This fundamentals textbook presents small chapter-units of about two dozen pages each and supplies brief answers to all exercises. This is a new edition including a chapter on Fourier integrals and transforms along with new problems and section updates along with an interactive study guide.

The printed book alone, while adequate as a course text with substantial augmentation from lecture, is much too sparse and broad for serious use by the independent reader outside of a reference. Consider the case of PDEs, germane to physics and engineering, which are covered in two pages as an outcome of the wave equation, with only D’Alembert’s solution given any depth.

While the wide scope forces brevity (for example, this text goes from introducing vectors and matrices to calculus fundamentals in less than one hundred pages), the authors’ approach is largely effective. Topics and definitions are introduced conceptually as motivations for the mathematics. For example, torque is broached as a mechanical action on page 23, then modeled as a vector product on page 31. At times this race through material with little time for depth causes much to be glossed over. A general theory of polynomials and their roots is not introduced until section 6.5.7 as a subtopic to partial fraction decomposition in the context of integration, after chapters on differentiation and functions of the trigonometric, logarithmic, and exponential variety. Much here is as illustrative as it is spare, making for elegant classroom capsules, such as the overview of power series in chapter eight.

What makes this a truly powerful package, however, is the interactive study guide downloaded from the publisher as a rich Adobe PDF package. This has two versions. The one intended for printing struck me as cumbersome:

To continue you must first print the chapter to be studied. Each page contains two frames of the study guide. You will find the first frames of the chapter in the upper part of the pages. While working through you will see the answers to a given question only after having turned over the page…

The screen version is as effective as — or better than — any analogous interactive and educational system I have seen, such as Aplia, MyMathLab, or Moodle. It is effectively an interactive Adobe PDF with 2,395 pages. So long as the computer system selected handles this well, the guided learning experience is engaging. Students are required to compose their answers offline and then advance in a direction depending on the accuracy of their self-assessed answers. Missing here are interactive widgets for graphing, symbolic notation, etc. While this means the tool cannot be reliably used as a homework solution, I find most student frustration from such tools comes from their unique solutions to answer entry, which is not involved here.

What makes this work stand out for me among fundamentals texts is the pedagogical motivation. It is plain that the authors are drawing on lecture experience for a student-friendly presentation. The material additions, especially the “screen version” of the Adobe PDF additions, brings this much closer to being an excellent study guide for the undergraduate. I say “study guide” because it would really take about a thousand pages of material, probably spread over three or four volumes, to be suitable to the ambitious scope.

I feel the authors would truly succeed if they refined their scope rather than adding any more material. (They could easily add the calculus of variations, complex variables, especially in the context of electrical engineering, and adequate coverage of PDEs.) But any student not guided by frequent lecture contact or tutor support will soon be at sea with the thinness of the coverage of each topic. Since any semester-long course will only cover only a portion of the gamut here, focusing the content will improve this work. In its present form, purchasers should consider it a helpful compact reference as well as adjunct text for the bulk of their undergraduate academic courses.

Tom Schulte teaches mathematics to some future engineers and maybe some physicists in the heart of Automation Alley at Oakland Community College’s Auburn Hills campus.