Let F be a vector field of class C1 in a neighborhood of the point P in R3, and let n be

Chapter 7, Problem 31

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Let F be a vector field of class C1 in a neighborhood of the point P in R3, and let n be a unit vector drawn with its tail at P. Let C be a simple, closed curve such that there is an orientable surface S bounded by C that contains P and such that n is normal to S at P. Orient S by using n, and orient C consistently with S. Following Proposition 3.5, show that, if A denotes the area of S, then n curl F(P) = lim A0 1 A C F ds. Here the limiting process is assumed to be such that C shrinks down to the point P. (See Figure 7.47.) S C n P Figure 7.47 Figure for Exercise 31.