TITLE: Reference for proof of Green's theorem
QUESTION [7 upvotes]: I'm looking for a rigorous proof of Greens theorem for piecewise smooth jordan curves and would appreciate if someone could link a reference text. The only proof I've seen works for regions which can be bounded by curves $\{(x,y)\in \mathbb{R}^2: a\leq x \leq b,\, \phi(x)\leq y \leq \psi(x)\}$.
The article on wikipedia seems to be lacking several details.

REPLY [2 votes]: I've yet to come across a rigorous definition of "piecewise smooth". Nečas (1967), Direct Methods in the Theory of Elliptic Equations (section 3.1.2) proves Green's theorem for sets in $\mathbb{R}^n$ with Lipschitz boundary, which includes the case where $\Omega$ has piecewise $C^\infty$ boundary and the turning angle at each corner is strictly between $-\pi$ and $\pi$. (Lipschitz boundary for $\Omega \in \mathbb{R}^2$ means for each $x \in \partial \Omega$, we can define coordinate axes centred at $x$ and choose some open rectangle $V \subset \mathbb{R}^2$ containing $x$, such that $\partial \Omega \cap V$ can be written as $\{(t,a(t)) : -\epsilon_1<t<\epsilon_1\}$ and $\Omega \cap V$ can be written as $\{(t,s) : -\epsilon_1<t<\epsilon_1,a(t)<s<\epsilon_2\}$ in the new coordinates.)