Let $C$ be the rectangle, with vertices at points $(0,0)$, $(0,1)$, $(2,0)$, $(2,1)$ and an anticlockwise orientation. Let $$F=(P(x,y),Q(x,y))$$ be a vector field with $$P=y^2+e^{x^2}+ye^{xy}$$ and $$Q=xe^{xy}$$Find the work done by F along C.
I have started by drawing up the region of the rectangle which is $0\to2$ in the $x$ direction and $0\to1$ in the $y$ direction. I then checked if the vector field F is conservative using $P_y = Q_x$. Calculating $P_y$ and $Q_x$ resulted $P_y \neq Q_x$ and hence the field being not conservative.
Would I be correct in applying Greens Theorem in this case?
$$\int\int_DQ_x-P_y=\int_0^2\int_0^1(e^{yx}+ye^{yx})-(2y+e^{xy}(xy+1))dydx$$
$$\nabla \times F=-2 y$$
Call the surface of rectangle $S$
$$\int _ {\partial S}F \cdot dl= \int \int _ S\nabla \times F\cdot dS$$