# Potential theory?

• April 4th 2011, 08:42 AM
craig
Potential theory?
Tricky one here, just not sure where to start.

$\displaystyle \mathbf{u} = \nabla \phi$, with

$\displaystyle \phi = \mathbf{U} \cdot \mathbf{r} + \frac{a^3}{2r^3}\left(\mathbf{U} \cdot \mathbf{r} \right)$

With $\displaystyle \mathbf{U} = U\mathbf{k}$.

Show that:

$\displaystyle u_i = U_i + \frac{a^3}{2}\left\{\frac{U_i}{r^3} - \frac{3U_j x_j x_i}{r^5} \right\}$

Now I presume that we have to work out $\displaystyle \nabla \phi$ first? Just not got a clue where to start.

Any pointers in the right direction would be much appreciated.

Start by determining $\phi$. What's $\mathbf{U}\cdot\mathbf{r}$?