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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.

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  • $\begingroup$ You should include the definitions of type I and type II. $\endgroup$ – eyeballfrog Mar 21 at 21:00
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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.

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  • $\begingroup$ Thank you Theophile, the example of the square is a basic but helpful example. $\endgroup$ – Keighleyite Mar 21 at 21:11
  • $\begingroup$ @J.Bry You're welcome. In general, any convex shape will be both Type I and II, although (as the Wiki example shows) convexity is not a requirement. $\endgroup$ – Théophile Mar 21 at 23:21

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