# A prove that if u x y satisfies the laplace equation

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a) Prove that if u ( x, y ) satisfies the Laplace equation (no boundary conditions are assumed), then u ( x, y ) has no nondegenerate minima or maxima, i.e., all nondegenerate critical points are saddle points. b) Prove that if u ( x, t ) satisfies the heat equation (no boundary conditions are assumed), then u ( x, t ) has no nondegenerate minima or maxima, i.e., all nondegenerate critical points are saddle points.

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c) Provide an example of a function u ( x, t ) that satisfies the wave equation AND it has a relative minimum and/or a relative maximum. As with the previous problems, no boundary conditions are assumed. 7) (10 points) You showed in Homework 2 that the Fourier series for f ( x ) = | x | on - π x < π is: π 2 - 4 π summationdisplay k =0 1 (2 k + 1) 2 cos(2 k + 1) x a) Use Parseval’s Identity on f ( x ) = | x | to come up with a neat infinite sum identity. b) Following the process you performed in 2.5-7 b) and c), use the value of ζ (4) to come up with the same infinite sum identity. c) Using the Fourier series itself, prove that summationdisplay k =0 1 (2 k + 1) 2 = π 2 8 8) (10 points) a) Let a = b = 1, f 1 ( x ) = 0, f 2 ( x ) = x , g 1 ( y ) = 0, and g 2 ( y ) = y . Solve the Laplace equation with boundary conditions u ( x, 0) = f 1 ( x ), u ( x, 1) = f 2 ( x ), u (0 , y ) = g 1 ( y ), and u (1
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