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\begin{center}
\textbf{Vector Fields}

\textit{\textbf{Conservative Fields}}
\end{center}

\textbf{Question}

For the following vector field, find whether it is conservative. If
so, find a corresponding potential

$\ds \un{F}(x,y) = \frac{x \un{i} - y \un{j}}{x^2+y^2}$


\textbf{Answer}

\begin{eqnarray*}
F_1 & = & \frac{x}{x^2+y^2}\\
F_2 & = & -\frac{y}{x^2+y^2}\\
\Rightarrow  \frac{\pa F_1}{\pa y} & = & -\frac{2xy}{(x^2+y^2)^2}\\
\frac{\pa F_2}{\pa x} & = & \frac{2xy}{(x^2+y^2)^2}.
\end{eqnarray*}
So $\un{F}$ cannot be conservative.

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