Proof of Cauchy's Theorem, differential forms, exterior derivative, wedge product

Question

Can anyone direct me to a proof of Cauchy's theorem working in terms of differential forms, and using the properties of the exterior derivative and the wedge product?

Answer 1

Let $K$ be a compact of $\mathbb{C}$ such that $\partial K$ is a regular curve. (One can discuss a lot about the smallest hypothesis about the curve) Using Stokes theorem, we have

$$\int_{\partial K} f(z)dz = \int_{K} d(fdz) = \int_{K} \frac{\partial f}{\partial z} dz \wedge dz + \frac{\partial f}{\partial \overline{z}} d\overline{z} \wedge dz =0.$$ Indeed, we have $dz\wedge dz =0$ and $\frac{\partial f}{\partial \overline{z}}=0$ since we assume $f$ to be holomorphic on an open set containing $K$.