TITLE: Is there any connection between Green's Theorem and the Cauchy-Riemann equations?
QUESTION [43 upvotes]: Green's Theorem has the form: 
$$\oint P(x,y)dx = - \iint \frac{\partial P}{\partial x}dxdy , \oint Q(x,y)dy = \iint \frac{\partial Q}{\partial y}dxdy $$
The Cauchy-Riemann equations have the following form:(Assuming $z = P(x,y) + iQ(x,y)$)
$$\frac{\partial P}{\partial x} = \frac{\partial Q}{\partial y}, \frac{\partial P}{\partial y} = - \frac{\partial Q}{\partial x}$$
Is there any connection between this two equations?

REPLY [3 votes]: That's Green's Theorem can be used to prove Cauchy's Theorem or a corollary of it, depending on your text, is in A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 4.25

Note that that $f'$ assumption is superfluous because $f$ is holomorphic, but this is not proven until Ch5.

Appendix
Green's Theorem is stated as:



Cor 4.20 is a corollary of Cauchy's Thm 4.18 for the authors and is stated as:



Cauchy's Thm 4.18 is stated as:



The authors acknowledge that Cauchy's Theorem is sometimes designated to be the statement in Cor 4.20 instead of their Thm 4.18