3.5 Calculating the divergence in higher dimension

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A First Course in String Theory

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Let \displaystyle{\vec f = f(r) \hat{\mathbf{r}}} be a vector function in \displaystyle{\mathbb{R}^d}.

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Derive a formula for \displaystyle{\nabla \cdot \vec f} by applying the divergence theorem to a spherical shell of a radius \displaystyle{r} and width \displaystyle{dr}.

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Volume and sphere area of an \displaystyle{n}-ball:

\displaystyle{ \begin{aligned} V_{n}(R)&={\frac {\pi ^{\frac {n}{2}}}{\Gamma \left({\frac {n}{2}}+1\right)}}R^{n} \\ S_{n-1}(R)&={\frac {2\,\pi ^{\frac {n}{2}}}{\Gamma \left({\frac {n}{2}}\right)}}R^{n-1} \end{aligned} \\ }

Divergence theorem:

\displaystyle{ \int_V \nabla \cdot \vec f dV = \int_S \vec f \cdot \hat n dS }

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Let \displaystyle{V} be a spherical shell of radius \displaystyle{r} and width \displaystyle{dr}. Then

\displaystyle{ \begin{aligned} \int_V \nabla \cdot \vec f dV &= \left. \nabla \cdot \vec f \right|_r \int_V dV \\ &= \left. \nabla \cdot \vec f \right|_r \left( S_{n-1}(r) dr \right) \\ \end{aligned} \\ }

and

\displaystyle{ \begin{aligned} \int_S \vec f \cdot \hat n dS &= f(r+dr) S_{n-1}(r+dr) - f(r) S_{n-1}(r) \\ \end{aligned} \\ }

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So

\displaystyle{ \begin{aligned} \left. \nabla \cdot \vec f \right|_r S_{n-1}(r) dr &= f(r+dr) S_{n-1}(r+dr) - f(r) S_{n-1}(r) \\ \end{aligned} \\ }

\displaystyle{ \begin{aligned} \left. \nabla \cdot \vec f \right|_r &= \frac{1}{S_{n-1}(r)} \frac{f(r+dr) S_{n-1}(r+dr) - f(r) S_{n-1}(r)}{dr} \\ &= \frac{1}{S_{n-1}(r)} \frac{d}{dr} \bigg( f(r) S_{n-1}(r) \bigg) \\ &= \frac{1}{r^{n-1}} \frac{d}{dr} \bigg( r^{n-1} f(r) \bigg) \\ \end{aligned} \\ }

— Me@2023-08-02 09:28:32 AM

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2023.08.02 Wednesday (c) All rights reserved by ACHK