Multivariable Calculus with Applications

Lax and Terrell’s sequel to their Calculus With Applications presents a first course in multivariable calculus that fits in just over 400 pages. Even instructors who use standard texts will find much of value in this refreshing first edition. The book is written with a wide range of STEM students in mind, and its exposition remains remarkably fluid without scarificing precision. Every section of each chapter ends with an excellent collection of exercises, which should be graciously welcomed by independent learners and instructors alike.

The first couple of chapters introduce basic linear and multilinear algebra, as well as functions of several variables and their continuity. The next three chapters expose the rudiments of differentiation (chain rule, inverse and implicit function theorems, extrema of functions of several variables) and applications to motion in space. These are followed by a trio of chapters covering the rudiments of integration, line and surface integrals, and the divergence and Stokes’ theorems. A more thorough discussion of electromagnetism and Maxwell’s equations would have brought the techniques to life. On the upside, there is a neat introduction to fluid dynamics (a topic largely absent from a number of multivariable calculus texts used in US universities) via describing the three basic conservation laws of mass, momentum, and energy of a fluid flow with an emphasis on one-dimensional flows. The final chapter on partial differential equations is a fitting coda to the methods and theory learned thus far. The authors derive the laws governing the vibration of stretched strings and membranes, and those governing heat propagation. They end with skimming the surface of the Schrödinger equation.

The authors often present proofs and examples in low dimensions before tackling more general cases, which is certainly a pedagogically sound strategy aimed at helping students better understand the material. Keeping in mind the struggles of many beginners that I have taught, certain sections (e.g., those on the inverse and implicit functions theorems) could benefit from the inclusion of a larger variety of worked examples.

The authors chose not to go further with their discussion of the fundamental theorem of calculus in higher dimensions, and ended that narrative arc with Stokes’ Theorem applied to surfaces in \(\mathbb{R}^3\) as a curved version of Green’s Theorem. I felt that a chapter (or even a section) covering calculus on higher-dimensional manifolds and applications thereof would have been useful. Perhaps we can hope for a sequel that does expose such content!

The text could certainly be further enhanced via the inclusion of projects that involve some coding and numerical methods (say using open source mathematics software like SAGE) at the end of each chapter. There was also a lack of footnotes or endnotes to guide interested students to further explorations. For instance, Section 9.2 (Vibration of a membrane) ends with the lines: “This explains why all musical instruments are essentially one-dimensional vibrating systems. Violins and cellos use vibrating strings to generate sound, wind instruments like flutes and clarinets use vibrating thin columns of air to generate sound. One can point to drums as a truly two-dimensional instrument; but the sound of a drum of is muffled, without a definite pitch!” Students who are aware of drums that can generate distinct pitches (e.g., tabla or timpani/kettledrums) may want references to where they could learn more about mathematical models of such instruments!

Minor quibbles aside, I find Lax and Terrell’s to be a marvelous text! I will certainly keep my copy close at hand when teaching courses on multivariable calculus and mathematical physics. I hope the authors will write us more sequels, for instance on ordinary and partial differential equations, and continue with their exposition of the “essential relationship between calculus and modern science”.

Tushar Das is an Associate Professor of Mathematics at the University of Wisconsin–La Crosse.