I am having issues to calculate the principal curvatures and directions of a hypersurface. I have the hypersurface defined as $S=f(\mathbf{x})$ with $\mathbf{x}\in R^n$, and I can compute all the first and second derivatives respect to $x_i$ (where $\mathbf{x} = [x_1,...,x_n]$).
I need to compute the principal curvatures and directions at a generic point $\mathbf{x}_0$.
From what I understood, I need to compute the eigenvalues and eigenvectors of the shape operator ( or Weingartner map ) but I still have problem to figure out how to obtaining the shape operator from the derivative respect to $x_i$.
I found several example when $\mathbf{x} \in R^2$ (therefore a surface $\in R^3$), but I have some difficulties to extend that notion in higher dimensions.
Thank you in advance for your help