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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

$$\nabla(\phi\psi) = \phi \nabla\psi + \psi\nabla\phi$$ 


\textbf{Answer}

\begin{eqnarray*}
\nabla(\phi\psi) & = & \frac{\pa}{\pa x}(\phi\psi) + \frac{\pa}{\pa y}
( \phi\psi) + \frac{\pa}{\pa z} (\phi\psi)\\
& = & \left ( \phi \frac{\pa\psi}{\pa x} + \frac{\pa\phi}{\pa x}\psi
\right )\un{i} + \cdots + \left ( \phi \frac{\pa\psi}{\pa z} +
\frac{\pa\phi}{\pa z}\psi \right ) \un{k}\\
& = & \phi \nabla\psi + \psi\nabla\phi
\end{eqnarray*} 

\end{document}