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\begin{center}
\textbf{Vector Calculus}

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

\textbf{Question}

The smooth vector field $\un{F}$ is irrotational and solenoidal on
$\Re^3$. Show that both the three components of $\un{F}$ and the
scalar potential for $\un{F}$ are harmonic functions in $\Re^3$.


\textbf{Answer}

div$\un{F}=0$ and curl$\un{F}=\un{0}$, $\Rightarrow
\nabla^2\un{F}=0$. So the components of $\un{F}$ are harmonic
functions.

If $\un{F}=nabla\phi$
$$\Rightarrow \nabla^2\phi = \nabla\bullet\nabla\phi = \nabla \bullet
\un{F} = 0$$
therefore $\phi$ is also harmonic.

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