Originally Posted by

**craig** Right bit of a horrible looking one here:

We've got the following:

$\displaystyle \mathbf{u} = U\left(1-\frac{a^3}{r^3}\right)\cos{\theta}\;\mathbf{\hat{r }} - U\left(1+\frac{a^3}{2r^3}\right)\sin{\theta}\;\mat hbf{\hat{\theta}}$ where $\displaystyle a$ and $\displaystyle U$ are constants.

We have already shown that $\displaystyle \nabla \cdot \mathbf{u} = 0$, and because of this we have a vector field $\displaystyle \mathbf{A}$ s.t. $\displaystyle \mathbf{u} = \nabla \times \mathbf{A}$. So assuming that $\displaystyle \mathbf{A} = A(r,\theta)\hat{\mathbf{\phi}}$:

Find 2 equations for $\displaystyle A(r,\theta)$, determine $\displaystyle A(r,\theta)$ and verify that they're consistent.

Right, so does the fact that $\displaystyle \mathbf{A} = A(r,\theta)\hat{\mathbf{\phi}}$ mean that $\displaystyle \mathbf{A}$ is of the form $\displaystyle (0,0,\phi)$? This sort of makes sense.

If so, then do we just complete the cross product and then have two differential equations, essentially $\displaystyle \frac{\partial \phi}{\partial r}$ and $\displaystyle \frac{\partial \phi}{\partial \theta}$.

If so, how do I go about formulating these two equations, I don't seem to be having much luck..

Thanks in advance