Books : reviews

Lara Alcock.
How to Think about Analysis.
OUP. 2014

Analysis—sometimes called Real Analysis or Advanced Calculus—is a core subject in most undergraduate mathematics degree programmes. It is elegant, clever and rewarding to learn. But students who are unprepared often find it incomprehensible at first, and even the best students find it challenging.

This book aims to help all students prepare for and enjoy the challenge. It is not a textbook containing standard content; rather, it is designed to be read before starting an Analysis course, or as a companion text once a course has begun. It provides a friendly and readable introduction to the subject by building on the student’s existing understanding of six key topics: sequences, series, continuity, differentiability, integrability and the real numbers. It explains how mathematicians develop and use sophisticated versions of these ideas, and provides a detailed introduction to the central definitions, theorems and proofs. Throughout, it highlights common sources of difficulty and confusion and explains how these can be overcome.

The book also provides study advice focused on the skills that students need if they are to build on this introduction and learn successfully in their own Analysis courses. It explains how to understand definitions, theorems and proofs by relating them to examples and diagrams, how to think productively about proofs, and how theories are taught in lectures and books on advanced mathematics. It also offers practical guidance on strategies for effective study planning. The advice throughout is research-based and is presented in an engaging style that will be accessible to students who are new to advanced abstract mathematics.