Green's Theorem

In Lecture 5 we introduced divergence and curl and stated Green's Theorem. In this lecture, we will give examples of how to use Green's Theorem and then sketch a proof of the theorem.

Green's Theorem

F=Mi+Nj. R is the region enclosed by C.

Example

Let F = 2xi - 3yj
C: r(t) = a cos t i + a sin t j, 0<=t<=2*pi

Using Green's theorem is much easier:

Now, to calculate the circulation without Green's Theorem, we find T, dot it with F, and integrate over C.

Again, Green's Theorem is much simpler:

Thus, Green's Theorem is a very good alternative to the line integral.

Another Example

One might notice that in the previous example, divF was constant, making the integral very easy. This is not always true.
Suppose F=2x^2i-3y^2j
divF=4x-6y

One More Example

Given an integral of the form

there are two ways to solve using Green's Theorem.
1. Let M=x^2 and N=x-y and plug into the Flux equation.
2. Let M=x-y and N=x^2 and plug into the Circulation equation.

Proof of Green's Theorem

We will not do a complete proof - we will just give a sketch of one way to prove it.
First, we will assume:
• F and its first partial derivatives are continuous on R.
• C consists of a finite number of smooth curves.
• C is a simple curve.

Special Case of C - paths where horizontal and vertical lines cross C only twice.

We must prove that:

We can prove for the dN/dx term in a similar way, though we will not show it here.
To extend this proof to other shapes:

1. Rectangular Shapes. This is not very difficult, but we will not show it here.
2. Note that in the second drawing, the two additional line segments cover the same distance and point in opposite directions. They cancel out so that the same result is reached from the two regions in the second drawing as from the entire region in the first drawing.
3. Again we can break the region into many acceptable shapes and sum to get the same result.

Lecture written by Aric Shafran on Monday, September 18, 1995
Edited by Lawrence C. Weintraub on Friday, September 22, 1995
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