Michael Spivak: Calculus (1967)

Edition: Addison-Wesley (World Student Series)
Review number: 1506

Calculus was the very first textbook I read for my university degree. As well as being a fine description of the basics of analysis (mostly real, with a toe in the deep water of complex functions), it is an excelent book to ease the transition from mathematics as taught at school level to the rigours of university mathematics.

Unlike many writers of textbooks in mathematics, Spivak makes a big effort to give more than a dry exposition: theorem - proof - next theorem etc. Considerable attention is paid to motivating the discussion, showing why each result is important (though mainly in the pure mathematics context, applications of calculus being mainly found in the problems at the end of each chapter). Of especial use to the budding mathematician are the points where Spivak discusses potential proof strategies for the theorems, often explaining the pitfalls that student taking a naive approach could fall into. There are even occasional jokes, both in the text and the index.

For students with an interest in how analysis can be used in apparently unrelated parts of mateematics, a number of advanced sections give proofs of such topics as the transcendence of the number e, and a construction of the real numbers from set theoretic principles.

Calculus was not just the first university textbook I read, but one of the best.

My rating: 10/10.