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\begin{center}
\textbf{Vector Calculus}

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

\textbf{Question}


Calculate $\textbf{\textrm{div}F}$ and $\textbf{\textrm{curl}F}$ for
the vector field

$\un{F} = \un{\hat{\theta}} = -\sin\theta \un{i} + \cos\theta\un{j}$


\textbf{Answer}

\begin{eqnarray*}
\textrm{div}\un{F} & = & \frac{\cos\theta \sin\theta}{r} - \frac{\cos
\theta \sin\theta}{r} = 0 \\
\textrm{curl}\un{F} & = & \left | \begin{array}{ccc} \un{i} & \un{j} &
\un{k}\\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} &
\frac{\partial}{\partial z}\\
-\sin\theta 7 \cos\theta & 0
\end{array}
\right |\\
=\left ( \frac{\sin^2\theta}{r} + \frac{\cos^2\theta}{r} \right )
\un{k} = \frac{1}{r} \un{k}\\
& = & \frac{1}{\sqrt{x^2+y^2}} \un{k}
\end{eqnarray*}

\end{document}