Multivariable Calculus Set Free

Abstract

A sequel to Calculus Set Free: Infinitesimals to the Rescue, this text explores the usual topics of multivariable calculus using recent developments in infinitesimal methods, notation, and procedures. A review chapter is included with a full explanation of infinitesimals, approximation, and single-variable calculus using these methods, for the benefit of readers with no prior experience with infinitesimals. The treatment of some topics may seem fairly standard, but infinitesimal methods are used where applicable. 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. Abundant use of diagrams aids the development of visual intuition and understanding. Exercise sets provide ample practice and range from the simple to the straightforward to the more interesting and exploratory, appropriate for a wide range of abilities.