This is the (hopefully accurate) translation of a test question on the elementary theory of surfaces (it was originaly written in my native language).
The context is that we are given the surface induced by $f(x,y)=xy$ and we are first asked to find the normal vector $\mathbf{n}(x,y)$ on it, give the matrices of the first and the second fundamental form and subsequently the Gaussian curvature at a random point $(x,y)$. All these are just fine, but...
The last question is — to me at least — incomprehensable; it reads:
‘Let $\gamma (\theta)$ be the plane curves generated by the intersection of the surface with vertical planes which pass through $(0,0,0)$ and form an angle $\theta$ with the positive $x$-axis. Find the maximum and the minimum curvature of $\gamma (\theta)$ at point $(0,0,0)$ and the respective values of $\theta$. Also, show that for all but two angles vector field $\mathbf{n}$ does not lie on the plane of intersection and thus the vertical vector field of the curve of intersection is linearly independent of $\mathbf{n}$.’
It's kinda personal now! Am I so dump that I don't understand that ‘easy’ differential geometry question, or does it realy need to be rephrased more clearly? Where is that so much celebrated mathematical rigour anyway?
$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Del}{\nabla}\newcommand{\Vec}[1]{\mathbf{#1}}\newcommand{\Brak}[1]{\left\langle #1\right\rangle}$If $\Vec{n}$ is a continuous (hence smooth) unit normal field on a regular surface $M$ in $\Reals^{3}$, and if $\Vec{v}$ is a tangent vector at some point $p$, then the covariant derivative $S(\Vec{v}) := \Del_{\Vec{v}} \Vec{n}$ lies in the tangent plane at $p$, and $$ \Brak{S(\Vec{v}), \Vec{w}} = \Brak{\Del_{\Vec{v}} \Vec{n}, \Vec{w}} = \Brak{\Del_{\Vec{w}} \Vec{n}, \Vec{v}} = \Brak{S(\Vec{w}), \Vec{v}} $$ for all tangent vectors $\Vec{v}$ and $\Vec{w}$ at $p$. That is, $S:T_{p}M \to T_{p}M$ is a symmetric linear operator, hence orthogonally diagonalizable.
A unit tangent vector $\Vec{v}$ (black) determines a normal section, the intersection of $M$ with the plane spanned by $\Vec{n}$ and $\Vec{v}$. The covariant derivative of $\Vec{n}$ along $\Vec{v}$, i.e., $S(\Vec{v})$ (green), is generally not proportional to $\Vec{v}$.
The eigenspaces of $S$ are spanned by tangent vectors $\Vec{v}$ for which $\Del_{\Vec{v}} \Vec{n}$ is proportional to $\Vec{v}$. The corresponding eigenvalue, a principal curvature, is minus the signed curvature of the corresponding normal section (blue).