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# 374chw6 - HW#6 Phys374Spring 2008 Due before class Friday...

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HW#6 —Phys374—Spring 2008 Prof. Ted Jacobson Due before class, Friday, March 28, 2008 Room 4115, (301)405-6020 www.physics.umd.edu/grt/taj/374c/ 1. If a function g satisﬁes Laplace’s equation 2 g = 0 in a compact volume V , and if g = 0 on the closed boundary surface V , then g = 0 everywhere in V . Prove this two ways: (i) Use what you know about maxima and minima of harmonic functions. (ii) Show using integration by parts that the integral R V g · ∇ g dV vanishes. Since g · ∇ g 0, this implies g · ∇ g = 0, which implies g = 0, which implies g is constant, which implies g = 0 everywhere in V , since g = 0 on the boundary V . [5+5=10 pts.] Uniqueness of solutions to Laplace’s equation : The previous result implies that so- lutions to Laplace’s equation in V are uniquely determined their values on V . For suppose f 1 and f 2 are solutions that agree on V . Then their diﬀerence g = f 2 - f 1 is a solution that vanishes on
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