Why is Green's theorem not valid for
$$\{(x, y):x^2 +y^2 \le 1\} \setminus \{0\}$$ for $$\vec{F}=\frac{-y\vec{i}}{x^2+y^2} - \frac{x\vec{j}}{x^2-y^2} $$
Clearly $dF/dx=dF/dZ$ then why is the integral equal to $2\pi$? Is it because the partial derivative is not continuous at the boundary?
I am not sure whether you do not understand what "Green's theorem" is or if you just have bad notation. First, obviously you mean the derivative with respect to y, not 'Z'. But you also should not have the partial derivatives of "F" but of the components of F.
Green's theorem says that "$\oint_C (Ldx+ Mdy)= \int_D\int \left(\frac{\partial M}{\partial x}+ \frac{\partial L}{\partial y}\right)dxdy$".
If $\frac{\partial L}{\partial y}= -\frac{\partial M}{\partial x}$ then the right side is 0 so the left side is 0.
However, as Mark Viola said (he got in before me- boo, hiss) Greens theorem requires that the partial derivatives be continuous inside the closed curve, C. But that curve encloses (0,0) at which the function itself is not continuous so the partial derivatives do not exist.