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\begin{center}
\textbf{Vector Calculus}

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

\textbf{Question}

It is given that $\phi$ and $\psi$ are scalar fields and $\un{F}$ and
$\un{G}$ are vector fields. They are all assumed to be smooth
functions. Prove the following identity

$$\nabla\bullet(\un{F} \times \un{G}) = (\nabla \times
\un{F})\bullet\un{G} - \un{F} \bullet(\nabla \times \un{G})$$


\textbf{Answer}

\begin{eqnarray*}
\nabla\bullet(\un{F} \times \un{G}) & = & \frac{\pa}{\pa x}(F_2G_3 -
F_3G_2 + \cdots\\
& = & \frac{\pa F_2}{\pa x}G_3 + F_2\frac{\pa G_3}{\pa x} - \frac{\pa
F_3}{\pa x}G_2 - F_3 \frac{\pa G_2}{\pa x} + \cdots\\
& = & (\nabla \times
\un{F})\bullet\un{G} - \un{F} \bullet(\nabla \times \un{G})
\end{eqnarray*}

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