Let $$\psi = 2\left(\frac{xy}{(x^2+y^2)^2} - \frac{xy}{x^2+y^2}\right)$$
Not very good with the software as you can see so help with that would be great...
Am I correct when I say "partial $\psi$ by partial $y$" is the following?:
$$\frac{2x(x^2+y^2) - 8xy^2 - 2x(x^2+y^2)^2 + 4xy^2(x^2+y^2)}{(x^2+y^2)^3}$$
$\psi$ is the streamfunction of a volvox cell, and our project is to model the swim of this cell. We are told $u$=partial $\psi$ by partial $y$, and $v=-$(partial $\psi$ by partial $x$).
I am not sure that this is easier, but:
$$ u=\frac{\partial \psi}{\partial y} = \frac{2 x \left(-x^4+x^2+y^4-3 y^2\right)}{\left(x^2+y^2\right)^3}$$
$$ v=-\frac{\partial \psi}{\partial x}=-\frac{2 y \left(x^4-3 x^2-y^4+y^2\right)}{\left(x^2+y^2\right)^3} $$