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.
Express the Laplacian of $u \ldots ∂$
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ordinary-differential-equations
1 Answers
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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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0Once 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 :-) – 2012-05-07