the vector field is actually converging on that point. Collapsing on that point. Moving into that point.0637

For example, if the vector field happens to represent the speed of some random fluid moving in a plane, it is the extent to which the fluid is actually moving away from that point, or the extent to which the fluid is moving into that point. That is what is happening.0648

So, a positive divergence implies that it is the vector field moving away, expansion. The vector field is going like that.0665

Negative divergence, that implies moving toward the point, and again, we are evaluating these divergences at specific points, like anything else, that is what it is.0686

We find the expression for the divergence, but we put the particular point in and it is telling us what is happening at that point at that instant... moving towards, which is contraction.0698

When you speak about specific vector fields in your respective engineering and physics classes, you will get a better idea of the behavior, what is happening -- electric field, heat field, fluid field -- like that.0710

So, a positive divergence, is going to be something like this. The vector field is moving away. Negative divergence, it is moving towards, at that point, at that moment.0724

Okay. There is flow away from the point, there is flow towards the point.0740

Now, the curl, this is a measure of the extent to which the vector field actually rotates around that point. The extent to which it curls around that point.0745

It is a measure of the extent to which the vector field, I will just say vf, rotates around that point. That is why you have the alternative rotation for curl.0760

So, if you are given a certain point, and what the curl measures when you take it and you evaluate it, it is a measure of the extent to which the vector field itself at that point or near that point is rotating in this direction, or rotating in that direction.0785

How it is circulating, how it is rotating around that point.0803

A positive curl, that means rotating counterclockwise, and this is based on a convention called the right hand rule, which you probably learned in your physics classes, and a negative curl, and I will describe it in just a second... this is rotation clockwise.0808

okay. So, the right hand rule... looking down at a page, the curl is actually a vector, it represent a direction.0839

We have not represented it as a vector, and we will when we talk about Stoke's theorem in 3-dimensions, but for right now it is best to think about it in this way.0855

So, rotating counterclockwise, based on the right hand rule, If I am looking at a vector field and a vector field happens to be rotating this way, counterclockwise... counterclockwise by thumb is pointing up, so the actual vector itself, this curl vector, is pointing out of the page, and my fingers are moving in the direction of the rotation.0865

If it is rotating clockwise, now my thumb is pointing down, so the curl vector is actually pointing down into the page, perpendicular to the page.0887

I personally do not think of curl as a vector. I mean, it is mathematically, but again it is best to sort of treat it the way we treat it formally as the definition, df2 dx - df1 dy, and just sort of remember that a positive curl is rotating counterclockwise, negative curl is rotating clockwise.0898

That is it. Okay. So, let us see what we have got here. For our example, we had a divergence of F, which was equal to -2.927, and we had a curl of F, which was equal to -0.879.0919

So, as far as the divergence being -2.927, that means the vector field at that point is actually flowing towards the point. There is contraction.0945

The curl of F was -0.879, negative curl that means at that point the vector field is actually rotating around that point clockwise.0957

Okay. So now that we have talked about divergence and curl, let us go ahead and talk about Green's theorem and the relationship between divergence and curl, the line integrals and Green's theorem itself.0969

Now we are going to state the two versions of Green's theorem. One is called the circulation curl form, one is called the flux divergence form.0980

I am going to state them. I am going to talk about the curl first, and then I am going to talk about the divergence later.0986

The two forms of curl are equivalent in the sense if you prove one, then you have proven the other. They are not the same thing in that they do not measure the same thing.0994

Clearly, you are going to see that the integrals are different. So, they are equivalent, but they are not the same thing. It is important to distinguish between the two.1000

Okay. So, it is important to start this on a new page. Well, that is okay, I can go ahead and do at least the first version here.1010

The two versions of Green's theorem. Okay. The first form is called circulation curl.1020

So, this is called the circulation curl form, and it looks like this... da -- that is ok, I do not need to... I can use c here... actually, you know what I think I am going to start on a new page here, simply because I want to avoid some of these lines that show up.1037

Let me go over here. So, the two forms of Green's theorem. One is circulation curl.1067

It says c · c' dt = curl(F) dy dx.1081

Okay. So, the circulation curl form says the following. It says that the line integral of a given vector field around a closed curve, this thing, which we already know, is equal to the double integral of the curl of the vector field over the area enclosed by that closed curve.1103

That is what this says. If I want to, I can either evaluate the line integral, or what I can do is I can integrate the curl of that vector field over the area enclosed by that path.1123

So, if I have some closed path, I can either take the line integral or I can integrate the curl of that vector field over the area enclosed by that curl. This is the circulation curl form of Green's theorem.1136

Now let us go ahead and talk about the flux divergence form. Again, what is important at this stage is... yes, we would like to have a good sense and understanding of what is going on, but we would like you to just be able to work formally.1150

We would like you to be able to construct the integrals, solve the integrals, and just work symbolically. If you do not have a complete grasp of what is going on, you will develop that as you do more problems.1164

But, definitely understand the symbolism here. That is what is important. Given F, given c, given c'... can you construct this integral, and then solve that integral?1176

The flux divergence form says the following. It says that the integral around a closed path c of F(c) this time · something called n dt, which we will talk about a little bit later... is equal to the divergence of F dy dx.1184

So, this one I am going to talk about a little bit later. This is the one that I want to concentrate on first. I want to get a sense of what the integral says, we want to get a sense of what this integral says, and then we will talk about the divergence.1210

So, notice that the circulation curl form is the one that you have actually already learned.1222

This thing right here, curl(F), well curl(F) is dF2 dx - dF1 dy, that is just the Green's theorem that you learned in a previous lesson, so this is just another way of actually writing it.1227

This is Green's theorem, but now we give it a specific name. It is called the circulation curl form. This is the circulation integral, this is the curl integral, and we will describe what these mean.1240

Now we will talk about what these mean... so given... let us say I have some closed path here and I am traversing this path in the counterclockwise direction... this is in the x, y plane, and of course I have some vector field that is defined in the x, y plane.1253

So, a vector field is just a bunch of vectors -- you know -- we do not know which way they point, some of them point this way... well because this path actually passes through the vector field, the points on this path are actually defined.1267

The vector field is defined on those points. Let us pick a specific point, and let us say... boom... we can form F(c), so here, this point is c(t), this vector right here is F(c(t))... I will just write it as F(c).1285

The vector field is defined on this x, y plane, there is a curve on the x, y plane so the point on that curve can be used in the vector field.1307

Now, we can form the following. I will write it up here. We can form c'(t), the tangent vector, we have already done it. Wherever we are given a c(t), we can form c'(t), the derivative. That just happens to be this, the tangent vector. That is c'(t).1315

We can form c'(t), which is the tangent vector to c(t) in the direction of traversal.1341

We are going in a counterclockwise direction, so the tangent vector is this way. That is it. It is just the derivative, we know this already... derivative.1367

Now, F(c) · c'(t) is the component, and I will put component in quotes, is the component of f in the direction of c' -- oops, see, this is what we did not want to happen here, with these crazy lines going here -- okay, in the direction c'(t).1376

In some sense, when we take the dot product of two vectors, you know that if one of the vectors that you are taking the dot product of is a unit vector, well that gives you the component of the big vector on top of the unit vector.1414

Well, the reason I put a component in parentheses, in quotes here, is that in some sense, this f(c) · c'(t), it is the component of this along this.1427

You will want to think of it as sort of a projection. It is not really a projection because this c'(t) is not a unit vector. So, in some sense, you want to think of it as the component of that vector in the direction of c'(t).1444

That is just the best way of thinking about it. It is really just a number, that is all it is.1457

Now, let us move on here. So, component, let me write component is in parentheses because in general c' is not a unit vector.1466

I will say recall, when we have some F in some u, okay... well F · u is the projection of F onto u, so it gives you the component of F in the direction of the unit vector.1496

Well, when we take F(c) · c', in sense we are taking the component of it. c' is not a unit vector, which is why I say component, but it is good to think about it that way.1514

So, like you to think of this F of c · c', which is the integrand and the line integral as the extent to which the vector field at that point is moving in the direction of c'(t).1528

This f(c) · c', it is the extent to which the vector field is actually moving in the direction of c'.1568

Now, when we take the integral of that, of all of these f(c) · c's, when we take the integral of f(c) · c' dt, we are adding all of those numbers up.1578

That is what an integral is. You are just adding up everything around the path, so we are getting the net extent to which F is circulating.1597

Circulating along the path, and this is what is important. We had this path, well, you know we had this vector field this way, we had the tangent vector, well we can evaluate this F(c) · c' at that point, and it is the extent to which it is moving in the direction of c'.1626

Well, we can do that for every single point moving along this path. When we add all of those up, which is what we are doing when we take the integral, we can actually measure the extent to which this vector field is actually circulating around this path in a given direction.1648

That is what is happening. It is called the circulation.1662

Now, what about the double integral? What about the double integral of a of the curl of F?1668

Well, here, now we are doing a double integral, so instead of evaluating the line integral along the path, what we are doing is we are actually taking the curl of all of the points inside the region contained by that closed curve.1681

We are calculating the curl of all of these individual points, and then we are adding up all of the curls. Green's theorem says that these two things are equal.1701

We said that the curl of F is a measure of the extent to which the vector field is curling, or rotating around a point.1711

Okay. When I integrate, when I do the double integral, when I integrate all of the curls, for all of the points that are contained in that region... when I integrate all of the curls for all of the points in the region bounded by the curve, I get the net extent to which the vector field is rotating.1750

That is it. When I take the line integral, I am getting the extent to which the vector field is actually circulating around the path.1810

When I take all of the curls of all of the individual points and I add them up, some are rotating this way, some are curling this way, when I integrate them, the double integral, When I add them all up, I am going to get the next rotation of the vector field over that region.1822

The extent to which the vector field is going this way, or it is going this way. That is what I am doing. Green's theorem is telling me that those 2 numbers are actually the same.1835

That is what is amazing. Green's theorem tells me these two things are equal. Hence the name circulation curl.1847

The line integral is the circulation integral, the double integral is the curl integral... circulation, curl. In and of themselves, they are not the same thing, but they end up being equal.1870

So, we have the integral over the boundary of a region of a vector field... c' dt over a.1883

The curl of F, dy dx. The integral of a vector field around the closed curve is equal to the integral of the curl of that vector field over the area enclosed by that curve. It establishes a relationship between the boundary of a region and the region that that boundary contains.1901

This is profound, this is the fundamental theory of calculus. Okay, let us go ahead and do an example to finish it off that way.1923

So, example 2, we use the same vector field as before, so we have F is equal to... we said it was cos(xy), and sin(x²y), okay.1932

Now, let us go ahead and do our c(t), so our curve c(t) is going to be 4cos(t), 2sin(t), which is the ellipse that has a major focal radius of 4 and a minor focal radius of 2 centered at the origin.1950

Or, if I write it in Cartesian coordinates, it will be x² + 4y² = 16, so this is just 2 different ways of representing the same thing... this is the parameterized version, this is the Cartesian version.1973

Well we found the curl already, the curl of F we found it already, that is equal to 2xy cos(x²y) + x × sin(xy).1990

Now, let us go ahead and solve this. Green's theorem says that the integral over a closed curve of this vector field dotted with c', in other words the circulation is equal to the double integral over that of the curl of F, it is often a good idea ot repeat the formula, write it over... dy dx.2014

Well, this is going to equal, let me go ahead and draw this region here. This is an ellipse like that, and this is minus 4, this is 4, this is 2, this is -2, so x, we are going to go from -4 to 4.2039

Again, we are solving a double integral, we need to find the region over which the double integral is.2063

x is going from -4 to 4, right? Now we are going to have to take little strips of y because we are going to go from y.2071

Now y, in this particular case, I am going to leave it in terms of x², y², because I am not doing the parameterization.2081

I am not doing the line integral, I am actually doing the area integral. So, x² + 4y² = 16. I get 4y² = 16t - x².2089

y² = 16 - x²/4, so y = + or - 16 - x²/2, so y is going from -sqrt(16 - x²)/2 to + sqrt(16 -x²)/2.2103

And, of course I have my function here, which is the curl... 2xy × cos(x²y) + x × sin(xy) × dy dx.2126

Then when I end up putting this into my mathematical software, I get 0.2147

So, this is what is interesting. Remember we found the curl at a certain point. The curl was negative, so despite the fact that we had the curl of f evaluated at (pi/2,1) was equal to -0.879, the net curl equals 0.2155

In other words, this vector field over this region is not rotating this way or this way. When the curl = 0, we call the vector field irrotational.2200

This vector field is irrotational. That is it. Straight mathematics.2216

Okay. So, when we meet again next time, we will talk about the flux divergence version of Green's theorem.2226

Thank you for joining us here at educator.com, we will see you next time. Bye-bye.2233