Using Green’s TheoremRegions with many holes Green’s

Chapter 16, Problem 39E

(choose chapter or problem)

Problem 39E

Using Green’s Theorem

Regions with many holes Green’s Theorem holds for a region R with any finite number of holes as long as the bounding curves are smooth, simple, and closed and we integrate over each component of the boundary in the direction that keeps R on our immediate left as we go along (see accompanying figure).

a.Let ƒ(x, y) = ln (x2 + y2) and let C be the circle x2 + y2 = a2 Evaluate the flux integral

b. Let K be an arbitrary smooth, simple closed curve in the plane that does not pass through (0, 0). Use Green’s Theorem to show that

has two possible values, depending on whether (0, 0) lies inside K or outside K.