I need help with this differential geometry problem dealing with Gaussian curvature and mean curvature
Given an ellipsoid with parametric representation $(a\cos u\cos v, b\cos u \sin v, c\sin u)$ compute its Gaussian curvature and mean curvature.
(Hint: Compute $E,F,G$ first and $L, M, N$ second then Weingarten map (Shape operator). The Gaussian curvature and the mean curvature are the determinant and the trace of the Weingarten map (Shape operator).
This is how I was taught.
If $M$ is a surface and a point $p\in M$ then let $\kappa_1, \kappa_2$ be the principal curvatures of the surface $M$ at the point $p$. Then the product of the principal curvatures is the Gaussian curvature $K = \kappa_1 \kappa_2$. The mean curvature is $A = \frac{\kappa_1 +\kappa_2}{2}$.
Now the principal curvature are the eigenvalues of the shape operator denoted by $S_p$ of a surface $M$ at a point $p$. To find the eigenvalues you must solve the equation $\text{det}(\lambda I - A) = 0$ where $A$ is the matrix of the shape operator $S_p$. To find the matrix of the shape operator you must compute the First and Second Fundamental forms.
Here is a Wolfram Alpha link of the Gauss curvature of an ellipsoid.
