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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

$$\nabla\bullet(\phi\un{F}) = (\nabla \phi)\bullet \un{F} + \phi(\nabla\bullet\un{F})$$


\textbf{Answer}

\begin{eqnarray*}
\nabla\bullet(\phi\un{F}) & = & \frac{\pa}{\pa x}(\phi F_1) +
\frac{\pa}{\pa y} (\phi F_2) + \frac{\pa}{\pa z}(\phi F_3)\\
& = & \frac{\pa\phi}{\pa x}F_1 + \phi\frac{\pa F_1}{\pa x} + \cdots +
\frac{\pa\phi}{\pa z}F_3 + \phi\frac{\pa F_3}{\pa z} + \cdots\\
& = & (\nabla \phi)\bullet \un{F} + \phi(\nabla\bullet\un{F})
\end{eqnarray*}

\end{document}
