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I have a Vector $F = \frac {\bf r}{\|{\bf r}\|}$ where $r = xi+yj+zk$ I want to find

$\iint F \cdot n dS$

using the divergence theorem, where S is a sphere of radius 2 centered at the origin.

Now, I know that $F = n$ (both are unit normal vectors), and when I take that I get

$\iint 1 dS $, which should be the surface area of the sphere.

But how do I do this problem using divergence theorem? I tried finding the divergence, and using spherical coordinates, but I get a $ln(0)$ term. How do I do this?

1 Answers 1

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First of all, I'm not sure what you mean by $r=x^2 i+y^2 j+z^2 k$. Assumedly you mean ${\bf r}=x{\bf i}+y{\bf j}+z{\bf k}$.

The divergence is best taken in spherical coordinates where ${\bf F}=1{\bf e}_r$ and the divergence is

$\nabla\cdot{\bf F}=\frac{1}{r^2}\frac{\partial}{\partial r}(r^21)=\frac{2}{r}.$ Then the divergence theorem says that your surface integral should be equal to $\int \nabla\cdot{\bf F}\,dV=\int dr\,d\theta\,d\varphi\,\,r^2\sin\theta\,\frac{2}{r}=8\pi\int_0^2 dr\,r=4\pi\cdot 2^2,$ which is indeed the surface area of the sphere.

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