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- Jan 26, 2012

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Thanks again to those who participated in last week's POTW! Here's this week's problem!

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\[\nabla^2u=\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} + \frac{\partial^2u}{\partial z^2}.\]

Show that the Laplacian $\nabla^2u$ in spherical coordinates $(\rho,\phi,\theta)$ is given by

\[\nabla^2u=\frac{\partial^2u}{\partial\rho^2} + \frac{2}{\rho}\frac{\partial u}{\partial\rho} + \frac{1}{\rho^2}\frac{\partial^2u}{\partial\phi^2} + \frac{\cot\phi}{\rho^2}\frac{\partial u}{\partial\phi} + \frac{1}{\rho^2\sin^2\phi} \frac{\partial^2u}{\partial \theta^2}\]

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It may be a good idea to convert from rectangular to cylindrical coordinates $(r,\theta,z)$ first, where the Laplacian is

\[\nabla^2u=\frac{\partial^2u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{1}{r^2}\frac{\partial^2u}{\partial\theta^2} + \frac{\partial^2u}{\partial z^2}\]

and then convert from cylindrical to spherical.

Remember to read the POTW submission guidelines to find out how to submit your answers!

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**Problem**: Let $u:\mathbb{R}^3\rightarrow \mathbb{R}$, and define the Laplacian $\nabla^2u$ in rectangular coordinates $(x,y,z)$ by\[\nabla^2u=\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} + \frac{\partial^2u}{\partial z^2}.\]

Show that the Laplacian $\nabla^2u$ in spherical coordinates $(\rho,\phi,\theta)$ is given by

\[\nabla^2u=\frac{\partial^2u}{\partial\rho^2} + \frac{2}{\rho}\frac{\partial u}{\partial\rho} + \frac{1}{\rho^2}\frac{\partial^2u}{\partial\phi^2} + \frac{\cot\phi}{\rho^2}\frac{\partial u}{\partial\phi} + \frac{1}{\rho^2\sin^2\phi} \frac{\partial^2u}{\partial \theta^2}\]

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**Suggestion**:\[\nabla^2u=\frac{\partial^2u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{1}{r^2}\frac{\partial^2u}{\partial\theta^2} + \frac{\partial^2u}{\partial z^2}\]

and then convert from cylindrical to spherical.

Remember to read the POTW submission guidelines to find out how to submit your answers!

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