Green's theorem for piecewise smooth curves Green's theorem is usually stated as follows:

Let $U \subseteq \mathbb{R}^2$ be an open bounded set. Suppose its boundary $\partial U$ is the range of a closed, simple, piecewise $C^1$, positively oriented curve $\phi: [0,1] \to \mathbb{R}^2$ with $\phi(t) = (x(t),y(t))$. Let $f,g: \overline{U} \to \mathbb{R}$ be continuous with continuous, bounded partial derivatives in $U$. Then
  $$ \int_U \left(\frac{\partial f}{\partial x}(x,y) - \frac{\partial g}{\partial y}(x,y)\right) dx dy = \int_{[0,1]} f(\phi(t))y'(t) dt + \int_{[0,1]} g(\phi(t))x'(t) dt. $$

Is there a complete rigorous proof of this theorem somewhere? Most texts (Rudin, Munkres, Spivak) proves the generalized Stokes' theorem first, and shows Green's theorem as a corollary. However, the "piecewise" criteria is never addressed, since the version of Stokes' proven in the texts above do not work for manifolds with corners. Hence the version of Green's theorem proven is always with full $C^1$ assumption.
Because Green's theorem is only on the plane, I'm wondering if there is an easy way to obtain the "piecewise" generalization from just the $C^1$ assumption. Spivak mentions

Green's theorem is true for a square ... can be proved by
  approximating the square ... by manifolds with boundary.

Is this easy to do in the context of just $\mathbb{R}^2$?

In "A First Course in Real Analysis" by Murray H. Protter and Charles B. Jr. Morrey Green's theorem is proved in paragraph 16.4 prior to proving Stokes' theorem. They prove Green's Theorem for so-called regular regions, which are regions whose boundary is given by piecewise-differentiable curves (along with some other conditions).
