Let F be a continuously differentiable vector field in R3, Q a point, and S a plane
Chapter 17, Problem 36(choose chapter or problem)
Let F be a continuously differentiable vector field in R3, Q a point, and S a plane containing Q with unit normal vector e. Let Cr be a circle of radius r centered at Q in S, and let Sr be the disk enclosed by Cr. Assume Sr is oriented with unit normal vector e. (a) Let m(r) and M(r) be the minimum and maximum values of curl(F(P)) e for P Sr. Prove that m(r) 1 r2 Sr curl(F) dS M(r) (b) Prove that curl(F(Q)) e = lim r0 1 r2 Cr F dr This proves that curl(F(Q)) e is the circulation per unit area in the plane S. 17
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