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\begin{center}
\textbf{Vector Calculus}

\textit{\textbf{Grad, Div and Curl Identities}}
\end{center}

\textbf{Question}

Find a vector potential for $\un{F}  = - y\un{i} + x\un{j}$.


\textbf{Answer}

\begin{eqnarray*}
\Rightarrow \frac{\pa G_3}{\pa y} - \frac{\pa G_2}{\pa z} & = & -y\\
\frac{\pa G_1}{\pa z} - \frac{\pa G_3}{\pa x} & = & x\\
\frac{\pa G_2}{\pa x} - \frac{\pa G_1}{\pa y} & = & 0
\end{eqnarray*}
Find a solution with $G_2=0$
$$\Rightarrow G_3 = -\int y \,dy = -\frac{y^2}{2} + M(x,z).$$
Try setting $M(x,z)=0$, $\Rightarrow G_3 = -\frac{y^2}{2}$. So now
\begin{eqnarray*}
\frac{\pa G_3}{\pa x} & = & 0\\
\textrm{and }G_1 & = & \int x \,dx = xz + N(x,y)
\end{eqnarray*}
As $\frac{\pa G_1}{\pa y}=0$, use $N(x,y)=0$.

So a (non-unique) vector potential for $\un{F}$ is given by 
$$\un{G} = xz\un{i} - \frac{1}{2} y^2 \un{k}.$$

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