Mathematical Analysis: A Very Short Introduction

Very Short Introductions #734

by Richard Earl

Very Short Introductions : Brilliant, sharp, inspiring

The 17th-century calculus of Newton and Leibniz was built on shaky foundations, and it wasn't until the 18th and 19th centuries that mathematicians--especially Bolzano, Cauchy, and Weierstrass--began to establish a rigorous basis for the subject. The resulting discipline is now known to mathematicians as analysis .

This book, aimed at readers with some grounding in mathematics, describes the nascent evolution of mathematical analysis, its development as a subject in its own right, and its wide-ranging applications in mathematics and science, modelling reality from acoustics to fluid dynamics, from biological systems to quantum theory.

ABOUT THE The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.

Genres: Science, Mathematics, Nonfiction

224 pages, Paperback

Published September 22, 2023

Reviews

[Beau · Rating 4 out of 5]

Brilliantly written by someone who really knows the ins and outs of all the topics presented: calculus, its history, and its people; differential equations--ordinary and partial; numerical analysis; vector calculus; the calculus of variations; the wave equation and boundary value problems; Fourier analysis; complex analysis; Lebesgue integration and measure theory; statistics; and the Riemann hypothesis. The author covers the tip of the iceberg on all of these topics--and a few more--and even offers some explanations and proofs along the way (sometimes in the appendix). The book was very well planned and organized. The only problem: who is the intended audience? If you haven't had calculus yet, you'll be very lost after the first couple dozen pages, and yet the author introduces the reader to calculus. I think the book is ideally read by a math major who has finished single and multivariable calculus, as well as a course in ordinary differential equations. Having a physics course would help too. Then it will be a pleasure to reminisce on these topics while learning something new about them, and the more advanced topics will be tantalizing, rather than abstruse. Nevertheless, it's hard to imagine a better very short introduction to mathematical analysis.

[Wing · Rating 4 out of 5]

I have close to zero idea what’s going on in the last chapter — which, fortunately, is pretty short. The earlier chapters are fantastic and, as an introduction for the lay reader, better than, say, The Calculus Lifesaver. Granted, they have different target audiences: the former is like a pamphlet, while the latter is almost a tome. But this Very Short Introduction actually allows the reader to understand how mathematical analysis works — and, more importantly, why it is so relevant to other branches of science.

For instance, readers will feel that they understand Schrödinger’s equation, because the author explains what partial differential equations are and why they can model natural phenomena. It should be clear that the actual mathematics — although described in some detail and quite profusely — are meant to be beyond certain readers (readers like me; I’m a medical doctor by profession). Those expecting comprehensive proofs and derivations will be disappointed. But then, any reasonable person wouldn’t expect them from a book like this. Four stars.

[cassie · Rating 3 out of 5]

I liked the opening on different types of infinities and how key figures developed modern mathematical analysis. Some cool examples, such as harmonics and the vibrations of strings, but partial derivatives are beyond me xx. Loved narrating to Richard Earls voice in my head, though, and it was a privilege to be lectured by him whilst reading.

[Dirk · Rating 4 out of 5]

A good introduction, it covers a lot of ground, it's more suited for people who like equations, maybe it could have covered less topics but could go deeper.