Editing Green's theorem (section)

== Validity under different hypotheses ==
The hypothesis of the last theorem are not the only ones under which Green's formula is true. Another common set of conditions is the following:

The functions <math>A, B:\overline{R} \to \R</math> are still assumed to be continuous. However, we now require them to be Fréchet-differentiable at every point of <math>R</math>. This implies the existence of all directional derivatives, in particular <math>D_{e_i}A=:D_i A, D_{e_i}B=:D_i B, \,i=1,2</math>, where, as usual, <math>(e_1,e_2)</math> is the canonical ordered basis of <math>\R^2</math>. In addition, we require the function <math>D_1 B-D_2 A</math> to be Riemann-integrable over <math>R</math>.

As a corollary of this, we get the Cauchy Integral Theorem for rectifiable Jordan curves:

{{math theorem|name=Theorem (Cauchy)|math_statement=If <math>\Gamma</math> is a rectifiable Jordan curve in <math>\Complex</math> and if <math>f : \text{closure of inner region of } \Gamma \to \Complex</math> is a continuous mapping holomorphic throughout the inner region of <math>\Gamma</math>, then <math display="block">\int_\Gamma f=0,</math> the integral being a complex contour integral.}}

{{math proof|proof=We regard the complex plane as <math>\R^2</math>. Now, define <math>u,v:\overline{R} \to \R</math> to be such that <math>f(x + iy) = u(x, y) + iv(x, y).</math> These functions are clearly continuous. It is well known that <math>u</math> and <math>v</math> are Fréchet-differentiable and that they satisfy the Cauchy-Riemann equations: <math>D_1 v + D_2 u = D_1 u - D_2 v = \text{zero function}</math>.

Now, analyzing the sums used to define the complex contour integral in question, it is easy to realize that
<math display="block">\int_\Gamma f=\int_\Gamma u\,dx-v\,dy\quad+i\int_\Gamma v\,dx+u\,dy,</math>
the integrals on the RHS being usual line integrals. These remarks allow us to apply Green's Theorem to each one of these line integrals, finishing the proof.}}