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/ [email protected]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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