In 1984, as a second year undergraduate at the University of Adelaide, I was taught a course in continuum mechanics by a young lecturer by the name of Tony Roberts. I remember the course being very clear, well structured and was inspirational enough that this first course lead me to a career in applied mathematics. It is with a certain irony that, eleven years later, I find myself lecturing a course on continuum mechanics and reviewing a book that has as its genesis, the course I saw as a student. Perhaps I should add that this year I successfully used part of this book on one-dimensional continuum mechanics as the introductory part of the course I teach. My only regret being that I didn't use Tony Roberts' text for more of the course.
This book is different from many other texts on continuum mechanics which often begin with three-dimensional dynamics and stress-strain tensors. Many of the important concepts of continuum mechanics can be taught using only one-dimensional models which are easier for the student to understand. The transition to higher dimensions is then straightforward.
This book is an excellent introduction to one-dimensional continuum mechanics with an emphasis on mathematical modelling. The first chapter considers the continuum mechanical assumption and the difference between Eulerian and Lagrangian representations. Conservation of mass is considered in the second chapter and applied in detail to the modelling of car traffic and then briefly to the aggregation of slime mold amoebae. Chapter three looks at the conservation of momentum with application to gas dynamics. Chapter four considers stress-strain relationships with applications to bending beams, p-waves and various models of visco-elasticity. The fifth chapter considers quasi-one-dimensional continua such as blood flow in arteries, water flow in rivers, wave propagation and solitons. The last chapter looks at some more advanced applications such as dispersion in fluid flow, solidification of alloys and the greenhouse effect.
This text is also a good introduction to a variety of mathematical methods. Chapters two and three deal extensively with the method of characteristics. In particular the section on car traffic modelling is an excellent introduction to characteristics with the mathematics complementing the model very well. This was my favourite section. More complicated characteristics are also dealt with in chapter three. Importantly this latter section has the warning to the reader `` This section is not needed for the development of continuum mechanics in the book, and is significantly harder.'' The concept of stability is introduced with application to slime mold amoeba and water waves.
One of the main features of this book is its honesty. Difficult sections are described as such. When a derivation is heuristic the author makes this statement clearly. Where models are approximate or incomplete the text clearly describes where the model is incorrect and why. At no stage in the book did I feel important points were glossed over or hidden from the reader. The style of writing employed by Professor Roberts is very easy to read as he writes as if he is talking to you (rather than at you). In this way the book is very much like a series of detailed lecture notes. Margin notes have also been used to good effect with comments such as `` The only theorem in the book'' adding to the conversational tone. The margins are also used to highlight `` Asides'' and also to describe the end of section questions as either Easy, Moderate or Hard . These questions are important parts of the text as they often give insight into extensions of the work. A highlight of this book is that it is written with student learning as a primary goal.
I have only a few minor criticisms about this book. Although a few references were given at the end of the book I would have liked to see more included in the text, particularly references which might show the more eager students where to extend their knowledge of a subject. I also felt the section on the aggregation of slime mold amoebae needed a little more justification and description. Some readers may feel that the book is a little brief in parts and there is certainly scope for some further additions and extensions to the text. Part of the beauty of this book is, however, its brevity. The extensive section on visco-elasticity was well written but could perhaps do with a physical example that students could relate to (as they can to car traffic modelling). The last example on the greenhouse effect was, perhaps, a little out of place with the rest of the book.
This book is a good example of how a successful lecture course can be transformed into a useful and informative text book. It is not designed as a comprehensive guide to one-dimensional continuum mechanics. It is, however, an excellent book that can be immediately used to teach a course on continuum mechanics. I would recommend it for any second year undergraduate course on continuum mechanics or as an integral part of a third year course. I would also recommend it as part of a mathematical modelling course. Certainly it is a book that students will be able to read, understand and learn a great deal from.
University College, UNSW
Australian Defence Force Academy
Occasionally a topic from mathematics goes beyond the usual mathematical boundaries and captures the imagination of the public at large. During the eighties, with the promise of a new way of looking at nature and with stunning graphical images, fractals and chaos theory achieved such prominence. The authors of the two books reviewed here contributed to this popularity by being a part of the immensely successful exhibit The Frontiers of Chaos: Images of Complex Dynamical Systems and with their books The Beauty of Fractals and The Science of Fractal Images . Since that time the subjects of chaos theory, dynamical systems and in particular fractal geometry have been filtering their way down the syllabi to the point where now they are reaching school level.
In the preface to the books the authors say that these two books grew out of the demand from members of the National Council of Teachers of Mathematics (NCTM) following an address to their annual meeting in 1988 (and subsequent lectures) by H-O. Peitgen. Thus, and as suggested by the title, these books are not for the specialist in the field but are aimed more at those involved in mathematics education, particularly teachers of upper secondary and lower tertiary classes. And for this audience these books will be a tremendous resource.
I say resource because the books are not really suitable as textbooks. They are too wordy to be used with students and cover far too much material. Also there are no exercises included in the books although apparently there are supplementary volumes of activities written by the same authors in co-operation with others from the NCTM. I have not seen these and therefore can make no comment. The authors claim that their objectives in writing these books is ``to give the reader a broad view of the underlying notions behind fractals, chaos and dynamics'' and ``to show how fractals and chaos relate both to each other and to many other of mathematics as well as to natural phenomena''. I would have to say that they have succeeded on both counts.
I found the books to be very well written. The explanations are very clear and, where appropriate, good use is made of analogies to explain the ideas. For example, iterated function systems are compared to a reducing photocopier that can output multiple images and the mixing behaviour of the quadratic iterator is explained in terms of stretching and kneading dough. There are a wealth of clear diagrams and examples to support the text. The writing is such that the books can read like a novel although there are many ``technical boxes'' which go more deeply into the mathematics. I must admit to finding these boxes, which can go on for several pages, a nuisance on first reading. This was simply because they are not clearly delineated and I could not find where the sentence I was reading continued. A minor point: of necessity, in a number of places I believe that the mathematics discussed in the technical boxes goes beyond the level of the advanced secondary student, that being the level that the authors claim to have written. However for a reference book this is not a problem. Also, for those who want to go further, plenty of references are given right throughout the books. One feature of the books that I particularly like are the many diversions into the personalities and the development of the ideas. Things like photographs of the people and extracts from the original journal articles are included. Each chapter concludes with a ``Program of the Chapter'' in which a program written in Basic is given to encourage the reader to play with some aspect of the ideas of the chapter. I had no trouble getting those programs I tried to work, although the program for Chapter 8 is much easier to do with LOGO.
As can be seen by glancing over the table of contents given below the coverage of the content is comprehensive. What is not so obvious from the contents is the diverse range of applications that are covered. I appreciated the fact that whenever applications are discussed, sufficient background is given to enable the reader to understand. Book One concentrates, apart from the introductory Chapter 1, on fractals. Book Two then concentrates on dynamics and chaos.
In summary, for the intended audience, these two books are thoroughly recommended. A list of table of contents follows.
Table of Contents:
Book One: The Backbone of Fractals: Feedback in the Iterator, Classical Fractals and Self Similarity, Limits and Self Similarity, Length, Area and Dimension: Measuring Complexity and Scaling Properties, Encoding Images by Simple Transformations, The Chaos Game: How randomness Creates Deterministic Shapes, Irregular Shapes: Randomness in Fractal Constructions
Book Two: Recursive Structures: Growing of Fractals and Plants, Pascal's Triangle, Cellular Automata and Attractors, Deterministic Chaos: Sensitivity and Mixing, Order and Chaos: Period Doubling and its Chaotic Mirror, Julia Sets: Fractal Basin Boundaries, The Mandelbrot Set: Ordering the Julia Sets
The University of Newcastle
The Matlab primer is a beginner's introduction to the powerful Matlab software system. It is perhaps best introduced by an excerpt from the book itself:
The first five sections cover the basics of Matlab command syntax. Examples are given for entering matrices and the operations that can be performed on them. The book moves on to logical control structures in script files, and row/column operations for creating and combining blocks within matrices.
User defined functions are then explained, as well as more information on scripting, such as: printing of text to the screen, counting the number of floating point operations for executed commands, execution of operating system commands and so on. Kermit manages to give a good background to the scripting language without resorting to lengthy explanations needed for a larger text or reference book. It should be noted that this might make it a little hard for programming illiterate people to understand. The scripting section concludes with some handy instruction on managing program files.
The next section launches into program output: firstly text, and then the realm of graphical visualisation. The graphing section covers Matlab's capabilities in the area of 2d and 3d plotting. The fun part was the manipulation of rendering attributes for the plots, such as colour maps, shading, viewing examples (rotating the colour map in HSV mode produces some dazzling effects!).
The instructional aspect of the Primer concludes with a short section on sparse matrices and a major section devoted to symbolic manipulation. The latter being a part of Matlab's Symbolic Manipulation Toolbox add-on. The book proper finishes with an appendix of functions grouped by subject area. This appendix has been lifted directly from the Matlab reference guide.
The Primer presents an abundance of useful examples/exercises/hints in small morsels, one new principle at a time. This facilitates easier learning. The book also contains useful pointers to in-built Matlab demos.
The writing style is lucid and succinct, yet it is not too formal (he calls a helix a `slinky'!). This makes it easier to read for students who would form a large part of his intended audience. He is also able to give a beneficial summary of Matlab within seventy pages, which is to be commended.
There are two points that would enhance the book's usefulness. Firstly, it should be spiral bound which would leave both hands free to type in the examples. Using the primer beside the computer is, after all, one of its major aims. Secondly, fonts, or even colour, should be used judiciously to highlight important sections. Some headings seemed to `lose' themselves in the homogeneous typesetting style.
In conclusion, this is a fine book which achieves its purpose of providing a short, example filled introduction to Matlab. Although I had no prior experience with Matlab, I received a good foundation in the system after using this book.
La Trobe University, Bendigo