If $M\subset\mathbb{R}^2$ is a plane domain and the image of parametrized surface $f:M\to \mathbb{R}^3$ is contained in \[\mathbb{R}^2=\{(x,y,z)\in \mathbb{R}^3\,\,|\,\, z=0\}\] then the defining equations of a conformal map \begin{align*}\left|f_u\right|&=\left|f_v\right| \\\\ \langle f_u,f_v\rangle &=0\end{align*} imply that $\left|f_v\right|$ arises from $\left|f_v\right|$ by … Continue reading Conformal Maps I: Holomorphic Functions