I know that a differential $dA=B(x,y)dx+C(x,y)dy$ is exact iff
$$\tag{1} \left( \frac{\partial B}{\partial y} \right)_x = \left( \frac{\partial C}{\partial x} \right)_y$$
I do not understand how to interpret this relation. The rate of change of $B$ with respect to $y$ equals to the rate of change of $C$ with respect to $x$?
I understand that if differential of a function $A(x,y)$ is exact, then the function $A$ is a state function, ie. independent of path. For example, for a closed loop
$$\oint dA=0$$
which implies
$$\tag{2} \oint B(x,y)dx=-\oint C(x,y)dy$$
It seems like another condition for exact differential. I am wondering how (1) is related to (2).