maximas, minimas, curvature and inflection points of multivariate functions

Question

I am an engineer by qualification, so I can easily understand the aforementioned concepts if I talk about functions with single independent variables, for example $f(x)$, but what about systems with two independent variables, for example $f(x,y)$? I need to work on a bit of curve fitting, hence I need this idea. For example, I think that if I have a function $$z=f(x,y)$$ Then I can write, $$\phi (x,y,z) = z-f(x,y)$$ and then use $$\nabla\phi = \phi_1\hat\imath + \phi_2\hat \jmath + \phi_3\hat k$$ and then if I use $\phi_2=0$ and $\phi_3=0$ then I think I would have a gradient facing vertically up or down and would hence end up with the position of either a maxima or a minima, but for maxima I would need $\nabla^2\phi<0$ and for minima greater than zero, but what about the points of inflexion? Would I need to have $\nabla^2\phi =0$? I mean this is all my intuition from which I am talking, but is there some literature which I can visit to learn about more of this stuff, kindly help.

Answer 1

The curvature of surfaces is much more complicated than the curvature of curves.

Imagine a point $P$ on your surface. Construct a line $L$ passing through $P$ that's parallel to the surface normal at $P$. Now think about the family of planes that containing the line $L$. Any one of these planes $\pi(\theta)$ will "slice" the surface and form a curve of intersection, $C_\theta$. As you vary $\theta$ (rotating the slicing plane around $L$), you will get different intersection curves $C_\theta$. You study the curvature of the surface by studying the curvature of these intersection curves $C_\theta$. Since there's an infinite number of them, all kinds of interesting things can happen.

The curvature of the intersection curve is called the "normal curvature" of the surface in the direction defined by the angle $\theta$. It turns out that, as the section plane rotates, the normal curvature will assume minimum and maximum values in directions that are at right angles. The minumum and maximum values are called principal curvatures, typically denoted by $\kappa_1$ and $\kappa_2$. It's often useful to study the mean curvature $\tfrac12(\kappa_1+\kappa_2)$ and the Gaussian curvature $\kappa_1\kappa_2$. Points on the surface are classified as elliptic, hyperbolic, or parabolic according to the signs of $\kappa_1$ and $\kappa_2$.

Look up terms like "normal curvature", "principal curvatures", "Gaussian curvature", etc. You could start with this page.