# Extending Green's theorem from very special regions to more general regions

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.

• You may consult Federer, Geometric measure theory, or (for the beginners) F. Morgan, same title. – Alexandre Eremenko Apr 16 at 2:29
• @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? – Nemo Apr 16 at 3:00
• Morgan's nook is aimed at advanced undergraduates. – Alexandre Eremenko Apr 16 at 10:03