Reference for proof of Green's theorem

Question

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.

Answer 1

$\newcommand{\curl}{\operatorname{curl}} \newcommand{\dm}{\,\operatorname{d}}$ I do not know a reference for the proof of the plane Green’s theorem for piecewise smooth Jordan curves, but I know reference [1] where this theorem is proved in a simple way for simple rectifiable Jordan curves without any smoothness requirement.

The key assumptions in [1] are

• a “geometrical” definition of the curl of a vector field as a limit of the value of the circulation integral around the contour of a contracting domain (precisely a $\varepsilon$-side length square $Q_\varepsilon$ centered at the point $(x,y)\in\Bbb R^2\equiv \Bbb C$) $$ \curl F(x,y)= \lim_{\varepsilon\to 0}\dfrac{\displaystyle\oint\limits_{\partial Q_\varepsilon}F\cdot\dm\ell}{|Q_\varepsilon|}= \lim_{\varepsilon\to 0} \frac{1}{\varepsilon^2}\oint\limits_{\partial Q_\varepsilon}F\cdot\dm\ell\label{1}\tag{C} $$ where $| Q_\varepsilon|$ is the area of the vanishing domain
• the hypothesis that the limit in \eqref{1} is uniform.

This allows the Author to prove the theorem in great generality but yet in a relatively simple way.

Notes

In reference [1], pp. 703-704, the author briefly reviews other proofs of the theorem found in contemporary literature. In particular he cites Spivak’s Calculus on manifolds for a proof of the result for piecewise smooth curves: however, Spivak proves the result in the form of a general Stokes’s theorem for $n$-dimensional manifolds and their (piecewise smooth) boundaries.

Reference

[1] Greenlee, W. M., “On Green’s theorem and Cauchy’s theorem”, Real Analysis Exchange, 30(2004-2005), No. 2, 703-718, MR2177428, Zbl 1098.53012.

Answer 2

Too long for a comment.

I googled a bit but, surprisingly, found no answer exactly matching your natural question. Nevertheless, according to Section 600 (§3 of Chapter XVI) of the book [Fich], Green’s theorem indeed holds for a domain (D) bounded by one or several piecewise-smooth contours. Unfortunately, the author skips some notations, so I had to guess on an exact form of the Green’s theorem he proves. I guess it is following. If the functions $P$, $Q$, $\frac{\partial P}{\partial y}$, and $\frac{\partial Q}{\partial x}$ are continuous in $(D)$ then $$\int_{(L)} Pdx+Qdy=\iint_{(D)}\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dxdy.$$ As I understood, the idea of the proof of the general case is to use approximations of the contour $(L)$ by broken lines surrounded by $(L)$ and by broken lines surrounding $(L)$.

Also I found paper [Zen]. It looks containing a detailed proof of Green’s theorem in the following form.

Making use of a line integral defined without use of the partition of unity, Green’s theorem is proved in the case of two-dimensional domains with a Lipschitz-continuous boundary for functions belonging to the Sobolev spaces $W^{1,p}(\Omega)\equiv H^{1,p}(\Omega)$, ($1\le p<\infty$).

References

[Fich] Grigoriy Fichtenholz, Differential and Integral Calculus, v. III, 4-th edition, Moscow: Nauka, 1966, (in Russian). According to Wikipedia “book was translated, among others, into German, Chinese, and Persian however translation to English language has not been done still”. The guessed proof of Green’s theorems consists of a bit less than four pages: 174, 175, 176, 177, and 178.

[Zen] Alexander Ženíšek, Green's theorem from the viewpoint of applications, Applications of mathematics 44:1 (1999), 55–80.

Answer 3

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.)