$F(x,y)=\langle y^2+e^x,x^2+e^y\rangle$. Using green's theorem in its circulation and flux forms, determine the flux and circulation of $F$ around the triangle $T$, where $T$ is the triangle with vertices $(0,0),(1,0),$ and $(0,1)$, oriented counterclockwise.
My Try:
$f=y^2+e^x$
$g=x^2+e^y$
$$\frac{\partial f}{\partial y}=2y$$ $$\frac{\partial g}{\partial x}=2x$$ $$\int F\cdot dr=\int\int_R(2x-2y)dA$$ $$\int^{1}_{0}\int_{0}^{1-x}(2x-2y)dydx=0$$ My quesiton: Is my above attempt correct? Because I got the answer as $0$
The result is correct indeed we have that the region is symmetric with respect to $y=x$ and the integral for $2x$ is equal to the integral for $2y$ that is
$$\int\int_R 2x\,dA=\int\int_R2y\,dA \implies \int\int_R(2x-2y)dA=0$$