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7. (10 points) For a function f(x,y,z), we define Af(x, y, z) to be the scalar function fxx + fyy + fzz. We call Af the Laplacian of f (Don't confuse this with the gradient, denoted by V). Laplace's equation is a second-order partial differential equation: Af = 0. If Af = 0, then f is called harmonic. Harmonic functions arise frequently, especially in situations concerning vibrations and waves. (a) Is g(x,y, z) = et siny + z a harmonic function? (b) Verify that h(x,y,z) = - Vx 2 +y 2 +2 2 is a solution to the Laplace's equation. (c) Verify that for any function f, Af = Div(Vf). (d) Let S be a surface bounding a solid region E in R3, oriented outward. Verify the following (e) Recall the integration by parts (in single variable calculus): So u du = uvla - So vdu. In multi- variable calculus, we have a similar conclusion: where S is a surface bounding a solid region E in R3, oriented outward. Explain why the following inequality must be true Ill , sasav s JJivs. as