\[\mathbf{r} f(r)\]
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
\[\begin{equation} \begin{aligned} \mathbf{\nabla} \cdot \mathbf{f} f(f) &=
\frac{1}{r^2} \frac{\partial}{\partial r}(r^3 f(r)) \\ &=
3 f(f) + \frac{df}{dr} r=0
\end{aligned} \end{equation}
\]
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.