Green's theorem with absolute value boundry

Question

I need to apply green's theorem with the field $F(x,y)= (x+y,x-y)$ on a positive oriented region bounded between the circle $x^2+y^2=9$ and $|x|+|y|=1$, but when I try to parametrize the boundry, I get more than one path because of the $y^2$ and the absolute values.

Answer 1

The vector field is conservative, so the line integrals along any closed contour is $0$. We look for a scalar function $f(x,y)$ such that $\nabla f(x,y)=\vec F(x,y)$.

$$\frac{\partial f}{\partial x}=x+y\implies f(x,y)=\frac{x^2}2+xy+g(y)$$

$$\frac{\partial f}{\partial y}=x-y=x+\frac{\mathrm dg}{\mathrm dy}\implies\frac{\mathrm dg}{\mathrm dy}=-y\implies g(y)=-\dfrac{y^2}2+C$$

$$f(x,y)=\dfrac{x^2}2+xy-\dfrac{y^2}2+C$$