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eliasds
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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 \varphi and the azimuthal angle is \theta.The divergence formula is easy enought to look up: DIV(F) = \nabla\bulletF =
\frac{1}{r^{2}}\frac{\partial}{\partial r}r^{2}F_{r}+\frac{1}{rsin\varphi}\frac{\partial}{\partial \varphi}\left( sin\varphi F_{\varphi}\right) + \frac{1}{rsin\varphi}\frac{\partial F_{\theta}}{\partial\theta}
And the volume of the little piece of a sphere is easy enough:
r^{2}sin\varphi \Delta r \Delta\varphi\Delta\theta
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 sin\varphi in the equation. Supposedly I'm supposed to multiply by a sin\theta but I don't see why.
What I end up with is:
\frac{\partial}{\partial r}F_{r}+\frac{1}{r}\frac{\partial}{\partial \varphi}\left( F_{\varphi}\right) + \frac{1}{rsin\varphi}\frac{\partial F_{\theta}}{\partial\theta}
\frac{1}{r^{2}}\frac{\partial}{\partial r}r^{2}F_{r}+\frac{1}{rsin\varphi}\frac{\partial}{\partial \varphi}\left( sin\varphi F_{\varphi}\right) + \frac{1}{rsin\varphi}\frac{\partial F_{\theta}}{\partial\theta}
And the volume of the little piece of a sphere is easy enough:
r^{2}sin\varphi \Delta r \Delta\varphi\Delta\theta
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 sin\varphi in the equation. Supposedly I'm supposed to multiply by a sin\theta but I don't see why.
What I end up with is:
\frac{\partial}{\partial r}F_{r}+\frac{1}{r}\frac{\partial}{\partial \varphi}\left( F_{\varphi}\right) + \frac{1}{rsin\varphi}\frac{\partial F_{\theta}}{\partial\theta}
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