Minimal surfaces in the small

Abstract

Given a surface

$$ \begin{array}{*{20}c} {S:x = x\left( {u,v} \right),} & {y = y\left( {u,v} \right),} & {z = z\left( {u,v} \right),} \\ \end{array} $$

it is convenient to use vector notation and write simply $\mathfrak{x}{\text{ = }}\mathfrak{x}\left( {u,v} \right),$ where $\mathfrak{x}\left( {u,v} \right)$ denotes the vector with components x (u, v) , y (u, v) , z(u, v).