Normal derivative is defined as:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \frac{\partial u}{\partial n} = \nabla u \;\cdot\; \hat{n} [/tex]

Where [itex]\hat{n}[/itex] is the unit outward normal of the surface of the sphere and for a small sphere with surface [itex]\Gamma[/itex], the book gave:

[tex]\int_{\Gamma} \frac{\partial u}{\partial n} \;dS \;=\; -\int_{\Gamma} \frac{\partial u}{\partial r} \;dS [/tex]

The book claimed on a sphere:

[tex] \frac{\partial u}{\partial n} = \nabla u \;\cdot\; \hat{n} \;=\; -\frac{\partial u}{\partial r} [/tex]

Where [itex]r [/itex] is the radius of the sphere. I understand [itex]\hat{n}[/itex] is parallel to [itex]\vec{r}[/itex] but [itex]r[/itex] is not unit length.

Can anyone help?

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# Normal derivative in a sphere.

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