SUNY Geneseo Department of Mathematics

Examples of Green’s Theorem

Friday, May 5

Math 223
Spring 2023
Prof. Doug Baldwin

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Green’s Theorem

Specifically, examples to complement “Circulation Form of Green’s Theorem” from section 5.4 in the textbook, and our discussion of it Wednesday.

Green’s Theorem “Works”

Suppose G(x, y) =⟨ -y, 2x ⟩. Does Green’s Theorem give the same value for the integral of G around the circle r(t) =⟨ cos t, sin t ⟩as integrating in the traditional way does?

Circle R of T equals vector cosine T comma sine T and field F of X and Y equals vector negative Y comma 2 X

Let’s start by stating exactly what it is that Green’s Theorem says, and therefore what it is that we’re checking. The Theorem says that the line integral of a vector field around a closed curve is equal to a double integral over the area inside the curve of the difference of certain derivatives of the field’s components.

Integral around closed loop of F of X and Y is double integral inside loop of D Q D X minus D P D Y

We calculated the traditional line integral on the left side of this equation first. It was tedious, but a mostly familiar process (except maybe where knowing an identity for cos2t was helpful):

Integral around circle of F of R of T dot R prime of T is 3 Pi

Then we calculated the area integral from the right side of the equation. Because we are integrating over a circle, we calculated the integral in polar form. But even so, it was much simpler to work out than the original line integral:

Integral over circle of derivative of Q with respect to X minus derivative of P with respect to Y is 3 Pi

A Multi-Part Curve

Evaluate the integral of sin(x2) dx + (4x - y) dy counterclockwise around the triangle with vertices (-1,1), (1,1), and (1,3).

Situations like this, where you have multiple pieces to the curve you’re integrating along, and thus potentially multiple line integrals to evaluate, are one of the places where Green’s Theorem is really handy. In this case we got the area integral set up, and then recognized it as just a constant times the integral for, literally, the area of the triangle. So we used the geometry formula (1/2 base times height) to find that area, and never really had to evaluate an integral at all:

Green's Theorem finds that a certain line integral around a triangle is just a constant times the triangle's area

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More examples of Green’s Theorem. There’s no new reading for this.

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