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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

$\un{F}(x,y,z) = e{x^2+y^2+z^2}(xz \un{i} +yz\un{j} +xy\un{k})$ 


\textbf{Answer}

\begin{eqnarray*}
F_1 & = & xze^{x^2+y^2+z^2}\\
F_2 & = & yze^{x^2+y^2+z^2}\\
F_3 & = & xye^{x^2+y^2+z^2}
\end{eqnarray*}
This gives
\begin{eqnarray*}
\frac{\pa F_1}{\pa y} & = & 2xyze^{x^2+y^2+z^2} = \frac{\pa F_2}{\pa
x}\\
\frac{\pa F_1}{\pa x} & = & (x+2xz^2)e^{x^2+y^2+z^2}\\
\frac{\pa F_3}{\pa x} & = & (y+2x^2y)e^{x^2+y^2+z^2} \ne \frac{\pa
F_1}{\pa z}.
\end{eqnarray*}
So $\un{F}$ cannot be conservative.

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