32. Harmonic functions A function f(x, y, z) is said to be harmonic in a region D in space if it satisfies the Laplace equation a²f ду? a²f a²f V²f = V • Vƒ = ax? dz? throughout D. a. Suppose that ƒ is harmonic throughout a bounded region D enclosed by a smooth surface S and that n is the chosen unit normal vector on S. Show that the integral over S of Vf•n, the derivative of f in the direction of n, is zero. b. Show that if f is harmonic on D, then fVf•n do = II \vs|² av. IV.

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