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\begin{center}
\textbf{Vector Calculus}

\textit{\textbf{Grad, Div and Curl Identities}}
\end{center}

\textbf{Question}

It is given that $\un{r} = x\un{i} + y\un{j} + z\un{k}$, with
$r=|\un{r}|$. It is also given that $f$ is a differentiable function
of one variable.

Show that
$$\nabla\bullet (f(r)\un{r}) = rf'(r) + 3f(r)$$
and find $f(r)$ if it is assumed that $f(r)\un{r}$ is solenoidal for
$r\ne 0$.


\textbf{Answer}

\begin{eqnarray*}
\nabla\bullet (f(r) \un{r}) & = & (\nabla f(r))\bullet \un{r} - f(r)
(\nabla \bullet \un{r})\\
& = & f'(r)\frac{\un{r}\bullet\un{r}}{r}+3f(r)\\
& = & rf'(r) + 3f(r)
\end{eqnarray*}
If $f(r)\un{r}$ is solenoidal, then $\nabla\bullet(f(r)\un{r}) = 0$,
so that $u=f(r)$ will satisfy
\begin{eqnarray*}
r \frac{du}{dr} + 3u & = & 0 \\
\frac{du}{u} & = & -\frac{3 \,dr}{r}\\
\ln |u| & = & -3\ln |r|+ \ln |C|\\
u & = & Cr^{-3}\\
\Rightarrow f(r) & = & Cr^-3
\end{eqnarray*} 
for an arbitrary constant $C$.

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