Question

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

Solution

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In this problem, we want to show that

\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 $\textbf{v}$ is $\textbf{v}=\textbf{r}r^{-1}$, where $\textbf{r}=(x,y,z)$ and $r=|\textbf{r}|$.

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