Given a function φ(z) with z = x+iy let U(x, y) = ½ [φ(x+iy) +...

Given a function φ(z) with z = x+iy let

U(x, y) = ½ [φ(x+iy) + φ(x-iy)] and V(x, y) = i/2 [φ(x+iy) –φ(x-iy)]

A) For φ(z) = z² find U and V and their induced vector fields E =▼U and

F =▼V also show that ▼²U = ▼²V = 0

B) Repeat for f(z) = z³

C) For f(z) = ln z we get U(x, y) = ½ ln (x²+y²) and V(x, y) = arctan (y/x) Find ▼U (electrostatic field) and ▼V (magnetic field) . Also, show that ▼²U = ▼²V = 0 for x²+y² ǂ 0.

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genius_generous answered 1 year ago

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