Higher dimensional stationary points

[abskebabs]

Hi everybody, this is a question that popped into my head while revising some maths. I have just covered the section describing stationary points for multple variable functions.

 

So for example if we have a function of 2 variables we can imagine in 3 dimensional cartesian axes that the variables are described by the x and y axes and the value of the function is represented by the z axes. The subsequent stationary points for the resulting shape can be found when the partial derivatives for both variables equal zero.

 

All this is fairly trivial, but we have been told that the 3 types of stationary points possible for this kind of function. These are maxima, minima and saddle points(kinda shaped like pringles:-) ). Also for a 2 variabled function we were told the conditions for each of these concerning one of the partia 2nd derivtivesl derivatives and a determinant made of the possible partial 2nd derivatives.

 

We were then shown how this extends to 3 variabled functions and then how it generalises to n variable functions and the resulting condtions for these types of stationary points for these functions.

 

What I was wondering was, we know that the saddle points we observe in cartesian axes cannot exist 2 dimensionally and this is an inherent characteristic. Geometrically though we can intuitively picture what one of these looks like 3 dimensionally(mmmm... pringles;) ).

 

My question is(I know, I took a long time to get to it!), couldn't there be other types of stationary points apart from the 3 I have mentioned, and inded are there? I know that once we have more than 2 variables it would be pretty much impossible to visualise these(at least with cartesian axes anyway!), but just because we cannot visualise them, that doesn't mean there isn't more does it?

 

I suppose however that these other types of stationary points may just happen to be called saddle points when in fact they just have different Nos of variables displaying -ve and positive partial 2n derivatives. Am I off the mark in suggesting this?

 

I would be very grateful for replies on this thread. Thanks in advance.

[insane_alien]

yes there would be more as there are different ways it can look. just that it would be wierd since they would require 4 spatial dimensions to be observed. still, that doesn't mean they don't exist in the psychotic world of maths.

[abskebabs]

yes there would be more as there are different ways it can look. just that it would be wierd since they would require 4 spatial dimensions to be observed. still, that doesn't mean they don't exist in the psychotic world of maths.

Would these many of these stationary points just be differen types of saddle points though? I have been told that the conditions for there being a minima or maxima, which depends on the sign of the determinants made up from an array of all the partial 2nd derivatives of a function. If all determinats from 1*1 to n*n are +ve we have a minimium. If we have the 1*1 -ve, 2*2 +ve, 3*3-ve etc... we have a maxmium. If otherwise I have been told we have a saddle point. My main point in posting was to verify whether this was correct, as I wasn't sure if it was necessarily. By no means do the 2ndary derivatives even have to be +ve r _ve(they could be zero). If so what do we have, and how can we determine this?

 

Also 3 dimensionally, wouldn't it be possible to have stationary points of inflection. To be specific I mean a 2 variable function with x and y as variables plotted on the z axis displaying a stationary point of inflection along one cartesian cross section(meaning perpendicular to either x or y axis), and another point of inflection observed from looking at another the other cross section? If this occurs, isn't it another type of stationary point in "ordinary" cartesian axes?

[Dave]

Classification of stationary points is not particularly simple, but at least for smooth functions I'm reasonably sure that there are only three types: those being local minima, local maxima and saddle point.

 

To answer the first part of your question: we know that if the Hessian of a function at a stationary point is positive definite then the point is a minimum, but the converse is not necessarily true. Essentially, if our Hessian is degenerate (i.e. at least one zero eigenvalue) then we'll have to resort to more rudimentary methods. Generally this involves considering the partial derivatives in each direction in a neighbourhood around the stationary point.

 

Finally, I'm pretty sure that a stationary point that is additionally a point of inflection is actually a saddle point. Check out the Wikipedia entry.

[abskebabs]

Thanks for the reply Dave. I read the conditions for a saddle point, and indeed a stationary point of inflection is a kind of saddle point. I still have an enquiry though. I read in the vector calculus section of my electrodynamics book that there was another type of stationary point described as a "shoulder" I could imagine this in 3d to be in a way either a max or minimium transposed orthognally on a pointo of inflection to produce a surface. Am I right in guessing this? Also is this also generally classified as just being another type of saddle point?

 

Related to this, how does the situation change when we start talking in terms of 2nd rank tensors, and represent functions using them, I mean do we get increased complexity in the behaviour or types of stationary points observed? I think I can kind of figure why General Relativity therefore is a nonlinear theory, from observing how tensors are represented and appear to be.

 

Also, I'm afraid I must confess my lack of knowledge of terminology; What is a Hessian?

[Dave]

Thanks for the reply Dave. I read the conditions for a saddle point, and indeed a stationary point of inflection is a kind of saddle point. I still have an enquiry though. I read in the vector calculus section of my electrodynamics book that there was another type of stationary point described as a "shoulder" I could imagine this in 3d to be in a way either a max or minimium transposed orthognally on a pointo of inflection to produce a surface. Am I right in guessing this? Also is this also generally classified as just being another type of saddle point?

 

I would guess from your description that it's probably just another type of saddle point.

 

Related to this, how does the situation change when we start talking in terms of 2nd rank tensors, and represent functions using them, I mean do we get increased complexity in the behaviour or types of stationary points observed? I think I can kind of figure why General Relativity therefore is a nonlinear theory, from observing how tensors are represented and appear to be.

 

I'm honestly not sure - I haven't done enough on tensor theory to be able to comment. Any physics guys care to answer?

 

Also, I'm afraid I must confess my lack of knowledge of terminology; What is a Hessian?

 

The Hessian of a function [math]f:\mathbb{R}^n \to \mathbb{R}[/math] is just the nxn matrix of all second order partial derivatives.