Quiz (Green's Theorem)

Show all work and justify each step.

(1) Evaluate the line integral



where is the boundary of the region between the -axis and the semicircle transversed counterclockwise (including the -axis).

(2) Evaluate



where is any Jordan curve whose interior does not contain the point

(3) If is a Jordan curve, show that



where is the region enclosed by

(4) Extend Green's theorem to triply-connected regions.

(5) Suppose is continuously differentiable in a doubly-connected region and that



throughout How many distinct values of are there for the integral



where is a piecewise smooth Jordan curve in

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