WEBVTT mathematics/multivariable-calculus/hovasapian
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Hello and welcome back to educator.com and multi variable calculus.
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Today's topic is going to be the divergence and curl of a vector field.
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Basically, what that means, the divergence and curl are types of derivatives for vector fields.
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Let us just jump into some definitions and see what we can do.
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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.
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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.
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It is just a -- you know -- unit vector notation vs. standard vector notation, that is all that is going on here.
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Okay. So let us go ahead and write down some definitions. Definitions.
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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.
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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.
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For right now, before we actually talk about what this means, it is just that. Just symbolically.
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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.
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That is it. Okay. We are also going to define the curl.
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Curl of f, that is equal to d1(f2) - d2(f1), or df2 dx - df1 dy.
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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).
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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.
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Note the curl f is exactly what is under the double integral of Green's theorem.
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Okay. So, let us just do an example of finding the divergence and curl and actually evaluating it.
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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).
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So, this is F1, and this is F2. Okay. So, let us go ahead and move to the next page here.
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Let me go ahead and rite the vector field one more time, so, F = cos(xy), and the sin(x²y).
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Now, let us go ahead and calculate the divergence. The divergence of F, we said it equals dF1 dx + dF2 dy.
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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.
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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.
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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.
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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.
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You have your divergence here, and you have your curl here. They are not the same. Clearly they are not the same.
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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 π/2 and 1.
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So, when x = (π/2,1), this is defined everywhere... this particular vector field. We want to know what the divergence and curl are at a given point.
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Let us go ahead and do the divergence first. So, the divergence of F, evaluated at π/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(π/2) + π/2² × cos(π/2)/4.
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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.
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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.
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So, the curl of F at this point (π/2, 1), again I put this into this expression here, the curl.
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It is going to be 2 × π/2 × 1 × cos(π²/4), right? yes. plus π/2 × sin(xy), which is π/2 × 1... and this is × 1 too.
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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.
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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?
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What do these mean? What do these numbers mean, and what is div() and curl().
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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.
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The divergence, it is a measure of the extent to which the vector field is moving away from that point.
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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.
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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.
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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.
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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.
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So, a positive divergence implies that it is the vector field moving away, expansion. The vector field is going like that.
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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.
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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.
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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.
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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.
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Okay. There is flow away from the point, there is flow towards the point.
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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.
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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.
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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.
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How it is circulating, how it is rotating around that point.
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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.
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okay. So, the right hand rule... looking down at a page, the curl is actually a vector, it represent a direction.
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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.
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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.
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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.
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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.
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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.
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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.
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The curl of F was -0.879, negative curl that means at that point the vector field is actually rotating around that point clockwise.
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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.
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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.
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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.
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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.
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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.
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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.
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The two versions of Green's theorem. Okay. The first form is called circulation curl.
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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.
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Let me go over here. So, the two forms of Green's theorem. One is circulation curl.
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It says c ⋅ c' dt = curl(F) dy dx.
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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.
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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.
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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.
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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.
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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.
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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?
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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.
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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.
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So, notice that the circulation curl form is the one that you have actually already learned.
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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.
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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.
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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.
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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.
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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).
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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.
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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).
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We can form c'(t), which is the tangent vector to c(t) in the direction of traversal.
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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.
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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).
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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.
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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.
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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).
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That is just the best way of thinking about it. It is really just a number, that is all it is.
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Now, let us move on here. So, component, let me write component is in parentheses because in general c' is not a unit vector.
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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.
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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.
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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).
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This f(c) ⋅ c', it is the extent to which the vector field is actually moving in the direction of c'.
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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.
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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.
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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'.
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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.
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That is what is happening. It is called the circulation.
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Now, what about the double integral? What about the double integral of a of the curl of F?
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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That is what is amazing. Green's theorem tells me these two things are equal. Hence the name circulation curl.
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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.
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So, we have the integral over the boundary of a region of a vector field... c' dt over a.
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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.
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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.
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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.
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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.
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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.
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Well we found the curl already, the curl of F we found it already, that is equal to 2xy cos(x²y) + x × sin(xy).
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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.
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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.
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Again, we are solving a double integral, we need to find the region over which the double integral is.
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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.
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Now y, in this particular case, I am going to leave it in terms of x², y², because I am not doing the parameterization.
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I am not doing the line integral, I am actually doing the area integral. So, x² + 4y² = 16. I get 4y² = 16t - x².
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y² = 16 - x²/4, so y = + or - 16 - x²/2, so y is going from -sqrt(16 - x²)/2 to + sqrt(16 -x²)/2.
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And, of course I have my function here, which is the curl... 2xy × cos(x²y) + x × sin(xy) × dy dx.
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Then when I end up putting this into my mathematical software, I get 0.
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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 (π/2,1) was equal to -0.879, the net curl equals 0.
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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.
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This vector field is irrotational. That is it. Straight mathematics.
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Okay. So, when we meet again next time, we will talk about the flux divergence version of Green's theorem.
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Thank you for joining us here at educator.com, we will see you next time. Bye-bye.