Let F = P , Q, R be a vector field defined on R3 such t
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Let F = P , Q, R be a vector field defined on R3 such that div(F) = 0. Use the following steps to show that F has a vec
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Let F = P , Q, R be a vector field defined on R3 such that div(F) = 0. Use the following steps to show that F has a vector potential. (a) Let A = f, 0, g. Show that curl(A) = ?g ?y , ?f ?z ? ?g ?x , ??f ?y (b) Fix any value y0 and show that if we define f (x, y, z) = ? y y0 R(x, t, z) dt + ?(x, z) g(x, y, z) = y y0 P (x, t, z) dt + ?(x, z) where ? and ? are any functions of x and z, then ?g/?y = P and ??f/?y = R. (c) It remains for us to show that ? and ? can be chosen so Q = ?f/?z ? ?g/?x. Verify that the following choice works (for any choice of z0): ?(x, z) = z z0 Q(x, y0, t) dt, ?(x, z) = 0 Hint: You will need to use the relation div(F) = 0.
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