$$\underbrace{[e^x\cos(\pi y^2)+ay-1]}\_{P}dx+\underbrace{[by\; e^x\sin(\pi y^2)+x+y]}\_Q dy$$ is given

I want to check if the given equation is exact, has potential, conservative.

**Question:** 

1. If I want to check exactness, I always begin with taking partial derivatives of $P\_y$ and $Q\_x$ and check whether they are equal or not. I guess if $P$ and $Q$ have continuous partials then we have conservaive and potential by Green's Theorem but what about exactness?
2. Without derivating $P$ and $Q$ just finding potential $\phi$ for the given equation how can I be sure about exactness?
3. What does exactness mean realy? Geometrically or intuitionaly ?

For $b=-2\pi$ and $a=1$
$$P\_y=Q\_x=-2\pi e^x\sin(\pi y^2+1)$$
Is it enough for exactness?

**Note:** I'm taking a course equivalent to vector calculus. Lecturer does not use a textbook and give us a refference book, he gives lectures in oldstyle. I want to learn the whole thing, any refference text would be appreciated.