2 dimensional Laplace's equation in polar coordinates

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The problem asks you to get Laplace's equation in 2 dimensions in polar coordinates using the fact that $\operatorname{div}(\cdot)$ in two dimensional vector field could be written as
$$\nabla \cdot u = \frac{1}{h_1 h_2} \left( \frac{\partial(u_1 h_2)}{\partial(q_1)}+\frac{\partial(u_2 h_1)}{\partial(q_2)} \right) $$

and the answer is
$\nabla^2\phi$=$\frac{1}{r}${$\frac{\partial}{\partial r}(r\frac{\partial\phi}{\partial r})+$$\frac{\partial}{\partial\theta }$($\frac{1}{r}$$\frac{\partial\phi}{\partial \theta}$)}

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1 Answer

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Hint:

1. Take $q_1 = r$ and $q_2 = \theta$.
2. Compute $h_1$ and $h_2$ based on that information.
3. Substitute in the given formula.

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