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We have $\mathbf{x}=\left(x_1,\cdots,x_n\right)\in\mathbb R^n$ and $\mathbf{v(x)=x}.$ How can we apply Gauss' divergence theorem to $\mathbf{v}$ over the unit ball $$B=\{ \mathbf{x}\in\mathbb R^n:\displaystyle\sum_{i=1}^{n}x_i^2\leq1\}$$.Thank you

MY TRY:Gauss' divergence theorem for multiple dimension is $$\int\cdots\int_{n\,\text{times}}\nabla\,\mathbf{F}\,dv=\int\cdots\int_{{n-1}\,\text{times}}\mathbf{F}\,\mathbf n\, ds$$.But I am unable to proceed further.

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    Can you calculate: 1. The divergence $\nabla \cdot \mathbf{v}$? 2. The dot product $\mathbf{v} \cdot \mathbf{n}$ along the unit sphere? Possibly also of interest: [Why is the derivative of a circle's area its perimeter (and similarly for spheres)?](http://math.stackexchange.com/questions/625) – 2017-01-05
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    $\nabla \cdot \mathbf{v}=n$ – 2017-01-05

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