Learning to Use Tensors in Maxima

In my previous post I used Maxima to derive the metric tensor of the 3-sphere using the coordinate system described by Mathview in his YouTube video. Now I'd like to continue exploring Maxima to calculate various other properties of S^3.

In a follow up video where he calculated the Christofell symbols for the 3-sphere, Mathview used new coordinate transformations that give a right-handed coordinate system instead of a left-handed one as in his previous video. It is very similar to the original, the difference being just the choice of the 3rd angle, theta3. This small change greatly simplifies the metric tensor and, as a consequence, all other derived tensors and properties. You can follow his derivation in the video below.

How do I get the Christoffel Symbols on the 3-Sphere?



Last time I used just the basic facilities of Maxima to derive the metric tensor from the transformations. I then said that it would be nice, in the future, to use one of the tensor packages that comes with Maxima to redo those calculations. Today I found out that you can start from the coordinate transformations and go directly to the metric, Christofells and other tensors. However, we can plug in the output of my original method into the ctensor package and continue from there. Although this gives a little more work it is more pedagogical as it reveals some of the details of how tensors are handled by this package.

So here we will quickly repeat the derivation of the metric tensor, only now using Mathview's new coordinate system. We assign the final matrix to "lg", for "lower-indices g". Maxima uses this convention to name some of the variables that it uses to hold tensors. The first letter in the name can be "l", "u" or "m" for lower, upper or mixed indices respectively.



I don't know why g[2,2] came out with (1 - sin(theta[3])^2) instead of cos(theta[3])^2. I tried other simplification functions but none did this transformation so I ended up doing it by hand.

Next we calculate ug, the inverse of lg, as it will also be needed.



Maxima comes with two packages that handle tensors: itensor and ctensor. The former does indicial tensor manipulation and the latter does component manipulation. ctensor is sometimes also referred to as the "curved space" tensor package. This is the one we want!

As I said, ctensor has a way to allow you to start directly from the coordinate transformations (ct_coordsys()). There is also a dialog based function that asks for the various components of the metric tensor (csetup()). But we have already calculated the metric tensor by our own device and the choice of names "lg" and "ug" was no accident: they are exactly what ctensor expects.

At this point we can see that Maxima implements component tensors as regular multi-dimensional arrays: [n,n] arrays (matrices) for rank 2 tensors, [n,n,n] arrays for rank 3 tensors and so on up to rank 5. Another interesting observation is that ctensor is not built in Maxima, but rather, it's a loadable package. In other words, it's implemented in the Maxima programming language, not in Lisp.

Therefore, we next load ctensor and give it the missing information it needs to do its calculations: the number of dimensions we'll be working on and a list of the variables that hold the coordinates.



Now we're ready for the fun! Equipped with the metric tensor and the list of variable names, ctensor can easily calculate the Christofell symbols, the Ricci tensor and many others. We'll start with the Christofell symbols of the 1st kind, which ctensor names as "lcs" because all the indices are downstairs:



Some observations are needed here. First, note that ctensor displays only the non-zero, unique elements in lcs[i,j,k]. In this exercise there are 6 non-zero elements, but only 4 are unique. The other two are duplicates due to the symmetry of lcs[]. Second, the convention that Mathview uses for the indices is different from the one used by Maxima. To convert between Mathview and Maxima, just reverse the order of the indices. So, what Mathview calls Gamma(3,1,1) is the same as lcs[1,1,3] which is -cos(theta[3]) * sin(theta[3]) * a^2. Finally, it's a pity that wxMaxima displays the Christofell symbols this way. It would have been nicer to use the proper Gamma character!

Now, just for fun we calculate the Christoffell symbols of the second kind (mcs), the Ricci tensor and the Riemann curvature tensor.



This post shows but a small portion of the capabilities of ctensor and Maxima. There are dozens of other interesting functions that you can learn about by reading the manual. Although there's no doubt about the power of Maxima for manipulating tensors, students learning the subject should make sure to go over the whole process of calculating the Christofell symbols and the Ricci and Riemann tensors by hand a few times. By watching Mathview's videos you'll see how much more insight this approach gives you as opposed to just seeing Maxima spit out the answers!

Turning the digital crank

Mathview has a series of very interesting videos on YouTube about Differential Geometry and related topics. In one of these videos he attacks the problem of finding the metric tensor for the 3-sphere using as coordinates a set of 3 orthogonal angles and a radius. Here's the first part of the video:

What's the Metric Tensor of the 3-Sphere?



This is a good exercise for those learning the subject as it gives us the opportunity to apply the theory to a concrete example. Of course, working with S^3 is not the best example of concrete mathematics because it requires dealing with 4 dimensions, but you know what I mean. Besides, the 3-sphere received a lot of attention lately due to the proof of the Poincaré Conjecture by Perelman.

The exercise is not intrinsically difficult, but it involves a lot of calculations in the form or partial derivatives and trigonometric simplifications. It is very easy to eat a minus sign here or use a sin(x) where cos(x) was meant and wreck the whole thing. Since this is basically a mechanical process, Mathview calls it "turning the crank".

Since I'm learning to use Maxima, which has powerful symbolic manipulation features, I thought it might be a good idea to exercise it trying to verify Mathview's result for the S^3 metric tensor. In other words, I'd like to "turn the crank" the digital way. I must say that it took me many hours to put all the pieces together, but in the end it was well worth the effort. Not only did I verify that Mathview's result was correct (not that I doubted it :-) but I also learned a few interesting things about Maxima.

I wanted to Read The Fine Manual entirely but that seemed like a bad idea since it's very large (900+ pages) and has lots of specialized packages that are not of my concern for now. Therefore I just skimmed over the main topics and tried to extract what I needed for this job. I immediately saw that it has a tensor package which knows about the metric tensor, Christofell symbols, etc. However, that was too complicated for a first encounter with Maxima, so I just used its generic features. As I learn more about Maxima I may revisit this exercise and try to do it with the itensor/ctensor packages.

We start by defining how the Cartesian coordinates relate to the polar coordinates. These relations were derived by Mathview at the beginning of his video. Moving from 3D to 4D with an orthogonal angle was a nice trick! Then we define an array 'X' with these definitions.



Two things to note here. First, Maxima knows about Greek letters like "theta" and wxMaxima (the GUI) renders them nicely as one would see in a well typeset book. Second, the square brackets used to denote the indices are transformed into subscripts, also as we would expect. We must use this notation here as it will allow us to refer to these variables generically via their indices.

As a first test, let's find the derivative of X with respect to theta1:



Now we define the metric tensor. It's the dot product of the partial derivatives of the coordinates. In this case we're only interested in the angles, so we leave out the radius, which we take to be constant (say, 1).



Note that this is not yet the tensor. It's just a "matrix function". For instance, if we want to know a particular element, it can be evaluated:



Here we learned that Maxima does not perform all the simplifications when presenting its results. Depending on what kind of calculations you're doing, there are different simplification functions and strategies. In our case, "trigsimp" did the trick and transformed that long trigonometric expression into r^2.

Finally, in order to create the g matrix we use "genmatrix", which will apply the function to all the indices. The result was not in a simplified form, but applying trigsimp after that reduced the matrix to the form that Mathview derived by hand!



I hope this has been a small but interesting introduction to Maxima. In future posts I will continue exploring it.

No program is an island

One of the first things you notice when you start either Maxima or Octave is that these programs run on a character-based terminal. It doesn't matter if you have the state of the art graphical user interface on Mac OS X, Linux or Windows, they will take input in the form of written commands and will display their results as strings of ASCII characters. This is not a bad thing in itself, it just shows that they have a lot of history behind them, especially Maxima. Make no mistake, these programs are very sophisticated and implement algorithms that incorporate a vast repository of mathematical knowledge.

In order to overcome their UI shortcomings, Maxima and Octave today are augmented with other programs that are well integrated in the graphical environments of modern operating systems. There are two categories of such helper programs.

The first one is the Integrated Development Environment (IDE). This is a replacement for the character terminal as the main interface with the program. Although the command line is still available, some commands can be activated by pushing buttons or clicking menus. Besides, you have a text editor for editing scripts, integrated online help and many other niceties. This concept, of course, is not new. It has been around at least since the early days of Visual Basic and Turbo Pascal. To be fair, even Emacs had such an integrated environment earlier than that, although not in a graphical environment. For Maxima, one usually uses wxMaxima while for Octave, QtOctave is a common choice.

The second category of helper program is the plotting utility. Although some IDEs can also graph functions, curves and surfaces, this task is usually relegated to a specialized program. There are some possible choices here (see the List of information graphics software in my previous post) but both Maxima and Octave use the services of Gnuplot.

But the story doesn't end here! Gnuplot itself does not know everything about graphical environments so it relies on yet other programs to do the low level display. The list of output devices is long, ranging from real graphical terminals to terminal emulators that run on top of graphical environments. If you're running on Linux, for example, XTerm comes up naturally, but on Mac OS X your best choice would probably be AquaTerm, which incorporates the native Mac look and feel. In some cases you might use one of these terminals to interactively work on a problem until you are satisfied with the resulting graphic. Then, if you want to publish it, you could output it directly into a file in a format that supports graphics, such as Encapsulated PostScript (.eps), Portable Document Format (.pdf) and others.


This architecture of interconnected programs is not new either. It more or less follows the early Unix motto that said that each program should do one thing well and one thing only. Nobody would think that Maxima and Octave do one thing only, but you can think of this rationale in a new level, where they do the core computations and other specialized programs do the input and output.

Now, as Monk would put it, this is a curse and a blessing. It's a curse for newcomers because they don't get a unified view of the program they're trying to use. Instead, they see a many headed beast that can be hard to correctly install, configure and understand. It also introduces new points of failure and makes it harder to keep all modules up to date in a compatible fashion.

The levels of complexity can run even deeper. When you run Maxima, for example, you don't see any process with this name (or similar). Maxima is written in Lisp so what you see is a Lisp interpreter which is executing the Maxima code. Which Lisp? Well, as you might guess, there are many implementations that can do the job.

On the other hand, for the experienced user it's a blessing, because he or she can, to a certain extent, pick and choose the best modules for each task. In part, it is this extreme flexibility that has attracted so many people to the world of Free and Open Source Software (FOSS).

Broadening the Horizon

In my previous post I superficially compared the most evident math programs from the commercial and free/open source fronts. Evidently, I was aware that such a small sample was not comprehensive but I was surprised when I found out how small that sample really was.

While searching Wikipedia for math related software I came across several interesting items. You will find, for example, good descriptions of all the packages we talked about, but what really caught my attention were three pages comparing the main features of a large number of programs. I think it would be rather difficult for a single person to collect all this information, sort out the details and display everything nicely in a table, not to mention keeping all the entries up to date. This is a true community effort!

The first page compares programs that can manipulate mathematical expressions symbolically. These are now called "Computer Algebra Systems" (CAS) and are epitomized by Macsyma:

Comparison of computer algebra systems

The second page is about numerical software. There is, of course, a lot of overlapping between these two categories (and the third one below), but the prototypical example here is Octave:

Comparison of numerical analysis software

Finally, this third page concentrates on programs that can create graphics. Again the overlap is big, but there is a table specifically comparing plotting programs, where our favorite example would be Gnuplot:

List of information graphics software

I'm sure there are many more programs out there that didn't enter these lists, either because they are too specialized or because they haven't reached maturity yet. Anyway, the landscape of math programs presented by Wikipedia is much wider than the one I painted last time!

Mathematical Software

So we want to experiment with mathematical software. At this stage we don't have any specific goal in mind. We just want to explore the possibilities and play with some programs a bit. We would like to do numerical calculations and also to evaluate symbolic expressions: factor, expand, integrate and differentiate. Finally, we would like to display the results of this work graphically. How easily can we plot curves and surfaces? How about data sets?

The commercial landscape for mathematical software is relatively small. It is basically dominated by two heavy weights: Mathematica and MATLAB. The first is the all-in-one solution to scientific computing. It can do numerics, symbolics, graphics and programming. The second, while probably equally capable, is seen as more oriented to matrix and numerical calculations. In a second tier, if you will, one finds two other strong contenders: Maple and Mathcad. Like their more famous friends these products can also be extended via numerous add-ons and packages to accommodate specific engineering and scientific needs.

In a third tier, considerably lower than the above products, one should not forget the ubiquitous Excel. Although we tend to associate this name to price lists and accounting, a spreadsheet can handle some interesting scientific tasks. I don't think you can evaluate symbolic expressions with Excel but you can do statistics, numerical integration and differentiation and, of course, draw nice graphics.

Judging by the existing literature about them, these commercial products are enormously successful. Each has a large number of users and (with the exception of Excel) could be used for learning and experimenting with mathematics. The problem, as we all know, is that they are also very, very expensive. Moreover, if you are curious about how they implement this or that function, you are out of luck. These products don't come with source code and you can only guess which techniques they use to implement your favorite feature.

So, without the need of further explaining, let's look at the Free/Open Software options!

Just a little searching around is necessary to reveal the two most famous mathematical programs in the free software world: Maxima and Octave. Perhaps I should put Octave first, given that it seems to be much more popular than Maxima, but I want to emphasize that they more or less mimic their commercial counterparts Mathematica and MATLAB.

Saying that Maxima tries to mimic Mathematica is, of course, a misstatement. Maxima descends from Macsyma, which takes its name not from the Apple computer but from the famous MIT Project MAC of the 60's, and as such is the granddaddy of all mathematical software. The "syma" suffix, as you may guess, suggests that this program was intended to do symbolic math, a revolutionary feature at that time. Mathematica has no direct link to Macsyma except, perhaps, as a source of inspiration. So the relation between these programs is rather tenuous but I think that today Maxima is still the closest free alternative to Mathematica.

On the other hand, Octave tries hard to emulate MATLAB in many areas. I'm still a newcomer to Octave, but a quick glimpse at its manual shows its authors' concerns about compatibility with the commercial cousin. There seem to be some small restrictions but the core functionality of MATLAB seems to be available in Octave. As "Macsyma", the name "Octave" is a bit misleading. It has nothing to do with music but comes, as we learn by reading the first paragraphs of its documentation, from a Professor close to the original author!

In the next posts I will try to report my own experiences dealing with these free mathematical programs and some other helper packages like gnuplot.