374bhw6 - F . (iii) Explain why the scalar potential must...

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HW#6 —Phys374—Spring 2007 Prof. Ted Jacobson Due before class, Wednesday, March 14, 2007 Room 4115, (301)405-6020 www.physics.umd.edu/grt/taj/374b/ jacobson@physics.umd.edu 1. Show that if v has vanishing divergence, then v is equal to the curl of w defined by the components w x ( x, y, z ) = Z z 0 dz 0 v y ( x, y, z 0 ) (1) w y ( x, y, z ) = - Z z 0 dz 0 v x ( x, y, z 0 ) + Z x 0 dx 0 v z ( x 0 , y, 0) (2) w z ( x, y, z ) = 0 . (3) This proves by explicit construction that every divergenceless vector field can be expressed as the curl of another vector field. [5 pts.] 2. (i) Show that F = yz ˆ x + zx ˆ y + xy ˆ z is both divergence free and curl free, so that it can be written both as the gradient of a scalar and as the curl of a vector. (ii) Find scalar and vector potentials for
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Unformatted text preview: F . (iii) Explain why the scalar potential must satisfy Laplaces equation, and verify that your scalar potential does so. [5+5+5=15 pts.] 3. Prove that solutions to Laplaces equation 2 f = 0 in a compact volume V are uniquely determined the values of f on the closed boundary surface V . [5 pts.] ( Hint : Suppose two solutions f 1 and f 2 agree on V . Consider the function g = f 2-f 1 , and remember what you know about maxima and minima of harmonic functions.) 4. Gravity waves on water supplement, Exercise a . [3+2=5 pts.]...
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This note was uploaded on 12/29/2011 for the course PHYSICS 374 taught by Professor Jacobson during the Fall '10 term at Maryland.

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