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Greens Theorem

Greens Theorem

The line integral of a closed curve going counterclockwise can be determined. The curve loops back to itself. The closed curve has the region D.

A simply connected region R bounded by the curve C

The integral sign is decorated with a little circel. That indicates to integragte along a closed curve.

  Example

Evaluate the line integral of . The region is bounded by the triangle which is determined by the vertices (0, 0), (1, 0), (1, 2).

So the diagonal line is .

The region of x is between 0 and 1. So the region of y lies between  and .

Transform the integral into a double integral and subtract the partial derivatives. Attention! In the difference the terms with dx and dy are swapped.

Evalute the double integral. Arrange the boundaries so that the numbers are outside. First integrate dy.

Put in the boundaries of y.

After integration put in the boundaries of x.