Hello - I'm supposed to derive the divergence formula for spherical coordinates by carrying out the surface integrals of the surface of the volume in the figure (the figure is a piece of a sphere similar to a box but with curves). The radial coord is r. The polar angle is [tex]\varphi[/tex] and the azimuthal angle is [tex]\theta[/tex].(adsbygoogle = window.adsbygoogle || []).push({});

The divergence formula is easy enought to look up: DIV(F) = [tex]\nabla\bullet[/tex]F=

[tex]\frac{1}{r^{2}}\frac{\partial}{\partial r}r^{2}F_{r}[/tex]+[tex]\frac{1}{rsin\varphi}\frac{\partial}{\partial \varphi}\left( sin\varphi F_{\varphi}\right)[/tex] + [tex]\frac{1}{rsin\varphi}[/tex][tex]\frac{\partial F_{\theta}}{\partial\theta}[/tex]

And the volume of the little piece of a sphere is easy enough:

[tex]r^{2}sin\varphi \Delta r \Delta\varphi\Delta\theta[/tex]

But when I try to set up the limits for each side as the volume goes to zero I never end up with the first and second [tex]sin\varphi[/tex] in the equation. Supposedly I'm supposed to multiply by a [tex]sin\theta[/tex] but I don't see why.

What I end up with is:

[tex]\frac{\partial}{\partial r}F_{r}[/tex]+[tex]\frac{1}{r}\frac{\partial}{\partial \varphi}\left( F_{\varphi}\right)[/tex] + [tex]\frac{1}{rsin\varphi}[/tex][tex]\frac{\partial F_{\theta}}{\partial\theta}[/tex]

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# Homework Help: Proof of Divergence Formula in Spherical Coordinates

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