laplacian - Phys374 Spring 2008 Prof Ted Jacobson...

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Phys374, Spring 2008, Prof. Ted Jacobson Department of Physics, University of Maryland Laplacian The Laplacian of a scalar function f is the divergence of the curl of f , 2 f = ∇ · ∇ f = 2 x f + 2 y f + 2 z f, (1) where the last expression is given in Cartesian coordinates, and 2 x f means 2 f/∂x 2 , etc. The textbook shows the form in cylindrical and spherical coordinates. There is a nice formula that is not in the textbook, and I have rarely if ever seen it written, but I think it’s sweet and useful so I record it here: V r 2 f dV = 4 πr 2 d dr [ f avg ( V r )] . (2) The integral on the left hand side is over a spherical ball V r of radius r , and the notation on the right denotes the average of f over the spherical boundary V r of this ball. Here’s the proof: V r ∇ · ∇ f dV = V r f · dS (3) = V r f · ˆ r r 2 d Ω (4) = r 2 V r r f d Ω (5) = 4 πr 2 d dr V r f d Ω / 4 π. (6) Here d Ω = sin θdθdϕ is the solid angle element on the sphere, whose integral
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