# Mathematical Food for Thought

Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.

• ## Meta

Colorful! Topic: Calculus. June 6th, 2007

Theorem: (Green’s Theorem) Let be a simply connected plane region whose boundary is a simple, closed, piecewise smooth curve oriented counterclockwise. If and are continuous and have continuous first partial derivatives on some open set containing , then

.

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Problem: Evaluate , where is the boundary of the region enclosed by and .

Solution: First, verify that this region satisfies all of the requirements for Green’s Theorem – indeed, it does. So we may apply the theorem with and . From these, we have and . Then we obtain

.
But clearly this integral over the region can be represented as , so it remains a matter of calculation to get the answer. First, we evaluate the inner integral to get

.

Then finally we have

.

QED.

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Comment: To me, Green’s Theorem is a very interesting result. It’s not at all obvious that a line integral along the boundary of a region is equivalent to an integral of some partial derivatives in the region itself. A simplified proof of the result can be obtained by proving that

and .

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Practice Problem: Let be a plane region with area whose boundary is a piecewise smooth simple closed curve . Show that the centroid of is given by

and .