## Possible Expression for a Function With Given Absolute Gradient

Suppose
$\mathbf{r} f(r)$
is sinusoidal and
$|\mathbf{\nabla} f(r)|_{(-1,1,0)} =5$
.
We want to find a possible expression for
$f(r)$
,
Since
$\mathbf{r} f(r)$
is sinusoidal
$\mathbf{\nabla} \cdot (\mathbf{r} f(r))=0$
. Evaluating
$\mathbf{\nabla} \cdot (\mathbf{r} f(r))$
gives
\begin{aligned} \mathbf{\nabla} \cdot \mathbf{f} f(f) &= \frac{1}{r^2} \frac{\partial}{\partial r}(r^3 f(r)) \\ &= 3 f(f) + \frac{df}{dr} r=0 \end{aligned}

This is a differential equation for
$f(r)$
. We rearrange to give
$3 \frac{dr}{r} + \frac{df}{f}=0$

Integrate to give
$3 ln r +ln (f(r))=A$

This can be rearranged to give
$f(r)=\frac{e^A}{r^3} =\frac{B}{r^3}$
$|\mathbf{\nabla} (f(r)|_{(-1,1,0)}= |\frac{\partial f}{\partial r} |_{(-1,1,0)}= \frac{3B}{r^4}|_{(-1,1,0)}=5 \rightarrow \frac{3B}{4} =5 \rightarrow B= \frac{20}{3} \rightarrow f(r)=\frac{20}{3r^3}$
is a posible solution.