A Cluttered Mind

$\newcommand\C{\mathbb{C}}$ I came up with the proof below while preparing a lecture for my complex analysis course. I learned from Sinan Gunturk that Peter Lax showed him the same proof back in 2007. Let \(O\subset\C\) …

$\newcommand\C{\mathbb{C}}$ This is a follow-up to Quick Proof of Cauchy Integral Formula. That proof is based on the standard proof, given below, of the homotopy invariance of the contour integral of a holomorphic function. Let …

$\newcommand\R{\mathbb{R}}\newcommand\C{\mathbb{C}}\newcommand\Z{\mathbb{Z}}$ Chain Rule for Map between Open Subsets of Euclidean Space Let $M$ be an open subset of $\R^m$, $N$ be an open subset of $\R^n$, and $O$ be an open subset of $\R^p$. Let …