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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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  • $\begingroup$ Do you know what $h_1, h_2, q_1, q_2$ are? This is half the answer. $\endgroup$ – Mark Fantini Apr 20 '14 at 21:00
  • $\begingroup$ OOH it's $ds_i$=$h_i dq_i$ guess the h is the size, and the q is the direction? $\endgroup$ – user3473612 Apr 20 '14 at 21:08
  • $\begingroup$ Yes. If you are dealing with polar coordinates, then $q_1 = ?$ and $q_2 = ?$... $\endgroup$ – Mark Fantini Apr 20 '14 at 21:11
  • $\begingroup$ Is the problem supposed to be telling the specific values for $q_1$and $q_2$??! It tells you nothing ;< $\endgroup$ – user3473612 Apr 20 '14 at 21:17
  • $\begingroup$ Not values, it is telling how to translate $q_1$ and $q_2$ to appropriate coordinates. When you use polar coordinates you take $q_1 = r$ and $q_2 = \theta$ (not necessarily in this order). Then you compute $h_1$ and $h_2$ based on other equations. $\endgroup$ – Mark Fantini Apr 20 '14 at 21:18
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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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