Problem $1$: Irrotational Flow

Show that the velocity potential $$ and the stream function $$ for the flows in $($a$)$ and $($b$)$

below satisfy Laplace's equation.

$($a$)$ A uniform flow in the $x$ direction:

$=V_{}x$

$=V_{}y$

$($b$)$ A two$-$dimensional source flow:

$=\frac{}{2}lnr$

$=\frac{}{2}$