Minimal Surface

5.5 Curvatures of Minimal Surface When a minimal surface $r(u,v) = (x(u,v),y(u,v),z(u,v))$ is represented by an isothermal coordinate system, we define $\phi_1(w), \phi_2(w), \phi_3(w), f(w)$ and a meromorphic function $g(w)$ as: $$ \phi_1(w) = \frac{\partial x}{\partial u} - i \frac{\partial x}{\partial v}, \quad \phi_2(w) = \frac{\partial y}{\partial u} - i \frac{\partial y}{\partial v}, \quad \phi_3(w) = \frac{\partial z}{\partial u} - i \frac{\partial z}{\partial v}, \quad f = \phi_1 - i\phi_2, \quad g = \frac{\phi_3}{\phi_1 - i\phi_2} $$ ...

In this section, we use the calculus of variations to find the equations of minimal surfaces, and before we begin, we need to understand what a calculus of variations is. For ease of understanding, the variational will be introduced in more general terms. Before we start to introduce calculus of variation, we need to recall what is differentiation. Suppose $f(x) : D \to \mathbb{R}$ be a continuous and differentiable function, When the independent variable $x$ undergoes a very small change $dx$, it is not difficult to see that the function value also changes due to the change in the independent variable; this is called differentiation of the function. ...