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\begin{center}
\textbf{Vector Calculus}

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

\textbf{Question}

Given that $\phi$ and $\psi$ are both harmonic functions, show that
$$\phi\nabla\psi - \psi\nabla\phi$$
is solenoidal.


\textbf{Answer}

Given that $\nabla^2\phi=0$ and $\nabla^2\psi=)$
\begin{eqnarray*}
\Rightarrow \nabla\bullet(\phi\nabla\psi - \psi \nabla\phi) & = &
\nabla\phi \bullet \nabla\psi + \phi \nabla^2\psi\\
& & -\nabla\psi\bullet\nabla\phi - \psi \nabla^2\phi =0
\end{eqnarray*}
Therefore $\phi\nabla\psi - \psi\nabla\phi$ is solenoidal.
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