Laplacians, Gradients and Divergence

I can never remember the Laplacians, gradients and divergence formulas in spherical and cylindrical coordinates. Usually I use Griffiths Introduction to Electrodynamics, however I don’t always carry the book with me. As such I am leaving them here as a useful place for me to reference. While there are many places to reference they are usually not front in center.

I hope you find them useful too.

 

Cylindrical Polar Coordinates

\nabla U = \frac{\partial U}{\partial r}\boldsymbol{e}_r + \frac{1}{r}\frac{\partial U}{\partial \phi} \boldsymbol{e}_\phi + \frac{\partial U}{\partial z} \boldsymbol{e}_z 

\nabla \cdot \boldsymbol{F} = \frac{1}{r}\left[\frac{\partial(rF_r)}{\partial r} + \frac{\partial(rF_\phi)}{\partial \phi}  + \frac{\partial(rF_z)}{\partial z} \right]

\nabla^2  U = \frac{1}{r}\frac{\partial}{\partial r}r \frac{\partial U}{\partial r} +\frac{1}{r^2} \frac{\partial^2 U}{\partial \phi^2}  + \frac{\partial^2U}{\partial z^2}

Spherical Polar Coordinates

\nabla U = \frac{\partial U}{\partial r}\boldsymbol{e}_r + \frac{1}{r}\frac{\partial U}{\partial \theta} \boldsymbol{e}_\theta +  \frac{1}{r \mathrm{sin}\theta}\frac{\partial U}{\partial \phi} \boldsymbol{e}_\phi

\nabla \cdot \boldsymbol{F} = \frac{1}{r^2 \mathrm{sin}\theta}\left[\frac{\partial (r^2 \mathrm{sin} \theta F_r)}{\partial r} + \frac{\partial (\mathrm{sin}\theta F_\theta)}{\partial \theta} + \frac{\partial(rF_\phi)}{\partial \phi} \right]

\nabla^2  U = \frac{1}{r^2}\frac{\partial}{\partial r}r^2 \frac{\partial(U)}{\partial r} + \frac{1}{\mathrm{sin}\theta} \frac{\partial}{\partial \theta}\left(\mathrm{sin}\theta \frac{\partial}{\partial \theta}\right) +\frac{1}{\mathrm{sin}^2\theta} \frac{\partial^2 U}{\partial \phi^2}  

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Laplacians, Gradients and Divergence

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