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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?

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    The last question is getting at the normal curvatures and their extrema, the [principal curvatures](https://en.wikipedia.org/wiki/Principal_curvature), a.k.a., eigenvalues of the [shape operator](https://en.wikipedia.org/wiki/Differential_geometry_of_surfaces#Shape_operator). In general, the eigenspaces (principal directions) are mutually-perpendicular because the shape operator is symmetric. Presumably your instructor wanted to be sure you understand the underlying geometry as well as the analytic formulas. (Here, the principal directions are $y = \pm x$.) – 2017-01-18
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    @AndrewD.Hwang WOW, you are Kobayashi's student. I love his book... – 2017-01-18
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    @AndrewD.Hwang It certainly looks like something related with the principal curvatures, but this thing involving the angle $\theta$ intrigues me... – 2017-01-18
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    We're talking about normal curvatures in the direction of a vector $\mathbf v$ that makes angle $\theta$ with a fixed vector (in this case, $(1,0,0)$) in the tangent plane. Think about Euler's formula. (Here, of course, as Andy pointed out, the $x$-axis is an asymptotic direction and not a principal direction. But it's the same idea.) – 2017-01-18
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    @Troy: _Foundations of Differential Geometry_? Kobayashi was a formidable, elegant author (and mathematician!). It's nice to hear you like his book(s). Working with him was a real privilege. (On a tangent, as you may know, Ted was a student of Chern.) – 2017-01-18
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    @AndrewD.Hwang Ahem...I be damned. Chern is a legend and hero to us Chinese. He said in a Chinese interview that he did everything so easily, and that he surpassed all the teachers that had taught him (well maybe except Eli Cartan...)...LOL. Its a shame that I am no mathematician material and is still struggling with understanding Chern class LOL...And yes, I meant Foundations of Differential Gometry, so beautiful...I am trying to use that to improve my knowledge of symmetric space. – 2017-01-18

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$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Del}{\nabla}\newcommand{\Vec}[1]{\mathbf{#1}}\newcommand{\Brak}[1]{\left\langle #1\right\rangle}$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).

Normal sections of a ruled saddle

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    Thank you Dr Hwang for your response. So the key point is that the curvature at $(0,0,0)$ of the curve obtained by the intersection of the surface with a vertical plane is given by (the opposite of) the normal curvature at $(0,0,0)$ towards the direction $\theta$ of the plane. Also, $\gamma (\theta)$ (what confused me most) is the family of all those curves and we are dealing with something like Euler's formula, as Dr Shifrin commented. Mystery solved! Our professor is indeed focussing on intuition too much... The lectures are full of (ill-defined) geometrical jargon and few real proofs! – 2017-01-19
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    Differential geometry has a particularly large gap between geometric motivations on one hand, and formal definitions and proofs on the other. It doesn't help that curves and surfaces are _sets_, but they're studied almost exclusively via _mappings_ (parametrizations). Compare the situation with elementary algebra, where the proofs are almost immediate from the definitions, or analysis, where mastering the triangle inequality and the definition of a limit suffices. Differential geometry done with a comparable level of mathematical rigor is surprisingly technical and non-geometric. – 2017-01-19
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    I know... I've also asked another question (http://math.stackexchange.com/questions/1991340/what-is-a-curve-definition) which actually had to do with seeing sets as parametrisations. Algebra has certainly the beauty of combining intuition with rigour, and I (as a humble student) prefer more algebraic/axiomatic approaches to geometry... – 2017-01-19