\( \newcommand{\bbc}{\mathbb{C}} \newcommand{\Int}{\operatorname{Int}} \)Navigation: 1. Exact, conservative and closed forms | 2. Locally exact forms and singular homology | 3. Applications to complex analysis Preliminaries Let \(F\) be a complex Banach space, let \(U\subseteq\bbc\) be an open set, and let \(f:U\to F\). Recall that the complex Fréchet derivative of \(f\) at \(z\in U\), if it […]

Navigation: 1. Exact, conservative and closed forms | 2. Locally exact forms and singular homology | 3. Applications to complex analysis As in part 1, \(E,F\) are Banach spaces and \(U\subseteq E\) is an open set. Recall that a form \(\omega:U\to L(E,F)\) is exact if \(\omega=Df\) for some \(f:U\to F\), closed if it is differentiable […]

Navigation: 1. Exact, conservative and closed forms | 2. Locally exact forms and singular homology | 3. Applications to complex analysis The line integral is a useful tool for working with vector fields on \(\mathbb{R}^n\), (co)vector fields on manifolds, and complex differentiable functions. However, it is often unclear how these different versions of the line […]