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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

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


\textbf{Answer}

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

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