Multivariable Calculus Set Free: Infinitesimals Ride Again

A sequel to the author's previous book, Calculus Set Free: Infinitesimals to the Rescue, this text explores all the key topics of multivariable calculus using infinitesimal methods. These recently developed infinitesimal procedures, based on the approximation relation, simplify limit-based computations and make many concepts more intuitive and accessible.

Limit-based computations utilize infinitesimals, simplifying computations and making a wider variety of limits accessible. Definitions of double integrals, triple integrals, line integrals, and surface integrals, as well as related geometric and physical applications, are based on infinitely many subintervals (or subregions) of infinitesimal size, resulting more clearly in a sum and simplifying definitions. Local linearity is captured in an easily understood approximation formula, demystifying the use of straight objects instead of curved objects when developing integration formulas. Throughout the text, abundant use of diagrams aids the development of visual intuition and understanding. Exercise sets provide ample practice, ranging from the simple to the straightforward to the more interesting and exploratory, appropriate for a wide range of abilities. Notes in the margins include comments meant to supplement understanding, including line-by-line commentary for many worked examples.

For the benefit of readers with no prior experience with infinitesimals, a review chapter is also included with a full explanation of infinitesimals, approximation, and single-variable calculus using these methods, setting the stage for their use in multivariable calculus. Without sacrificing academic rigor, Multivariable Calculus Set Free offers an engaging, storytelling style that helps readers solidify their understanding of an otherwise difficult subject, written by an author with over three decades of experience working with calculus students.

C. Bryan Dawson, ᏣᎵ ᏓᏐᏂ, University Professor of Mathematics, Union University

C. Bryan Dawson received his PhD in mathematics in 1992 at the University of North Texas and has over three decades experience teaching calculus. In recent years, Dawson's scholarly efforts have focused on infinitesimal analysis. Recognizing that his newly developed notation and procedures are beneficial to beginning calculus students, Dawson began writing textbooks incorporating the methods, beginning with Calculus Set Free: Infinitesimals to the Rescue and culminating with Multivariable Calculus Set Free: Infinitesimals Ride Again. He received the Union University Faculty Researcher of the Year award in 2023. He belongs to several academic societies including Indigenous Mathematicians.