Use Green\'s Law to calculate (in picture below) If F (x, y) = (-yi + xj)/(x2 +

Question

Use Green's Law to calculate (in picture below)

If F (x, y) = (-yi + xj)/(x2 + y2), show that F middot dr = 2 pi for every positively oriented simple closed path enclose the origin. Since C is an arbitrary closed path that encloses the origin, it's difficult to compute the given integral directly. So let's consider a counterclockwise-oriented circle C' with center the origin and radius a, where a is chosen be small enough that C' lies inside C. Let D be the region bounded by C and C'. Then its positively oriented boundary is C U (-C') and so the general version of Green's Theorem gives We now easily compute this last integral using the parameterization given by r(t) = a cos ti + a sin t j, 0 t 2 pi. Thus

Explanation / Answer

r(t)= acosti+asintj

F.dr=(sin2tcost+cos2tsint)dt = sin3tdt

integrating 0 to 2pie

we get 0

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