I have two questions related the the surface $S$, $$x(u,v) =(u,v,u^2 + uv)$$ which I was hoping somebody might be able to help with:
- Give the normal curvature of $S$ at the point $x(0,0)$ in the direction $dx/du + dv/du$.
I think I need to use Meusnier's theorem but I'm not sure how.
- Find the principe directions of $S$ at $x(0,0)$.
I have calculated the first and second fundamental forms at this point to be $E=1$ , $F=0$, $G=1$, $L=2$, $M=1$, $N=0$, this gives me a matrix for the Weigengarten map to be $$\begin{pmatrix} 2 & 1\\ 1 & 0 \end{pmatrix}.$$
Are the eigenvectors of this matrix the directions I am looking for?
Thanks for your help