Isothermal Parameterization

A parameterization is isothermal if, for $\zeta\equiv u+iv$ and

$$\phi_k(\zeta)={\partial x_k\over\partial u}-i{\partial x_k\over\partial v},$$

the identity

$${\phi_1}^2(\zeta)+{\phi_2}^2(\zeta)+{\phi_3}^2(\zeta)=0$$

holds.

See also Minimal Surface, Temperature