My book says "A differential expression $M(x, y)dx+N(x, y)dy$ is an exact differential in a region $R$ of the $xy$-plane if it corresponds to the differential of some function $f(x, y)$ defined on $R$"
What I understand is that $M(x, y)dx+N(x, y)dy$ is called an exact differential if it is equal to a function that is integrable (its indefinite integral exists). Is my understanding correct?
That the differential $Mdx+Ndy$ is exact means that for some $f$, $\frac{\partial f}{\partial x}=M$ and $\frac{\partial f}{\partial y}=N$, i.e. $df=Mdx+Ndy$. In general, if $f:\Bbb R^n\to\Bbb R$ is differentiable, $df=\sum \frac{\partial f}{\partial x_i}dx_i$. On the other hand, the differential is closed if $d(Mdx+Ndy)=0$ that is if $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$. Exact differentials are always closed. It is a theorem of Poincaré that the converse holds in star shaped regions in $\Bbb R^n$ or more general when the space is "contractible to a point."