Calculus and Analysis in Euclidean Space

ISBN-13: 9783319493121 / Angielski / Twarda / 2016 / 505 str.

Jerry Shurman

Opis

The graceful role of analysis in underpinning calculus is often lost to their separation in the curriculum. This book entwines the two subjects, providing a conceptual approach to multivariable calculus closely supported by the structure and reasoning of analysis. The setting is Euclidean space, with the material on differentiation culminating in the inverse and implicit function theorems, and the material on integration culminating in the general fundamental theorem of integral calculus. More in-depth than most calculus books but less technical than a typical analysis introduction, Calculus and Analysis in Euclidean Space offers a rich blend of content to students outside the traditional mathematics major, while also providing transitional preparation for those who will continue on in the subject. The writing in this book aims to convey the intent of ideas early in discussion. The narrative proceeds through figures, formulas, and text, guiding the reader to do mathematics resourcefully by marshaling the skills of

• geometric intuition (the visual cortex being quickly instinctive)
• algebraic manipulation (symbol-patterns being precise and robust)
• incisive use of natural language (slogans that encapsulate central ideas enabling a large-scale grasp of the subject).

Thinking in these ways renders mathematics coherent, inevitable, and fluid. The prerequisite is single-variable calculus, including familiarity with the foundational theorems and some experience with proofs.

Recenzje

"Shurman (mathematics, Reed College) has succeeded in presenting a text that encompasses multivariable calculus, advanced calculus, and an introduction to point-set topology. In short, this book covers aspects one should know about the elementary analysis, geometry, and topology of Euclidean space. ... Summing Up: Recommended. Upper-division undergraduates and above; researchers and faculty." (J. T. Zerger, Choice, Vol. 54 (11), July, 2017)

"The author's writing style is clear and easy to follow, but, more than that, it is exceptionally well-motivated and contains some useful pedagogical ideas. In addition, throughout the book, the author notes issues that are likely to cause trouble to beginning students, and takes the time and effort to single them out and discuss them thoroughly. There are lots of exercises, many of them quite illuminating. ... It is highly recommended." (Mark Hunacek, MAA Reviews, maa.org, March, 2017)

"This book contains a clear and well-planned lecture discussing the most important issues of differential and integral calculus. ... The big advantage of this book are nice, transparent and often colourful drawings illustrating some considerations. A nice complement to mathematical statements are explanations and comments." (Ryszard Pawlak, zbMATH 1357.26002, 2017)

Spis treści

Preface.- 1 Results from One-Variable Calculus.- Part I Multivariable Differential Calculus.- 2 Euclidean Space.- 3 Linear Mappings and Their Matrices.- 4 The Derivative.- 5 Inverse and Implicit Functions.- Part II Multivariable Integral Calculus.- 6 Integration.- 7 Approximation by Smooth Functions.- 8 Parameterized Curves.- 9 Integration of Differential Forms.- Index.

O autorze

Jerry Shurman is professor of mathematics at Reed College, Portland, OR, USA. He is also the co-author of GTM 228 "A First Course in Modular Forms."