"Mathematics Education needs to be reformed as we now pass into the new millennium", so reads the very first sentence in the preface to volume 1 of *Applied Mathematics: Body and Soul*. But readers will be relieved to hear that this refers specifically to the field of applied mathematics (for science and engineering students) with no mention of any need to restructure mathematics curricula internationally.

The process of reform, initiated by the authors, has its roots in work that began 20 years ago in courses in computational mathematical modelling (CMM), and they say that their philosophy has "permeated through their system of basic maths education in calculus and linear algebra". The overall aim is to develop a complete programme for mathematics education in science and engineering up to graduate level.

Currently, the programme is under-pinned by a series of books and a variety of software, freely available online at http://www.phi.chalmers.se/bodysoul/. This website is meant to be an interactive forum for the exchange of ideas, to which students and teachers are invited to submit material for the purpose of its ongoing development.

As for the books, I can vouch for the excellence of the above three volumes, but the earliest in the series appeared in 1996 under the title *Computational Differential Equations* (CUP), whose authors are the same as those above but with the inclusion of P. Hansbo. In addition to that, it is intended to publish volume IV under the same title as volumes I-III above, and the following three volumes are scheduled to appear in the near future: V *Dynamical Systems*; VI *Solid Mechanics*; VII *Electromagnetics*.

It must be pointed out that the online software is an integral part of the scheme, and the various packages are supplementary to the books. Thus, when specifying the omission of material (e.g., curve sketching techniques) one has to bear in mind that there may be computerised alternatives available via the website.

The pedagogical basis of the programme is clarified by twelve stated aims, salient members of which are paraphrased below :

- To synthesise mathematics, computation and applications.
- To employ constructive mathematical methods, including computational methods for differential equations arising from mechanics, physics and chemistry.
- Emphasis on problem solving, project work and presentation.
- To cover traditional material on analysis and linear algebra.
- Understanding of the real number system, Cauchy sequences, Lipschitz continuity and constructive tools for the solution of algebraic/differential equations.

I have to say that the contents the latest volumes are compatible with this overall philosophy and that they are quite different from any applied maths books of my acquaintance. For one thing, there is much interesting material pertaining to the history of mathematics, mainly constituted of nicely illustrated biographical descriptions of relevant mathematicians. Another distinguishing feature is the way in which the authors have successfully avoided a rigid pure/applied dichotomy and these books could certainly be recommended for maths majors generally.

Concepts and techniques are rigorously worked out from first principles and there are many excursions into the realm of foundational mathematics, with initial discussion as to the nature of mathematics and subsequent consideration of matters such as the Cantor-Kronecker conflict (formalism vs. constructivism) etc. In the section devoted to logical paradoxes, there is a picture of Bill Clinton, who is quoted in the caption as saying "I am not lying" and, many pages earlier, a verse from what sounds like a 1940s love song, perhaps providing a clue as to the vintage of the authors (and maybe some indication as to what Bill was not lying about).

#### Vol 1: Derivatives and Geometry in R^{3}

Volume 1 consists of 425 pages, divided into twenty-six chapters, the first eleven of which can possibly be described as foundational in the sense that they address questions such as: What is mathematics? What is calculus? What is number? What is a function? and so on.

The authors really get down to business in chapter 12, which introduces the notion of Lipschitz continuity, initially based upon discussion of functions whose domains are subsets of **Q** (the rational numbers). One of the reasons given for this approach to real analysis is that pathological difficulties encountered by use of Cauchy ε-δ techniques are best avoided when teaching non-maths majors and the authors maintain that this is possible by use of Lipschitzian techniques.

Chapter 13 provides a cogent discussion of rational sequences and series, although it does actually invoke the classical N-ε definition of convergence. There is, however, a quick return to the idea of Lipschitz continuity when it comes to proving results like: lim f(a_{n}) = f(lim a_{n}), where f is Lipschitz continuous. The chapter concludes with coverage of the various techniques for computing limits and there is a good collection of practice exercises at the very end.

The construction of irrational numbers and the real number system begins in chapter 14 and is further developed in chapter 15. Following some coverage of the bisection algorithm, rational Cauchy sequences make their first appearance and irrational numbers are then defined to be those with non-periodic decimal expansions, meaning that any irrational number, x, can be considered as the limit of a Cauchy sequence, {x_{i}}, of its truncated decimal expansions. Based upon this definition, rules for addition/subtraction of real numbers are given in terms of decimal addition and subtraction and by using results such as:

x + y = lim x_{i} + lim y_{i} = lim [(x_{i} +y_{i})]In order to extend the operations on **R** to include multiplication and division, they are defined as functions from **Q**x**Q** to **Q** and shown to be Lipschitz continuous.

In a way, chapter 15 is key to the whole book, because, apart from constructing the real number system, it proceeds to combine notions of Cauchy sequences and Lipschitz continuity for the purposes of extending functions f: **Q** → **Q** to f: **R** → **R**. The chapter concludes by deliberating on the pros and cons of Cauchy continuity as opposed to Lipschitz. Excellent stuff!

Computational methods arise by means of more work on the bisection algorithm (ch 16) and by an introduction to fixed points and contraction mappings in ch 19. Further treatment is given in the last two chapters (25 and 26), where there is some mathematical modelling of physical ideas (emanating from Galileo, Newton, Hooke, Malthus and Fourier). These chapters constitute a very nice synthesis of previous ideas in a historical setting.

The title of this volume refers principally to Derivatives and Geometry yet, apart from the preliminary sketch in ch 4, differentiation (the art of ascertaining the derived function) is the subject of the thirty-eight pages that go to make up chs 23 and 24, which is less than 10% of the entire contents. After some initial discussion on rates of change, and two numerical examples, there are three pages of analytical discussion, culminating with this definition (I've changed the notation slightly to render it in html):

The function f(x) is said to be *differentiable* at a if there are constants m(a) and K_{f}(a) such that, for x close to a, f(x) = f(a) + m(a)(x - a) + E_{f}(x,a), with |E_{f}(x, a)| ≤ K_{f}(a)|x - a|^{2}

The remainder of ch 23 and all of ch 24 employ this definition to obtain derivatives for simple polynomial functions and to formulate the standard rules for differentiating sums, products, quotients etc.

Personally, I really liked this treatment, and later discussion of things such as uniformly differentiable functions, but I have just one quibble regarding the chapter on differentiation. This concerns the brief introduction to Leibniz notation, on p. 367, where objects like df and dx are loosely described as finite differences (not even infinitesimals or linear transformations). Yet a few pages later, the symbol δf is used to denote the same thing as df (there's scope for a biographical vignette about Bishop Berkeley here!). Apart from a section obscurely buried within volume 3 in the chapter on differentiability of functions of the type f: **R**^{n} → **R**^{m}, there is no further clarification as to the meaning of differentials (because subsequent emphasis is placed on derivatives).

Treatment of geometry begins in chapter 8 with a brief review of some aspects of Euclid. Then, in chapter 20 (Analytic Geometry in **R**^{2}), the basic ideas vector algebra are explained. Included in this are introductions to scalar and vector products and certain transformations such as rotations and reflections and the matrix representation of such. Apart from this, and some work on the calculation of areas of simple 2d figures, the emphasis is very much on linear algebra. Chapter 21 (Analytic Geometry in **R**^{3}) extends the work of ch 20 but it is decidedly more geometric and introduces more concepts and techniques related to 3-dimensional linear algebra and matrices. Methods for the solution of systems of linear equations emerge quite clearly from this framework.

One final observation concerns the treatment of number. I feel that, had the authors introduced the notion of equivalence relation, it would have led to a less ambiguous derivation of rational numbers as equivalence classes of ordered pairs of integers (and real number as equivalence class of rational Cauchy sequences). Of course, this also applies to the Russell definition of natural number, which is the basis for correct teaching methods for the formation of children's number concept. Discussion of rational numbers would also have benefited from explicit mention of density, for it is this phenomenon that intuitively leads one into believing that there can be no rational gaps in the number line.

#### Vol 2: Integrals and Geometry in R^{n}

In this volume, the constructivist approach to analysis is put forth as a means of rigorously developing the theory and methods of integration. For example, the first chapter (27) launches into a seven-page formal proof of the fundamental theorem of calculus that makes G. H. Hardy's classical analytic exposition seem like light reading. The starting point for this is defined in terms of solving the differential equation u' = f for u: [0,1] → **R**, where f is a given real-valued, Lipschitz continuous function with the same domain. The proof itself invokes use of a variety of previously explained concepts such as Cauchy sequences, contraction mappings etc, and it evinces mathematical rigour of an aesthetically pleasing kind.

However, whilst working through all this, I couldn't help wondering about possible reactions from the intended beneficiaries of such analytical exposition; to wit, science and engineering students! Moreover, I'd imagine that mathematics specialist (majors) would find it rather daunting — unless, of course, prior concepts had been well and truly assimilated and more intuitive work on integration had been provided.

What of volume 2 as a whole? Here's a brief sketch of what's included in some of the ensuing chapters:

**Chapter 28: Properties of the Integral:** Piecewise integration of Lipschitz continuous functions, linearity, triangle of inequalities, methods of integration, Taylor's theorem.
**Chapter 29:** The logarithmic function defined as the solution f, to the differential equation f '(x) = 1/x for x > 0 and f(0) = 1.
**Chapter 30:** Numerical quadrature; Riemann sums, numerical integration and adaptive quadrature.
**Chapter 31:** Construction of the exponential function as being the solution to the equation u'(x) = u(x), for x > 0 and u(0) = 1. More challenging analysis here, with u(x) defined as being the limit of a sequence of piecewise linear functions, culminating with the revelation that u is the inverse of log.
**Chapter 32:** Circular functions arising from u''(x) = -u(x), for x > 0, u(0) = u_{o} and u'(0) = u_{1}. The usual range of identities then emerges, together with definition of tan, cot and the inverse circular functions. Hyperbolic functions are defined in terms of e^{x} and their relevance to the catenary is then revealed.
**Chapter 33:** exp(z), log(z), sin(z) and cos(z) where z is in **C**.
**Chapter 34:** Further techniques of integration (partial fractions etc.).

Take a break at this point to ask "How is all this being presented?" In my view, the pedagogical style is not strongly based upon heuristic principles, whereby understanding evolves from short amounts of written exposition frequently interspersed with developmental exercises/problems. Granted that the narrative does contain motivational observations and the literary standard is high, but the reader is often working his/her way through long pieces of mathematical narrative and is only directed towards specific activities at the end of each chapter. Moreover, the nature of the problems is such that tutors who employ these texts may wish to provide additional routine exercises for those chapters in which the problems are predominantly open-ended and designed to extend some of the ideas previously introduced.

**Chapters 35, 38, 39 and 40** provide specific coverage of differential equations. Use of integrating factors and solution of Euler's equation are nicely explained in ch 35 followed by discussion of scalar autonomous initial value problems in ch 38. Then there are two short chapters that consider separable initial value problems.

There is a great richness of ideas contained in this approach to differential equations, particularly in the last of these chapters, where there is discourse upon the nature of determinism and materialism, predictability and computability and concluding with some discussion of numerical methods. But most of this is introduced with a dearth of practice at modelling real situations. One has to wait to the very end of this volume to be rewarded by interesting accounts of Lagrange's principle of least action, N-body systems, electrical circuits, string theory, two-point boundary value problems etc. (chapters 47 to 53).

In the meantime, there is **chapter 42**, dealing with "analytic geometry in **R**^{n}," in which the geometrical content is subsidiary to the discourse on linear algebra. It considers linear independence, bases, scalar product and orthogonality, linear transformations and matrices. Applications include Cramer's solution for non-singular systems of linear equations followed by least squares methods, with the truly geometric ideas being confined to an explanation of how the use of determinants achieves a measure of volume in **R**^{n}.

Ch.43 reveals the mysteries of eigenvalues and eigenvectors leading to the arcane world of the spectral theorem for symmetric matrices. (Eureka! There's a misprint in the first line of the statement of this theorem on p648).

"Why are we doing all this?" scream incipient engineers and budding scientists. But once they reach chapter 44 and they'll see the pay-off in the form of applications to the numerical solution of systems of linear equations. Here, human ingenuity in all its guises is brought to bear on such problems, with a myriad of techniques constituting the mathematical panoply (Gaussian elimination, pivoting, iterative methods, conjugate gradient methods etc). Finally, there is a rich collection of practice exercises to assist with the consolidation of the many ideas it has introduced.

#### Volume 3: Calculus in Several Dimensions

In this volume, there are thirty-four chapters, whose contents are so rich and diverse that it is difficult to paint a reliable picture of what they contain and how the material is presented. (even in a review already as long as this one). But the authors, quite correctly, describe the contents as follows:

This volume presents calculus in several variables, including partial derivatives, multi-dimensional integrals, partial differential equations and finite element methods, together with a variety of applications modelled as systems of partial differential equations.

Of course, such a compacted account needs considerable embellishment, for which I am running out of space to provide. So let's whisk through the chapter headings:

Vector-valued functions in **R**^{n}. Level curves/surfaces and gradient. Linearization and stability of initial value problems. Adaptive solvers for IVPs. Lorenz and the essence of chaos. The solar system. Optimisation. Div, rot, Laplacian. Meteorology and Coriolis forces. Curve integrals. Double integrals. Surface integrals. Multiple integrals. Gauss' theorem & Green's formula in **R**^{n}. Stokes' Theorem. Potential fields. Centre of mass and Archimedes' principle. Newton's nightmare. Laplacian models. Chemical reactions. *Calculus toolbag*. Piecewise linearity in **R**^{2} & **R**^{3}. FEM boundary value problems. Inverse problems. Optimal control. *Differential equations toolbag*. *Applications toolbag*. Analytic functions. Fourier series. *Analytic functions toolbag*. *Fourier analysis toolbag*. Fourier transforms. Incompressible Navier-Stokes.

Those chapters described as "toolbags" (italicized above) serve the purpose of summarising the mathematical techniques used in particular parts of the book. For instance, the contents of that labelled "Applications Toolbag" contains the following techniques: Malthus Population Model, The Logistic Equation, Mass-Spring-Dashpot System, LCR-Ciruit, Laplace's Equation for Gravitation, Heat Equation, Wave Equation, Maxwell's Equations, Schrödinger's Equation, etc.

In my opinion, volume 3 captures the true essence of contemporary applied mathematics. I say this because of the diversity of the mathematical ideas that it covers and the fascinating range of applications to which they are applied. As for presentation, this volume lives up to the extremely high standard that's in evidence in vols 1 & 2. There is the historical commentary of course, and an abundance of illustrations, all serving to enhance the appeal of the rich mathematical content. All this is complemented by provision of many interesting and often amusing quotes just below each chapter heading, like this one (from James Joyce's *Finnegans Wake*) that appears in Ch 56:

*The logos of somewhome to that base anything, when most characteristically mantissa minus, comes to nullum in the endeth: orso, here is nowot badder than the sin of Aha with his cousin Lil, verswaysed or coversvised,and all that's cosecants and cotangincies...*

Oh, by the way, I suggest immediate purchase of all three volumes!

Peter Ruane ([email protected]) was Senior Lecturer in Mathematics Education at Anglia Polytechnic University, England. His research interests lie within the field of mathematics education and the history of geometry.