ChapterZero

Yesterday, I started reading a book on geometry for scientist and engineers. It covers affine, projective, and differential geometry to differing degrees, with an algorithmic mind-set. What I found surprising is that it claims that all geometries can be embedded in projective geometry, yet I’ve never even heard that much about it. I thought projective geometry was rather useless up to now.

Another interesting thing is the concept of a barycenter, which seems like the mathematical equivalent of a center of mass. Apparently a polynomial can be expressed as the set of barycenters of a finite number of points, something I would never have expected.

I just looked up Lagrangian mechanics on Wikipedia, which seems to be a reasonably useful resource for math, much to my surprise. The article by that title has an excellent exposition of the derivation of the Euler-Lagrange equations, sadly the first I’ve ever seen! Books that I’ve picked up here at UH either are so long winded that I never get to the ‘good stuff’, or just pick up with the definition of T and U (I think those are the quantities, but I’m probably wrong). So, I’m going to go study this and think about it. One exciting fact I’ve noticed is that the Euler-Lagrange equations look an awful lot like the Euler characteristic equation for extremals.

The calculus of variations is the mathematical theory concerned with finding functions which are stationary with respect to a given functional (a mapping from functions to reals). For example, it deals with the classical problem of finding the boundary which maximizes the enclosed area, given that the boundary must have a fixed length. Incidentally, the answer to this problem is given by the Isoperimetric inequality:

where is a simple closed curve, is the length (period) of the curve, and is the area of the interior of the curve, and equality holds only if is a circle.

Three problems characteristic of the calculus of variations are:

• What plane curve connecting two given points has the shortest length?
• Given two points and in a vertical plane, find the path which the movable particle will traverse in shortest time, assuming that its acceleration is due only to gravity.
• Find the minimum surface of revolution passing through two given fixed points, and .

The first problem in the calculus of variations is to find a function which minimizes/maximizes the integral . By analogy to the case of finding the point at which the maximum of a continuous function occurs by finding the critical points, we can find a differential equation which must satisfy in order to be a relative maximum/minimum of this integral expression.

To do so, we will assume that possess the relavant partial derivatives necessary for application of the MVT for functions of several variables. We will only consider arcs from to for which can be determined, or admissible . The problem is further proscribed by requiring that the values of at the endpoints be predetermined.

Assume is a function which minimizes/maximizes , also known as a stationary point. Let be another admissible arc, where is an arbitrary constant independent of and is a function independent of . With that restriction, is said to be subjected to weak variation. The important point of weak variations can be seen from the fact that , so as approaches , .

Define

Now apply the MVT:

more to come

References

A Brief Survey of the History of Calculus of Variations and its Applications. James Ferguson. University of Victoria.
An Introduction to the Calculus of Variations. Charles Fox.

About a week ago, Ben, one of the people I did work with this summer on characterizing finite tight sampling sets over intervals in the bounded subsets of , let me know that we were accepted to present a poster at the AMAA meeting in January. That’s cool, but we don’t have anything much: just some easily derivable characteristic equations, some examples of tight sets, some almost obvious theorems, and one interesting theorem. I had wanted to work on that this semester, but it looks like I have too much on my plate as it is already.

One of my friends is taking a probability class, and he gave me this problem to solve:

Entities and play game such that takes the first turn and can win with probability 1/4. goes second and can win with probability 1/3. What is the probability will win before ?

I’m not sure I understand it…

Clarification Now I understand it; consider that and are playing a game of darts. Then and , the probability that will make their next throws, do not necessarily sum to unity. And if alternate throws, then alternate between being zero and nonzero.

So here’s my solution:

since for even , this reduces to

.

The first term is calculated as

The second term is calculated as

Therefore, the desired probability is

.

I’m attempting to read way too many books at one time. I had to give up on Solid Shape; it seems that it would be more beneficial to read that book after I already understand the concepts that it attempts to present in a non-mathematical manner. I would rather gain the technical knowledge and then build the intuition, as opposed to culture an incorrect intuition that interferes with learning the technical details.

Anyhow, I found yet another nice book on Differential Geometry: A Comprehensive Introduction to Differential Geometry, by Michael Spivak; I think there are 5 volumes. I have to say, I ordered the first volume from the library without having a clue what to expect. I thought that, since it is comprehensive, it would cover the basic curve and surface theory stuff that we’re doing in class— which, incidentally, is why I was ordering a dg book in the first place— there was only one good one in the library (Elementary Differential Geometry by Springer), and I turned that back in so a friend who needed it more than I (i.e. was taking the class for credit) could check it out. Turns out it doesn’t seem to have that kind of intro dg stuff at all, but it still seems pretty interesting. It is the first book, dg or not, that I’ve seen that has a straight forward, non-intimidating definition of a manifold, offered on page 1:

… a manifold is a metric space with the following property: if , then there is some neighborhood of and some integer such that is homeomorphic to .

No mention of derivatives, or the implicit function theorem! Yeah! It’s nice that he didn’t jump right into that stuff, which I personally find very confusing. Speaking of the implicit function theorem, at UH, we learn about that in the Advanced Multivariable Calculus class, or at least we’re supposed to. When I took it, we spent maybe a month on basic vector calculus: identities, box product, etc. Nothing even as complicated as determinants, which we covered in a day. But we spent maybe two weeks on the implicit function/inverse function theorems. Considering how important it seems to be, I’m suprised and disappointed by that. But whatever— now, I’m getting the idea of finding out when those topics will be covered in that class this semester, and sitting in on those lectures.

This weekend, when I went to buy a calculator to replace the one I lost last week, I also made a pit stop by MicroCenter. A guy in one of my classes said that they had a large shelfful of math books, something which is unusual even in large dedicated booksellers— what’s more, these were the eminently affordable Dover math books. Surprisingly, he
was exaggerating; there was a large selection of math books. I just barely managed to buy five, for less than $70; a good deal— that’s the price of at most two Springer math books. Unfortunately, as part of the restraint I forced upon myself, I didn’t get a nice book by Halmos on finite dimensional linear algebra (I wanted it because it had a more algebraic, as opposed to matrix theoretic, approach, and a nice overview of finite dimensional operator theory— seemed a lot
like the book Linear Algebra Done Right— but that judgment is based on a quick flip through it), because I figured the material wasn’t new enough to me to merit buying a book on it. It’s certainly a book I would buy at anyother time for recreation or reference value, but I have a strong enough background in linear algebra that I could look up any topic the
book covered in a more advanced book, and follow along. There was also a thin book on a type of non-Euclidean geometry called taxi-cab geometry, under which the equivalent of a circle is a square; supposedly this type of geometry is useful for urban planning. I wanted it because I’ve never heard of a simple (i.e. not requiring involved differential geometric concepts, like the “metric tensor”) non-Euclidean geometry, much less had the opportunity to watch it be developed from first principles. That would be interesting, but in the end I settled for more practical books: one on Mechanics, one on
Dynamics, The Absolute Differential Calculus by Levi-Civita, and two introductory to intermediate type treatises on relativity. As can be inferred from the topics of these books, I’m trying to understand the theory of relativity. Auditing a differential geometry class just isn’t cutting it for me— we have yet to be introduced to the meaty stuff, like tensors and
whatever else is used in Relativity. So I figure a little self study won’t hurt.

In my economics class, we’re discussing price elasticity, a measure of how the quantity of a good demanded varies with the price of that good. Specifically, if is the demand curve, the quantity demanded of a good as a function of the good’s price, the price elasticity between a point and a point on is defined as

This is an interesting definition because it is not symmetric; the price elasticity going from to is not the same as the price elasticity going from to . Economists view this as a drawback for obvious reasons, so they replace the divisors and with the respective midpoint values, and call this is the midpoint price elasticity:

Not surprisingly, considering the fact that we’re dealing with the ratio of the percentages of two quantities, and not the quantities themselves, even for a linear , the elasticity between two points a fixed distance apart varies depending on the points.

The first thought that came into my mind when I heard about price elasticity was differential geometry: why not take the limit of or even more interestingly, as , and then investigate it as a property of curves?

Just shooting from the hip, here are some problems I thought of (here, elasticity refers to either or ): what does a curve of constant elasticity look like? Can it be found as a function of curvature, torsion, etc.?

Update: I took the limit, and found — I think I’ve seen this somewhere before. More later.

The book Solid Shape by Jan Koenderink is a rather weird, but certainly thought provoking book on the mathematics associated with solid shapes— it has an unusual, operative approach to geometry.

I’m only on the third chapter, yet I’ve come across a wealth of interesting ideas, all presented in a nonmathematical, yet logical manner. For instance, the idea of defining a metric on the space of shapes:

1. First I define the “-neighborhood” of a point as the spherical region of radius centered on that point.
2. Then define the “spherical -neighborhood” of an arbitrary region as the union of all -neighborhoods of the points belonging to that region. You may think of the -neighborhood as a slightly “thickened” shape.
3. With these preliminaries understood, you can proceed to define the “distance” between two shapes: the distance between two shapes is the smallest value of for which it is the case that each shape is contained within the -neighborhood of the other, perhaps after some arbitrary displacement.
4. As a result you are all set to define the equivalence class of a given shape for a given resolution: shapes within a distance given by the resolution are considered equivalent.

The last point there is illustrative of another nice feature of the text: the author repeatedly points out that in any practical applications of geometry, we are dealing with discrete values; there is no real continuity. Therefore we must define a level of resolution at which to investigate shapes: a cloud at one level of resolution appears to be uniform, and all detail at lower levels of resolution are consigned to what we call texture; at another level of resolution, the cloud has a random, chaotic structure. We can pick either level to consider the cloud at.

The point is also made that the notion of a surface area, or arc length, is meaningless unless the resolution is identified.

What a good book! Like all good books, it is giving me many more ideas and thoughts to consider than I could ever write down…