Question

The vector v is defined by v=rr1v=r r^{-1}, where r = (x, y, z) and r = | r |. Show that (v)\nabla(\nabla \cdot v) \equiv grad div v=2r3rv=-\frac{2}{r^{3}} r

Solution

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Answered 1 year ago
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In this problem, we want to show that

(v)graddiv=2r3r\begin{aligned} \nabla\left(\nabla \cdot v \right)\equiv \mathrm{grad}\mathrm{div}&=-\frac{2}{r^{3}}\textbf{r} \end{aligned}

We know that the vector v\textbf{v} is v=rr1\textbf{v}=\textbf{r}r^{-1}, where r=(x,y,z)\textbf{r}=(x,y,z) and r=rr=|\textbf{r}|.

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