Consider an isothermic parameterization with metric $$(g_{ij}) = \begin{bmatrix} \lambda^2 & 0 \\ 0 & \lambda^2\end{bmatrix}$$ with $\lambda = \lambda(u^1,u^2)>0$. Then the gaussian curvature can be calculated by $$K=\frac{\lambda_1^2+\lambda_2^2}{\lambda^4} -\frac{\lambda_{11}+\lambda_{22}}{\lambda^4}$$
I know that I can get the curvature as the determinant of the Weingarten map or as the quotient of the determinant of the second and first fundamental form, but I don't quite know how I can get to this just from the given metric.
Can anyone point me to the right direction with this? Thanks.
You need the Gauss equation, most conveniently in an orthogonal parametrization. See most any differential geometry text, including my own :)