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  • The proof of Green's Theorem for rectangle is very simple.
  • The proof for triangle is not too bad, and the general case follows ( at least intuitively ) from the triangle case.
  • But I'm lazy.
  • So is there a quick way to deduce the general case, or at least the triangle case from the rectangle case ?.

An attempt to make the question more precise:

  • Suppose for each smooth Vector Field $\mathbf{F}$ and each piecewise smooth simple closed curve $C$ we have a number $\langle\mathbf{F},C\rangle$, and the map $(\mathbf{F},C)\mapsto\langle\mathbf{F},C\rangle$ satisfies various nice properties such as "additivity" over regions, and $$\langle\mathbf{F},C\rangle=\iint_R \left(\frac{\partial F_y}{\partial x}-\frac{\partial F_x} {\partial y}\right)dxdy$$ whenever $C$ bounds a rectangle $R$, must the formula be true for arbitrary $C$ ?.
RobPratt
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new account
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  • (1) Any shape can be approximated by a large number of very small rectangles.

    (2) The dot product on the inner sides of the rectangles cancel each other out to reduce to the larger loop alone.

     – Paul Jul 08 '24 at 00:39
  • Once you have it for triangles, you can generalize to piecewise linear shapes by using vertex subdivision. Then you can use continuity to go to a more general class of curves. – Lee Fisher Jul 08 '24 at 00:39
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    @Paul I hesitate to approximate a shape by small rectangles, because it doesn't work for line integral of scalar function. That it works for line integral of vector field is truly a miracle in my opinion. Approximation by triangles does work, but that's exactly what I want to avoid here... – new account Jul 08 '24 at 00:43
  • @newaccount point 2 does not apply to the scalar function case. – Paul Jul 08 '24 at 00:54
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    @Paul I mean look at those zig-zag curves in the link. Their lengths certainly don't converge to the circumference of the circle, but their line integrals of a vector field $\mathbf{F}$ do converge to the line integral along the circle. How to explain the difference? – new account Jul 08 '24 at 00:59
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    I wouldn’t recommend the argument by approximation via rectangles. The best way is to use partitions of unity; this is just a slight shift in philosophy where rather than chopping up the domain, you chop up your vector field (or more generally, differential forms). – peek-a-boo Jul 08 '24 at 01:04

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