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Mathematica

Subject area
General science.

Description
A mathematics package combining interactive calculation, visualisation tools and a programming environment.

Authors
Wolfram Research.

Suppliers/Distributors
Wolfram Research Europe Ltd, 10 Blenheim Office Park, Lower Rd, Long Hanborough, Oxon OX8 8LN.
www.wolfram.co.uk

Date/Version
2000/Version 4.0.

Level
Undergraduate, research.

Type of Package
Calculation, simulation, programming.

Price
Student version £75. Full professional version £1190.

Hardware required
A PC, Mac, Unix or Linux computer.
For more details see supplier's web site.

Software required
For PC: Windows 95/98/NT/2000.
For PowerPC: Mac OS 7.5.3 or later.
For 680x0: Mac OS 7.1 or later.
For more details see supplier's web site.

Review

This is a difficult review for a variety of reasons. Mathematica has been around for a long time (1988) and has gathered such a well-deserved reputation for excellence that it is hard to say anything new. On the other hand, the program is so vast and versatile that even an expert will not have explored all of its potential. It is impossible, in this short review, to encompass the whole scope of this package, which has grown beyond the physical sciences and now pervades any modelling environment whether it is in financial analysis or computer art.

Let me start at the beginning. Mathematica arrived in the usual enormous box that seems to be obligatory packaging these days. However, in this case it was necessary since the CD is accompanied by an enormous 1500 page hardback manual plus a 500-page description of add-on packages and a substantial `Getting Started' brief. The installation presented me with the only negative aspect of this review in that none of the custom fonts appeared to be present. Since these are supremely important in what is basically a mathematical program, it was difficult to proceed. The on-line help uses these fonts (as does every aspect of this program) so it was impossible to understand the instructions for solving the problem and it was (shock, horror) necessary to read the manual. Unfortunately, none of the remedies were effective and neither was any number of re-installations so I E-mailed the UK agents for support. Sad to say, despite a half dozen attempts I received nothing, not even an acknowledgement, and I had to try support in the US. I received a holding response within a couple of days followed by detailed advice about a week later. None of their suggestions solved the problem and I was beginning to suspect a corrupt CD when I decided to look in detail at the custom fonts. The US support desk had provided me with a full list of these and I suspected that for some reason they had gone astray. In fact, they were all present as expected in the Fonts directory but a close examination revealed that their author was me! The mathematical, Greek etc. symbols were all present but the copyright and sundry information were identical to those written by me for a custom font of my own design. Removal of this font instantly restored all the Mathematica fonts. I have no explanation for this as others (e.g. c-cubed.ttf) do not cause a problem.

The program has five main strengths, all interlinked:-

Numerical Computation

Including matrix operations of any size, ordinary and partial differential equations, Fourier transforms, data manipulation and fitting, statistics etc., all with unlimited numerical precision.

Symbolic Computation

Including simplification, polynomial factoring, integration, and equation solving for both algebraic and differential equations.

A Programming Language

This is a fully featured symbolic language, which is easy to understand with even a smattering of knowledge of other languages.

Notebook

This is essentially a word processor incorporating text, equations, graphics and sound plus all the features you would expect such as spell checking with a scientific dictionary. The text can be exported to many other formats including HTML and LaTeX and is fully platform-independent in its own native format.

Graphics and Sound

This comprises a very full specification for 2D, 3D, contour and density plots with animation and sampled sound and export to all formats. To my mind, this is one of the most unexpected aspects of the program. I did expect to see beautiful images of the usual suspects _ fractals and complex mathematical topologies, I did not expect to see circuit diagrams, geological maps, diagrams of apparatus or even recognisable portraits.

I have used Mathematica a great deal for numerical computations, particularly for solving numbers of large polynomials but never seem to have needed the graphics capabilities. For this review I decided to give it a go and chose something very simple to see how easy it was. I used Equation 1 to look at the probability density in the particle in a square box problem:

Equation 1: 

At the outset it must be realised that Mathematica is extremely fussy about syntax and you must type in exactly what it expects to see. In this formula, the instruction Plot3D has a self-evident meaning but plot3D and Plot3d won't work (neither will sin, cos, pi, Plotpoints and so on). You must read the instructions thoroughly and be very careful about case, use of spaces, use of brackets (crucial _ there are three very different ones above with specific meanings) etc. In this example we are plotting an un-normalised function, with unit box lengths (x and y go from 0 to 1), for quantum numbers of 2 in both directions using 55 points per variable. Figure 1 shows the result.

There are commands for producing some stunning results. The program allows full 16 bit colour to achieve the shading and you have total control of viewpoint, lighting position, colour table, axis appearance, hidden surface removal and so on.

Once a function has been defined and drawn, subsequent manipulations are simple using the % reference. It is not necessary to re-enter the function. For example:

produces a contour map of equation 1. I find myself in two minds when it comes to the educational uses of Mathematica especially within a typical UK chemistry department. When I first encountered this program over 10 years ago, I was enthusiastic about its potential and this was reinforced by the introduction of the student edition. I was disappointed that students were not similarly excited and even the more mathematically able, despite recognising its power, strongly resisted it on the grounds that it was mathematics and not chemistry. I am aware of several chemistry departments using this program as part of their `remedial' mathematics courses and I admire them for it but I cannot see it working for most of us. In an era when an A-level mathematics qualification is a rarity and maths is viewed with suspicion and fear (and as a punishment), I can see very little scope for using this program. I am afraid that the majority of us are unlikely to encounter students who can make the effort required to learn even the simplest applications and psychological barriers will prevent them from trying. The manual states …"Mathematica is heavily used in education and hundreds of courses from high school to graduate school are based on it". The collapse of maths teaching in the US and the concomitant decline in the physical sciences has led to chemistry courses featuring much less in this scenario. Teachers in the US are disappointed that we have followed their trend so quickly and so well.

In conclusion, a brilliant program, which I find myself using constantly, and which would both entertain and help any teacher and researcher but will only find limited application in UK university chemistry undergraduate courses.

Steve Walker
Department of Chemistry
Liverpool University
Liverpool   L69 7ZD
September 2000