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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

$$\nabla \times (\nabla \phi) = \un{0}$$


\textbf{Answer}

By equality of mixed partials
\begin{eqnarray*}
\nabla \times (\nabla \phi) & = & \left | \begin{array}{ccc}
\un{i} & \un{j} & \un{k}\\
\frac{\pa}{\pa x} & \frac{\pa}{\pa y} & \frac{\pa}{\pa z}\\
\frac{\pa\phi}{\pa x} & \frac{\pa\phi}{\pa y} & \frac{\pa\phi}{\pa z}
\end{array} \right |\\
& = & \left ( \frac{\pa}{\pa y} \frac{\pa\phi}{\pa z} - \frac{\pa}{\pa
z} \frac{\pa\phi}{\pa y} \right )\un{i} + \cdots = \un{0}
\end{eqnarray*}

\end{document}
