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### Course: Multivariable calculus>Unit 5

Lesson 4: 2D divergence theorem

# 2D divergence theorem

Using Green's Theorem to establish a two dimensional version of the Divergence Theorem. Created by Sal Khan.

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• Should it not be the double integral over the region R of (-dQ/dy - dP/dx) * dA? Since the vector field is f = P(x,y) i + Q(x,y) j ?
• Excellent question!
The short answer: no

Although the vector field F = P(x,y) i + Q(x,y) j, the unit normal vector n = (dy i - dx j)/ds. (You divide by ds because n is a unit vector)
Thus, when you take F ⋅ n, you get (P(x,y) * dy - Q(x,y) * dx)/ds

So, ∳(F ⋅ n) ds = ∲[(P(x,y) * dy - Q(x,y) * dx)/ds] * ds = ∳(P(x,y) * dy - Q(x,y) * dx

Remember Green's Theorem? ∳P(x,y) * dx + Q(x,y) * dy = ∫∫ (∂Q/∂X - ∂P/∂Y) dA
Notice that the dx term is added to the dy term ^^. We can take our F ⋅ n line integral and rearrange it to match Green's Theorem (right now, our F ⋅ n line integral has the dy term subtracted by the dx term):
∳(P(x,y) * dy - Q(x,y) * dx = ∳(-Q(x,y)) * dx + P(x,y) * dy

Now that our line integral is in the correct dx + dy form, we can finally apply Green's Theorem properly:
∳(-Q(x,y)) * dx + P(x,y) * dy = ∫∫∂P/∂X - (-∂Q/∂Y) dA = ∫∫(∂P/∂X + ∂Q/∂Y) * dA

I hope this clarified things :)
• The transcript is completely broken. None of the timecodes are right. Can fix it?
• Does it matter if "F" is capital?
• As a general rule, when a function is referred to with a capital F it is an integral, as opposed to f being used for a function that isn't an integral. This isn't a hard and fast rule but a matter of convenient notation.
• What is Flux that Sal mentions?
• The formula that this video is about int(F*n ds) is the formula for flux, which has common applications in Physics and Engineering. His discussion relating to divergence gives an intuitive meaning behind the term, rather than simply using the term.
• At , sal was trying to say "how many", but he didn't and I thought he will say "how many" in the beginning. I want to know, is F-(vector field) dot n-hat represent an acceleration in somehow? and how would be possible to know how many? Should I know the total, for example, gases bounded in that contour first, then, minus the escaped-time multiplied by "how fast", to get how many? What does how fast mean if I am talking about bees exiting from a hole in bee house? Is it the divergence?
• But isn't that what we're doing? Since we're equating the integral on the left to an integral of the divergence, on the right?
• It appears to me that if Sal had picked a point (say, below), where the slope of the tangent was in the positive direction, then the normal would work out with a POSITIVE j component and there would be no negative to cancel the other negative so the partials would have been dP/dx - dQ/dy and the divergence would have been incorrect. I must be missing something. help!
• yi + (-x)j is always 90 degree clockwise so it does work at all points.
• Does anyone know if there is a place to discuss the Divergence Theorem with R^3??
• I am taking from the question above--what is the physical analog to the curve--diffusion through a membrane is fairly intuitive (3d flux), but why would evaluating diffusion through a curve be interesting--maybe as an exercise or maybe some 3d phenomena reduces in dimension.
• As I understand it, there aren't really that many direct applications (Although Luke's Frog Race answer to the question above would qualify), but a lot of the time when dealing with 3d phenomena, it can be easier to reduce it to two dimensions and work from there.
• Since we're using the unit normal vector, does that mean that the specific parametrization of the curve doesn't matter? (ie, y=sin(t), x=cos(t) would produce the same result as y=sin(2t), x=cos(2t))