The velocity components for a flow field are as follows;

$\displaystyle \displaystyle v_{x} = \frac{\partial \psi}{ \partial y} = a( x^{2} - y^{2}) $

and

$\displaystyle \displaystyle v_{y} = -\frac{\partial \psi}{\partial x} = -2axy $

a) Prove that it statifies the continuity equation ; $\displaystyle \displaystyle \frac{\partial v_{x}}{\partial x} + \frac{\partial v_{y}}{\partial y} = 0 $

b) Determine the stream function $\displaystyle \psi $

Not sure if my method for part 'a' is correct, I just started rearranging the equations

$\displaystyle \displaystyle v_{x} = \frac{\partial \psi}{ \partial y} $

$\displaystyle \displaystyle v_{x} \partial y = \partial \psi $

sub this into $\displaystyle \displaystyle v_{y} = -\frac{\partial \psi}{\partial x} $

$\displaystyle \displaystyle v_{y} = -\frac{v_{x} \partial y}{\partial x} $

$\displaystyle \displaystyle \frac{ v_{y}}{\partial y} = -\frac{v_{x}}{\partial x} $

$\displaystyle \displaystyle \frac{ v_{y}}{\partial y} + \frac{v_{x}}{\partial x} = 0 $

It's not the exact desired equation but I am not sure how to get the $\displaystyle v_{x} and v_{y} $ as partial derivatives?

Also any help on part 'b', do I integrate?

Thank you.