$\vec{F}=(9y-y^3)\vec{i}+e^{\sqrt{y}}(x^2-3x)\vec{j}$ and $C$ is the square given by the points $(0,0)$, $(0,3)$, $(3,0)$, and $(3,3)$, oriented counterclockwise.
I have found out that $\vec{F}$ is not conservative, so we cannot just assert that it is $0$. Furthermore the book suggests showing $\vec{F}$ is orthogonal to the edges along the square, but I do not know how to do that. Any hint is appreciated.
Hint: The edge between $(0,0)$ and $(0,3)$ has $x=0$ and $0\le y\le 3$, and can be parametrized as $t\vec j$, $0\le t\le 3$. What is $\vec F$ along this line segment? You should be able to work out the other edges similarly by yourself.