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\begin{center}
\textbf{Vector Calculus}

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

\textbf{Question}

Verify that $\un{F}\times\un{G}$ is solenoidal for smooth and
conservative vector fields $\un{F}$ and $\un{G}$. Also find a vector
potential for $\un{F}\times\un{G}$.


\textbf{Answer}

\begin{eqnarray*}
\un{F} &amp; = &amp; \nabla\phi\\
\textrm{and }\un{G} &amp; = &amp; \nabla \psi\\
\Rightarrow \nabla \times \un{F} &amp; = &amp; \un{0}\\
\textrm{and }\nabla \times \un{G} &amp; =&amp; \un{0}\\
\Rightarrow
\nabla\bullet(\un{F}\times\un{G}) &amp; = &amp; (\nabla \times \un{F})\bullet
\un{G} + \un{F} \bullet (\nabla \times \un{G})\\
&amp; = &amp; 0
\end{eqnarray*}
and so $\un{F}\times\un{G}$ is solenoidal.

It can be seen that
$$\nabla\times(\phi\nabla\psi) = \nabla \phi \times \nabla \psi =
\un{F}\times\un{G}.$$
so $\phi\nabla\psi$ is a vector potential for $\un{F}\times\un{G}$, as
is $-\psi\nabla\phi$.

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