Question

1. Let g : R30,0,0)-R be given by g(x, y, z) 2. 3 (a) Compute Vg(x, y, z) (b) Show that V2g V (Vg) for all (x, y, 2) (c) Verify by direct calculation that (0,0,0) for any sphere S centered at the origin. d) Why do (b) and (c) not contradict the divergence theorem? 2. Let f be C2 on R3 and satisfy Laplace's equation ▽2f-0. Such functions are called harmonic. (a) Applying Green's formulas to f and g from question l over R-{(z, y, z) | ε !(z, y, z)| r), show that the mean values off on the spheres I(x, y, z)| = r and I(x, y, z)| = ε are equal. (b) Conclude that the mean value of f on any sphere centered at the origin is equal to the value of f at the origin. (Side remark: there is nothing special about the origin here. Applying the result to f(x)-f(x+a), we see that the mean value of a harmonic function over any sphere is its value at the center of the sphere.)

Answer 1

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