Are there more types of critical points beyond maxima/minima/saddle points for higher dimensions?

Question

I had a course on single variable calculus and at that point, we had minima and maxima. Now on several variables calculus, there is maxima, minima and saddle points. Certain books of several variables calculus point only those three types of critical points but there is actually no guarantee that these are the only ones - other types could be more advanced and/or irrelevant for the study of calculus, perhaps?

• So are there more kinds of critical points for higher dimensional calculus?

• Supposing there are only these three, how can I know for sure that there are only these three? Supposing that there are more than three, how can I know that there are more than three?

Answer 1

Let $f : \mathbb{R}^n \to \mathbb{R}$ be a smooth function. A point $p \in \mathbb{R}^n$ is called a critical point if $$\frac{\partial f}{\partial x_1}(p) = \dots = \frac{\partial f}{\partial x_n}(p) = 0.$$

A critical point $p$ is called non-degenerate if the Hessian matrix at $p$

$$\left[\frac{\partial^2 f}{\partial x_i\partial x_j}(p)\right]$$

is non-singular (i.e. invertible).

Morse Lemma: Let $p$ be a non-degenerate critical point of $f : \mathbb{R}^n \to \mathbb{R}$. Then there is an neighbourhood $U$ of $p$ and coordinates $(y_1, \dots, y_n)$ on $U$ such that $y_1(p) = \dots = y_n(p) = 0$ and for every $q \in U$, we have $$f(q) = f(p) - y_1(q)^2 - \dots - y_k(q)^2 + y_{k+1}(q)^2 + \dots + y_n(q)^2.$$

The number $k$ is called the index of $f$ at $p$. It is a well-defined quantity (i.e. you can't get a different value of the index by using a different coordinate system). In fact, $k$ is equal to the number of negative eigenvalues of the Hessian matrix at $p$. Note that $0 \leq k \leq n$.

Non-degenerate critical points in $\mathbb{R}^n$ are classified by their index. Indeed, if $n = 2$, there are three possibilities for the index: $0$, $1$, or $2$. These correspond to a local minimum, saddle point, and local maximum respectively.

Index $0$: For all $q \in U$, we have $$f(q) = f(p) + y_1(q)^2 + y_2(q)^2 \geq f(p),$$ so we see that $p$ is a local minimum.

Index $1$: For all $q \in U$, we have $f(q) = f(p) - y_1(q)^2 + y_2(q)^2$.

Along the curve $y_1 = 0$, $$f(q) = f(p) + y_2(q)^2 \geq f(p),$$ so $p$ is a local minimum along $y_1 = 0$.

Along the curve $y_2 = 0$, $$f(q) = f(p) - y_1(q)^2 \leq f(p),$$ so $p$ is a local maximum along $y_2 = 0$.

Therefore, $p$ is a saddle point.

Index $2$: For all $q \in U$, we have $$f(q) = f(p) - y_1(q)^2 - y_2(q)^2 \leq f(p),$$ so we see that $p$ is a local maximum.

It is also worth considering the case $n = 1$ from this point of view. All non-degenerate critical points are either local minima (index $0$) or local maxima (index $1$), and zeroes of the derivative which are neither, sometimes called stationary points of inflection, are degenerate critical points.

As in the specific cases $n = 1, 2$, a critical point with index $0$ is a local minimum and a critical point with index $n$ is a local maximum. For $0 < k < n$, you can intuitively think of the critical point as having $k$ independent directions along which $p$ is a local maximum, and $n - k$ independent directions along which $p$ is a local minimum.

The study of non-degenerate critical points can also take place on a smooth manifold. The relation between such points and the topology of the manifold is known as Morse Theory.

Answer 2

That depends on how you classify them, that is what criterion you have for a critical point being of different type.

When you state that there are only these three in two dimensions you must be inclusive in what you consider saddle point. Basically what they do is to define the saddle point as a critical point which is not a local maximum or minimum - then of course you can never have a fourth type.

You can have an somewhat more detailed classification, by what relative values it takes in around the critical point. You have then strict minimum (only takes larger values), nonstrict minimum (takes both larger and equal values, but no lower), saddle point (take both larger and lower values), nonstrict maximum, strict maximum.

If you on the other hand imagine a saddle point looking like a saddle (but then the observation that there are only those three is basically incorrect) then you get into the question about regions around the critical point where the function takes larger and lower values and here is a slight difference in how these regions may be formed (assuming the function is continuous there):

In two dimensions you will around the critical point have directions in which the value increases and directions where it decreases (and directions inbetween where the value is constant). But you are not limited in having just two regions where it decreases and two where it increases (that is being a saddle point in the normal sense - having the shape of a saddle), you can also have situations where the function increases in on general direction and decreases in another (fx f(x,y)=x^3 at origin).

In some sense what we have in two dimensions that differs from higher is that in two dimensions we can only have one or even number of such regions. This is because if we traverse all directions we may start with non-increasing and then if we leave that we will reach a region where it's non-decreasing and so on alternating until we get back.

In higher dimensions this no longer applies - for example $f(x,y,z) = x^2+y^2-z^2$ has a critical point at origin where you have increasing behavior when $x^2+y^2 > z^2$ (one region), but decreasing when $z^2 > x^2+y^2$ (two non-connected regions).