Taylor (mathematics, U. of Utah) has developed a textbook from notes for a two-semester undergraduate course he teaches for students who have completed three semesters of calculus and one semester of linear algebra. For many students, he says, this is the first course in which everything is proved rigorously, and they are expected to create proofs as well as understand them. He presents a rigorous development of the calculus, beginning with a study of the properties of the real number system. The topics include sequences, the derivation, infinite series, convergence in Euclidean space, differential in several variables, and vector calculus. Annotation ©2012 Book News, Inc., Portland, OR (booknews.com)