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What the book is trying and failing to explain is called cohomology in general. In this particular case, the problem is that when there's a hole in your domain, such as if you didn't consider the origin, then you can have differentials where integrating along a path between two points depends on whether the path goes around the hole or not.
For instance, the function 1/(x+iy) in the complex plane is not defined at (0, 0), but if we ignore (0, 0) and look at everywhere else, 1/(x+iy) is well-defined. However, if we integrate from (1, 0) to (1, 0) along a path that goes counterclockwise around the origin once, we end up with 2πi, even though normally integrating along a closed path would give 0. A path that goes around twice would would give us 4πi, and a path that goes around clockwise around the origin once would give -2πi, and so on.
So we want to know when an integral is path-dependent, and when it isn't, and that's where exact differentials come in. The integrals of exact differentials are not path-dependent. The Poincare lemma gives us a way to tell if a differential is exact when our domain doesn't have any holes in it (more specifically, when the domain is contractible).
Your book really shouldn't be using df for its differential as that suggests right off the bat that the differential is exact.
It should be using g(x,y)dx + h(x,y)dy as the general differential, and thus saying that if (∂g/∂y) = (∂h/∂x) then we get that d(g(x,y)dx + h(x,y)dy) = 0, so g(x,y)dx+h(x,y)dy is closed, and hence by the Poincare lemma we get that on a contractible open space (basically a portion of R² without any holes in it), g(x,y)dx+h(x,y)dy = df (x,y) for some f, i.e that g(x,y) = (∂f/∂x)(x,y) and (∂h/∂y)(x,y).
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