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\begin{center}
\textbf{Vector Fields}

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

\textbf{Question}

The function $\un{F}$ is given by $\un{F} = r^2\cos\theta \un{\hat{r}}
+ \alpha r^{\beta} \sin\theta \un{\hat{\theta}}$. For what values of
the constants $\alpha$ and $\beta$ is $\un{F}$ conservative? For these
values find a corresponding potential. 


\textbf{Answer}

As $\un{F} = r^2\cos\theta \un{\hat{r}} + \alpha r^{\beta} \sin\theta
\un{\hat{\theta}} = \nabla \phi (r, \theta)$ we must have
$$\frac{\pa \phi}{\pa r} = r^2\cos\theta, \ \ \
\frac{1}{r}\frac{\pa\phi}{\pa\theta} = \alpha r^{\beta} \sin\theta.$$
\begin{eqnarray*}
\Rightarrow \phi(r,\theta) & = & \frac{r^3}{3} \cos \theta +
C(\theta)\\
\textrm{and } C'(\theta) - \frac{r^3}{3} & = & \frac{\pa \phi}{\pa
\theta}\\
& = & \alpha r^{\beta +1} \sin\theta.
\end{eqnarray*}

This can be solved for a function $C(\theta)$ which is independent of
$r$ if $\alpha = -1/3$ and $\beta=2$.

In this case, $C(\theta) = C$, with $C$ being a constant. $\un{F}$ is
conservative is the two constants $\alpha$ and $\beta$ have the above
stated values. A potential for $\un{F}$ is $\phi = \frac{1}{3} r^3
\cos\theta + C$.

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