I want to calculate the flux of the vector field $$X(x,y)=y\partial_x-x\partial_y$$ in $\mathbb R^2$ written in polar coordinates ($\partial_x:=\frac{\partial}{\partial x}$ and so on).
Step 1: transform X to polar coordinates
It is $r=\sqrt{x^2+y^2}$ and $\phi=atan2(y,x)$, so that we get: $$\partial_x=\partial_r \frac{\partial r}{\partial x}+\partial_\phi \frac{\partial \phi}{\partial x}=cos\phi \partial_r - \frac{1}{r}sin\phi \partial_\phi$$ and $$\partial_y=\partial_r \frac{\partial r}{\partial y}+\partial_\phi \frac{\partial \phi}{\partial y}=sin\phi\partial_r+\frac{1}{r}cos\phi\partial_\phi.$$ So using $x=rcos\phi$ and $y=rsin\phi$ and knowing $\partial_x$ and $\partial_y$ we can deduce that $$X(r,\phi)=-\partial_\phi.$$
Step 2: calculate the flux
This is the point where I dont know how to continue. I know that I have to find an integral curve by solving differential equations.