Book reviews: Petro-Luciano Buono; Advanced Calculus. Differential Calculus and Stokes’ The- orem, De Gruyter Textbook, De Gruyter, Berlin/Boston 2016, x+303 p., (ISBN 978-3-11-043821-5/pbk; 978-3-11-043822-2/ebook.

Abstract

The book is based on the notes of a one-semester Calculus III course at the University of Ontario Institute of Technology starting with 2012. Its aim is to give a unified treatment of Green’s, Stokes and Gauss’ theorems (in R2 and R3), paving the way to more advanced topics from differential geometry. The approach proposed by the author has a geometric flavor, the tangent space being introduced early in the study of differentiability of functions of one variable, differential forms and pullbacks. The main advantage of this approach, based on tools from linear algebra, consists in the possibility to define the differential of a function properly, as acting on tangent vectors, and from there the study of differential forms and pullbacks in the context of line integrals. As the author mentions in the Preface, one starts with the introduction of terminology in the context of curves (one-dimensional geometric objects, easier to understand) and then, after the introduction of the differentials of vector functions of several variables and of the Jacobian, one extends the differential form concepts to higher dimensions.