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prob2 - A = A μ dx μ compute F = dA and show that its...

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A.V. Manohar Ph225A: General Relativity Problem Set 2 1. You are given a vector field ( A x , A y , A z ) in Cartesian coordinates. Compute ( A r , A θ , A φ ) and ( A r , A θ , A φ ) in spherical polar coordinates. 2. Convert the one-form ω = x dy - y dx to polar coordinates. Compute d ω in both coordinate systems, and verify that the polar coordinate result is the same as converting the Cartesian d ω . 3. Given a function f = r cos θ , compute df and ddf. 4. Given a one-form vector potential A = A μ dx
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Unformatted text preview: A = A μ dx μ , compute F = dA and show that its components are the electric magnetic felds. Compute d± (without assuming F = dA), and show that d± = 0 are the source-Free Maxwell equations. 5. Let ω = sin θ d φ . VeriFy Stokes’ theorem i M d ω = i ∂ M ω where M is the Northern hemisphere oF a unit sphere. Repeat For ω = cos θ d φ . Why does Stokes’ theorem Fail in this case? 1...
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