Gaussian curvature $K$ of of orthogonal parametrization $X$ Let $X$ be an orthogonal parametrization of some surface $S$.  Prove that the Gaussian curvature $K = - \frac{1}{2 \sqrt{E G}} ((\frac{E_{v}}{\sqrt{E G}})_{v} + (\frac{G_{u}}{\sqrt{E G}})_{u})$, where subscripts denote partial differentiation of the quantity with the subscript with respect to the terms within the subscript, and where $E$, $F$, and $G$ give the first fundamental form of $S$ by $X$.

Orthogonal parematrization means that the first fundamental form has $F=0$.  We assume sufficient niceness of the surface $S$ (so that we never divide by $0$ and all functions are infinitely differentiable in all arguments, etc.).
We first derive two related results, where $\Gamma_{i,j}^{k}$ denotes Christoffel symbols of the first kind:
$\Gamma_{1,1}^{1} F + \Gamma_{1,1}^{2} G = X_{u,u} \cdot X_{v} = (X_{u} \cdot X_{v})_{u} - \frac{(X_{u} \cdot X_{v,u})}{2} = F_{u} - \frac{1}{2} E_{v}$.
Likewise, $\Gamma_{1,2}^{1} F + \Gamma_{1,2}^{2} G = X_{u,v} \cdot X_{v} = \frac{1}{2} G_{u}$.
Next recall the formula $K = \frac{1}{\sqrt{E G - F^2}} (\frac{\partial}{\partial v} (\frac{\sqrt{E G - F^2}}{E} \Gamma_{1,1}^{2}) - \frac{\partial}{\partial u} (\frac{\sqrt{E G - F^2}}{E} \Gamma_{1,2}^{2}))$.
From these equations, constraining $F=0$, it follows immediately by substitution:
$K = \frac{1}{\sqrt{E G}} (\frac{\partial}{\partial v} (-\frac{1}{2} \sqrt{\frac{G}{E}} \frac{E_{v}}{G}) - \frac{\partial}{\partial u} (\frac{1}{2} \sqrt{\frac{G}{E}} \frac{G_{u}}{G})) = - \frac{1}{2 \sqrt{E G}} ((\frac{E_{v}}{\sqrt{E G}})_{v} + (\frac{G_{u}}{\sqrt{E G}})_{u})$. QED

Bonus information
The isothermal formula for Gaussian curvature $K$ follows immediately.  The isothermal case is a special case of orthogonal parametrization ($F=0$) in which $E = G= \lambda \dot{=} \lambda (u,v)$.
In this case: $K = -\frac{1}{2 \sqrt{\lambda^2}} ((\frac{\lambda_{v}}{\lambda})_{v} + (\frac{\lambda_{u}}{\lambda})_{u}) = -\frac{1}{2 \lambda} ((log({\lambda})_{v})_{v} + (log({\lambda})_{u})_{u}) = -\frac{1}{2 \lambda} (log({\lambda})_{v,v} + log({\lambda})_{u,u}) = -\frac{1}{2 \lambda} (\frac{\partial^2}{(\partial u)^2} + \frac{\partial^2}{(\partial v)^2})(\log({\lambda})) = -\frac{1}{2 \lambda} \Delta(\log\lambda)$, where the penultimate equation uses a formal sum of derivative (left) operators for convenience and the cleaning up of notation and where the last equation further simplifies it by denoting this formal sum as the Laplacian operator $\Delta$.
