Are these hypothesis of a Green's theorem's proof too restrictive?

Question

I found a proof of Green's theorem on a pdf and tried to understand it. Becuase the proof in itself is clear to me, I'm wondering if the hypothesis are too restrictive or if it can be a "good" general proof. Let $D$ be a limited subset of $R^2$, simple with respect to the $x-axis$, that is $D=[(x,y)\in R^2: a\le y\le b, f_1(y)\le x\le f_2(y)]$, and simple with respect to the $y-axis$, that is $D=[(x,y)\in R^2: c\le x\le b, g_1(x)\le y\le g_2(x)]$, with its boundary, $\partial D$, being the union of (the supports of) a finite number of regular curves. We also say that its boundary, $\partial D$, is positively oriented if its oriented in the counter-clockwise direction and we write in this case $\partial D^+$. Let $F(x,y)=(F_1(x,y),F_2(x,y))$ be a vector field on $D$ with $F\in C^1(A)$ where $A$ is an open set containing $D$. With these hypothesis, we have $\oint_{\partial D^+}F\cdot \tau=\iint_{D}(F_{2x}-F_{1y})dxdy$ which is Green's formula.

Answer 1

Theorem 1.7 - Let $D$ be a bounded domain in $\mathbb{R}^2$ that's simple with respect to both axes. If $\mathbf{F} = P\mathbf{i}+Q\mathbf{j} \in C^1(D)$, then the formula holds:

$$ \iint_D (Q_x-P_y)\,\text{d}x\,\text{d}y = \int_{\partial^+D} P\,\text{d}x+Q\,\text{d}y\,. \tag{1.6} $$

Formula (1.6) holds for much more general domains than those considered so far. We limit ourselves to observing that this class includes domains $D$ which can be expressed as a union of simple domains, which intersect only along the boundary. The positive orientation of $\partial D$ is obtained by orienting the single curves that compose it in such a way that, following them, the domain is left to one's left. The typical case is that in which $\partial D$ is the union of a $\gamma$ curve and $N$ curves $\gamma_1,\gamma_2,\dots,\gamma_N$ internal to $\gamma$ as in the figure:

$\quad\quad\quad\quad\quad\quad\quad\quad$

from which:

$$ \iint_D (Q_x-P_y)\,\text{d}x\,\text{d}y = \int_{\gamma^+\cup\gamma_1^+\cup\gamma_2^+} P\,\text{d}x+Q\,\text{d}y\,. \tag{1.8} $$

This is the simplest version of the Gauss-Green theorem in the plane that I copied from Matematica - Calcolo Infinitesimale e Algebra Lineare - Bramanti, Pagani, Salsa - Zanichelli (first edition: September 2000). It's also the only one I know and that has always been enough for me in decades of application work.

---

EDIT: I recommend also consulting the excellent Analisi Matematica II - Teoria ed esercizi - Canuto, Tabacco - Springer (second edition: November 2014) where they directly prove the version of the theorem referring to a G-admissible domain, i.e. of the type of figure.