I am going through a proof of Green's Theorem for a simple region and I understand the mathematics taking place but do not understand the origins.
'Regions that are simultaneously of type I and II are “nice” regions' is a statement I don't fully understand.
I have proved mathematically that: $$-\iint_G \Bigg(\frac{\partial Q}{\partial x}\Bigg) \cdot dA = \int_{\partial G}Q \cdot dy$$ and $$-\iint_G \Bigg(\frac{\partial P}{\partial y}\Bigg) \cdot dA = \int_{\partial G}P \cdot dx$$ But I do not understand the two types of region being discussed and how a region can be simultaneously both.
Any help understanding this concept would be greatly appreciated.
A Type I region is essentially one that can be glued together with adjacent vertical strips. A Type II region can be glued together with horizontal strips. A square region would be both Type I and Type II, for example.
As an example that is Type I but not Type II, consider the region bounded by $x=0$, $x=100\pi$, $y=-1$, and $y=\sin x$. It can be decomposed into vertical strips, but horizontal slices would be disconnected.
The Wikipedia article on Green's Theorem goes into more detail.