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I'm in Calc III without any linear algebra experience, so differential forms are a bit too high level for me right now. But, I want to know the reason behind all these theorems.

For example, why does $\dfrac{\partial Q}{\partial x} - \dfrac{\partial P}{\partial y}$ appear as the integrand in Green's Theorem? Could you derive this and the integrands in the other theorems without using the exterior derivative of differential forms?

Also, if you know of any books/resources for this type of level please let me know.

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  • $\begingroup$ There is no need for abstract differential forms and exterior derivatives to obtain Green's formula, all can be done completely within the one-dimensional calculus. $\endgroup$ – A.Γ. Oct 16 '15 at 11:16
  • $\begingroup$ In fact, the concept of differential forms is natural; need not be intimidated by the fact that differential forms usually appear later. $\endgroup$ – Megadeth Oct 16 '15 at 11:37
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Here is an outline for the proof of greens theorem:

  1. Prove greens theorem for a square. (This means, just compute it in this simple case using the fundamental theorem of calculus.)
  2. Chop up the region you want to apply greens theorem to into little squares, with some margin of error (if your squares are sufficiently small, you can almost tile the region, but not quite - think about trying to tile the bottom of a circular pool with tiny square tiles...). Apply the case of greens theorem in part one to turn the integral over the area into the sum of the line of integrals.
  3. Notice that many of the integrands around the edges are going to cancel. You will be left with a version of greens theorem on a region very close to your original one.
  4. Let the squares get really small to get greens theorem for your region.
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