$\newcommand{\Ga}{\Gamma}$


I am trying to find a proof of Green's formula for rectifiable Jordan curves $\Ga$ (and the corresponding interior regions $R$). There is a proof by Ridder, followed by very similar proofs by Verblunsky and Potts. The idea in those three papers, which looks quite natural, is to subdivide $R\cup \Ga$ into small regions $R\_i\cup \Ga\_i$ by vertical and horizontal lines. The problem is how to deal with the "partial" regions $R\_i\cup \Ga\_i$ that are not rectangles; more precisely, with the line integrals over the corresponding $\Ga\_i$'s. To deal with such line integrals rigorously, one first of all needs to parametrize the boundaries $\Ga\_i$. However, none of the mentioned three authors (or any other ones known to me) even mentions such a parameterization. 


In [Mathematical Analysis: a Modern Approach to Advanced Calculus, 1957, by Apostol](https://www.abebooks.co.uk/servlet/BookDetailsPL?bi=22745766642&searchurl=tn%3Danalysis%26sortby%3D20%26an%3Dapostol%2Btom%26fe%3Don&cm_sp=snippet-_-srp1-_-title1), an apparent attempt is made to make Ridder's approach rigorous. The Green formula in question is stated there as Theorem 10--43. The crucial part in the proof of Apostol's Theorem 10--43 is Theorem 10--42, which tries to deal with the mentioned "partial" regions $R\_i\cup \Ga\_i$. However, there are a few places in the proof of Apostol's Theorem 10--42 that I don't understand. 


Let $\Ga$ be a rectifiable Jordan curve bounding the corresponding interior region $R$. First, Apostol takes a lowest and highest points, $p\_1$ and $p\_2$ on $\Ga$. These two points are then separated by a horizontal line, say $\ell$. Then $p\_1$ and $p\_2$ are joined by "arcs" $C\_1$ and $C\_2$ to points $q\_1$ and $q\_2$ in $R$ close to $p\_1$ and $p\_2$, respectively, so that $q\_1$ is below $\ell$ and $q\_2$ is above $\ell$. 



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> **Question 1.** Apostol says such "arcs" (possibly non-rectifiable) exist, without proof. I do not immediately see how to prove this, even though it's probably simple. (The $C\_j$'s are never mentioned again in the proof, after they are introduced.) 
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Then the points $q\_1$ and $q\_2$ are connected by a polygonal line $C\subset R$, which must have an intersection point with $\ell$. Without loss of generality, there are finitely many such points. 
For each such point $p$, let $[u,v]$ be the maximal segment on $\ell$ such that the open interval $(u,v)$ is contained in $R$. Then Apostol says there must be at least one such maximal segment $[u,v]$ that has an odd number of intersections with the polygonal line $C$; I guess this follows because the total number of the intersections of $C$ with $\ell$ can be proved to be odd, by induction. But then Apostol just says: "Using such a segment, it is clear that $L[u,v]$[$=[u,v]$ -- I.P.] forms, with $\Ga$, two rectifiable Jordan curves $\Ga\_1$ and $\Ga\_2$, one of which contains $p\_1$ (call this one $\Ga\_1$) and the other contains $p\_2$. Moreover, these curves form the boundaries of two regions $R\_1$ and $R\_2$, whose union is $R$. The positively oriented boundaries $\Ga(R\_1)$ and $\Ga(R\_2)$ of $R\_1$ and $R\_2$ such that $\Ga(R)=\Ga(R\_1)+\Ga(R\_2)$." 


There is a picture there, making the above claims plausible. Yet, here is my main question: 



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> **Question 2.** Nothing in that passage is clear to me. How does this "forming" of curves $\Ga\_1$ and $\Ga\_2$ occur? Are they Jordan curves? If so, why? How are $\Ga\_1$ and $\Ga\_2$ parameterized? Why is the union of $R\_1$ and $R\_2$ equal to $R$? What precisely are the roles of the points $q\_j$ and the "arcs" $C\_j$ in this proof? 
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Can one help me decipher this proof? Or, perhaps even better, is there a completely rigorous, fully detailed proof of this obviously important and natural result? (Of course, there is a general Stokes theorem for manifolds, but it does not seem at all obvious how to build a bridge from there to Jordan curves.)



Proof of Green's formula for rectifiable Jordan curves
One can circumvent the technical difficulties as follows. Consider a large ball $K$ containing $\Gamma$ and any $p>2$. Given a function $f\in L^p(K)$, we can define its Cauchy transform 
$$
\left(\mathcal{C}f \right)(z)=\frac{1}{\pi}\int\_K\frac{f(w)}{z-w}.
$$
By Hölder inequality, this is a continuous operator from $L^p(K)$ to $C(K)$. This operator inverts the $\bar{\partial}$ operator, namely, $\mathcal{C}\bar{\partial}\varphi\equiv\varphi$ for any $\varphi \in C^1\_0(K)$; see the computation after the equation (8) in Chapter 5 of "Lectures on quasiconformal mappings" by Ahlfors (1966).


Now, the composition of $\mathcal C$ and integration over $\Gamma$ is a continuous linear functional on $L^p(K)$:
$$
L(f):= \frac{1}{2i}\int\_\Gamma \left(\mathcal{C}f\right)(z)dz=\int\_K f(z)\psi(z),
$$
for some $\psi \in L^{p'}(K)$. It follows that for any $\varphi \in C^1\_0(K)$, we have 
$$
\frac{1}{2i}\int\_\Gamma \varphi(z) dz = \int\_K\psi(z)\bar{\partial}\varphi(z).
$$Hence, it remains to check that 
$\psi\equiv 1$ inside $\Gamma$ and $\psi \equiv 0$ outside; since $\Gamma$ itself is rectifiable, its area is zero. To this end, it suffices to check that $L(f)=\int\_{\text{inside }\Gamma} f$ for any $f\in C\_0(K\setminus\Gamma)$. For such an $f$, we can exchange the integrals:
$$
L(f)=\frac{1}{2\pi i}\int\_\Gamma dz\int\_K \frac{f(w)}{z-w}=\int\_K f(w)\frac{1}{2\pi i} \int\_\Gamma \frac{dz}{z-w}.
$$
The inner integral can be interpreted as the increment of $\log (z-w)$ along $\Gamma$, and thus it is equal to the winding number of $\Gamma$ w. r. t. $w$. Therefore, the result follows from the topological statement that any Jordan curve has winding number $\pm 1$ w. r. t. a point inside it, and $0$ w. r. t. a point outside. Some references for this statement are discussed [here](https://mathoverflow.net/q/231852/56624).


