Single and Multivariable Calculus

This open source calculus textbook covers both single and multivariable calculus. The book carries a “Creative Commons Attribution-NonCommercial-ShareAlike” license, which means it is freely redistributable and that others are free to edit the text for their own purposes. Between the two of us, we have taught both single and multivariable calculus courses using it and find it to be a well-written and complete textbook. This is all the more remarkable since it is entirely free to use.

Though hardcopies are available at cost from the on-demand publishing site Lulu.com, the electronic version of the text (in pdf format) is free and is a real pleasure to use. The chapters and sections are all indexed in the pdf for ease of navigation. In addition, all references to figures, theorems, examples, and exercises are internally hyperlinked, leading the reader directly to referenced information with the click of a mouse. Another advantage of the electronic version of the book is the inclusion of Java- and Sage-based figures that the interested reader can view and manipulate. This feature is particularly valuable in the multivariable chapters of the text.

The ordering and treatment of single-variable calculus is quite standard and traditional. It begins with a chapter reviewing precalculus/analytic geometry. After this, it does differential calculus, proceeding through limits, derivatives, transcendental functions, and standard applications of derivatives (finding extrema, related rates, linear approximations, Newton’s method). It then starts integral calculus, covering, in order, integration and the fundamental theorems of calculus, methods of integration, and standard applications of integration (area, volume, arc-length, surface area, work, average value, center of mass, probability). This is followed by a chapter on polar coordinates and parametric equations, including how to do calculus (slopes/areas) with them. Last comes a chapter on sequences and series, treating the usual topics and leading up to Taylor’s Theorem with remainder.

There were a few single-variable topics that the text omitted or treated very lightly in the version reviewed here. More recently, however, some of those have been incorporated into the current version of the book, e.g., the section on differential equations has been expanded, and moved to a later chapter. As another example, the book does not discuss at all numerical integration methods such as midpoint, trapezoidal, or Simpson’s rule. However, given the evolution of the text so far, we would not be surprised if the author were to introduce such a section upon request of instructors.

The arc of the multivariable calculus sections is also fairly traditional. It starts with an introduction to vectors, lines, and planes, moves on to vector-valued functions and their derivatives, then to functions of several variables and partial derivatives, then to multiple integration, and finishes with Green’s, Stokes’, and the divergence theorems. The narrative content is well-supported with concise figures and displayed equations and identities. The text is also good at presenting the “big ideas” without getting mired down in process.

The discussions and explanations are succinct and to the point, in a way that pleases mathematicians who don’t like calculus books to go on and on. This allows the instructor flexibility to choose which technical points to expand on, without distracting the student. It may be perceived as a disadvantage for instructors who want a book that adjudicates every technical point in advance. A section that we found admirable in its concision is Section 2.5 “Adjectives for functions.” This section discusses notions such as continuity and differentiability (at a point, and on an interval) very briefly, leaving the instructor freedom to say more, if desired, about issues of endpoints, right-handed and left-handed continuity, etc. Student feedback reveals that there are also many students who enjoy this sort of brevity, but others who want more discussion, and in particular, more examples.

Although the book doesn’t provide as many examples as a typical thicker text and doesn’t contain as many exercises at the end of each section as most calculus texts, it still manages to fit the bill in the following ways. The single-variable sections where one desperately wants a huge selection of problems, namely max-mins and related rates, are exactly the two sections that deviate from the rest by having a large selection of exercises — more than enough. In the rest of the book, we found that there were also usually just enough problems to choose from adequately. Although some students will want to work more exercises, this is easily remedied by suggesting that the students buy any of the many inexpensive and readily available calculus exercise books.

The book does not include the “frill” of historical discussions. This seems consistent with the concise and terse style that originally attracted us to the text. It’s not that we don’t enjoy the nice historical discussions that one finds either interspersed or in the appendices of some standard calculus texts; we just don’t enjoy them enough to insist on having them in the textbook! Perhaps in future versions relevant hyperlinks to some of the many fine math history websites could be included at strategic locations throughout the text.

There is a noticeable typographic issue in the book: the author gives important facts/rules/laws/definitions as either numbered theorems or definitions, e.g., “Theorem ?.??” or “Definition ?.??.” Standard calculus texts often have the same type of information boxed or highlighted in the center of a line for emphasis and to help readers scan and review. Some students do comment on this, and we found it a (minor) disadvantage in the exposition. An example of this shortcoming can be found in Section 12.6 “Other Coordinate Systems.” The spherical coordinate transformations appear only in the running text of an example. This makes it difficult for the student to quickly scan back through the section and find the formulas. (It should be noted that the placement of these formulas has improved in the current version, but they are still not as visible as they might be.)

A real strength of the book is the author’s adoption of open-source methods to incorporate changes and material suggested by instructors that have used the book (the open-source model). Typically, the author updates the book once a year based on his own ideas and feedback received from others throughout the academic year.

This is a well-written, thoughtful, and carefully-edited calculus book. It may lack the bells and whistles of more standard texts, but is still quite competitive. This book is proof that sometimes you get much more than you pay for.

Victor Reiner is a Professor in the School of Mathematics at the University of Minnesota. Albert Schueller is a professor in the mathematics department at Whitman College in Walla Walla, WA.