Question

1. To write Laplace’s equation, U_xx + U_yy = 0, in polar coordinates, we begin with

U_x = (∂U/∂r)(∂r/∂x) + (∂U/∂θ)(∂θ/∂x)

where r = √(x²+y²), θ = arctan (y/x), x = r cos θ, y = r sin θ. We get

U_x = (cos θ) U_r – (1/r)(sin θ) U_θ , U_xx = [∂(U_x)/∂r] (∂r/∂x) + [∂(U_x)/∂θ](∂θ/∂x)

Carry out this computation, as well as that for U_yy. Since U_xx and U_yy are both expressed in polar coordinates, their sum gives Laplace in polar coordinates.

Answer 1