Green's theorem

Let $C$ be a positively oriented and consists of a finite union of disjoint,piecewise smooth simple closed curve in a plane, and let $D$ be the region bounded by $C$. If $P$ and $Q$ are functions of $(x, y)$ defined on an open region containing $D$ and have continuous partial derivatives there, then $$\oint_C (Pdx+Qdy)=\iint_D (\frac{\partial Q }{\partial x}-\frac{\partial P}{\partial y})dxdy$$where the path of integration along $C$ is anticlockwise.

In almost all textbooks, authors only proved it when $D$ is a simple region or the region $D$ can be cut up into finitely many simple regions without common interiors.My questions is how to generalize those special regions to more general regions as Green's theorem gives? I need a rigorous complete proof.Likewise Green's theorem,the divergence theorem and Stokes's theorem have the same situation requiring some rigorous proofs from very special regions to more general regions.

  • $\begingroup$ You may consult Federer, Geometric measure theory, or (for the beginners) F. Morgan, same title. $\endgroup$ – Alexandre Eremenko Apr 16 at 2:29
  • $\begingroup$ @AlexandreEremenko Professor, thanks for your references.But is there a rigorous proof only applying the knowledge/tools of real analysis, such as partition of unity? $\endgroup$ – Nemo Apr 16 at 3:00
  • $\begingroup$ Morgan's nook is aimed at advanced undergraduates. $\endgroup$ – Alexandre Eremenko Apr 16 at 10:03

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