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Let $u=u(x,y)$. Now make the change of variables $x=x(r,θ)=r\cos θ$, $y=y(r,θ)=r\sin θ$. Express the Laplacian of $u$: $∂^2 u/∂x^2 + ∂^2 u/∂y^2$ in terms of derivatives of $u$ with respect to $r$ and $θ$ and everything should be in terms of $r$ and $θ$ with also the assumption that all partials are continuous.

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It's a boring but standard application of the chain rule. For example, $$\frac{\partial u}{\partial x} = \frac{\partial u}{\partial r} \frac{\partial r}{\partial x}+\frac{\partial u}{\partial \theta} \frac{\partial \theta}{\partial x}.$$

See this link and this link for the final formula.

A general, but complicated, approach to the Laplace operator in different coordinate systems is outlined here. Actually, $\mathbb{R}^n$ can be endowed with several riemannian metrics, and we get different expressions for the Laplace-Beltrami operator. When we use the flat metric, we get the standard Laplace operator. If we introduce curvilinear coordinates, then the expression changes.

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...and the product rule, I think. And shouldn't one also ask whether the bottom line has an intuitive justification that might suggest other ways of proving it? – Michael Hardy May 6 '12 at 17:25
Once upon a time, I learned that there is a powerful formula in Riemannian geometry, and that this formula contains almost every reasonable change of variables: polar coordinates, cylindrical coordinates, etc. It uses the coefficients of the metric in local coordinates, but I'm an analyst and I can't remember the most general expression :-) – Siminore May 7 '12 at 9:27

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