1. In the plane, let D be a simply-connected region whose boundary OD is a piecewise C, simple, closed curve, oriented counterclockwise. Let în be the outward unit normal vector to D. Given two functions, f(x,y) and g(x, y), both C² on an open set containing the domain D, show that: (a) · ds = - · dš, C C for any piece-wise C' closed curve C in the domain D. (b) II, uv°9) dædy = f (s79 i ·în ds – ||. (Fs . ÿ9) dædy where V²g = V · Vg = gxx + Iyy· (c) 72 f) dadau - (E,

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