A function g(x, y,z) is said to be harmonic at a point (x0, y0,z0) if g is of class C2

Chapter 7, Problem 2

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A function g(x, y,z) is said to be harmonic at a point (x0, y0,z0) if g is of class C2 and satisfies Laplaces equation 2g = 2g x 2 + 2g y2 + 2g z2 = 0 on some neighborhood of (x0, y0,z0). We say that g is harmonic on a closed region D R3 if it is harmonic at all interior points of D (i.e., not necessarily on the boundary of D). Exercises 24 concern some elementary vector analysis of harmonic functions.Assume that D is closed and bounded and that D is a piecewise smooth surface oriented by outward unit normal field n. Let g/n denote g n. (The term g/n is called the normal derivative of g.) Use Greens first formula with f (x, y,z) 1 to show that, if g is harmonic on D, then D g n d S = 0.