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\textbf{Vector Fields}

\textit{\textbf{Conservative Fields}}
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\textbf{Question}

The function $\un{F}$ is given by $\un{F} = r\sin 2\theta \un{\hat{r}}
+ r \cos 2\theta \un{\hat{\theta}}$. Show that $\un{F}$ is
conservative, and find a corresponding potential.


\textbf{Answer}

As $\un{F} = r\sin 2\theta \un{\hat{r}} + r \cos 2\theta
\un{\hat{\theta}} = \nabla \phi (r, \theta)$ we must have
$$\frac{\pa \phi}{\pa r} = r \sin(2\theta), \ \ \ \frac{1}{r}
\frac{\pa\phi}{\pa\theta} = r\cos(2\theta).$$
These are both satisfied if
$$\phi(r, \theta) = \frac{1}{2}r^2 \sin(2\theta) + C.$$
So $\un{F}$ is conservative, having $\phi$ as a potential.


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