So this is a very specific assignment. I Need to show that G is rotation free.
$F(x,y) = (P_F , Q_F) =( \frac{e^x(x\cos(y)+y\sin(y))-x}{x^2+y^2}, \frac{e^x(-x\cos(y)+y\sin(y))-y} {x^2+y^2} )$
$G(x,y) = (P_G , Q_G) = \begin{cases} F(x,y)& \text{if } (x,y) \neq (0,0)\\ (1,0) & \text{if } (x,y) = (0,0)\\ \end{cases}$
I have already shown that $F(x,y)$ is rotation free, by showing.
$Rot(F) = \frac{\partial Q_F}{\partial x} - \frac{\partial P_F}{\partial y} = 0$
But how do I show the same for $G(x,y)$ when it is a function seperated like so?
Calcuate the partial derivatives $\frac{\partial Q_G}{\partial x}$, $\frac{\partial P_G}{\partial y}$ using the definition of partial derivative. $$\frac{\partial Q_G}{\partial x}(0,0)=\lim_{x_\to 0}\frac{Q_G(x,0)-Q_G(0,0)}x=\cdots$$ $$\cdots$$