Calculus and Analysis in Euclidean Space

Authors: Jerry Shurman



Concentrates on n-dimensional Euclidean space

Uses multivariable calculus to teach mathematics as a blend of reasoning, computing, and problem-solving, doing justice to the structure, the details, and the scope of the ideas

Contains figures, formulas, and words to guide the reader to do mathematics resourcefully by marshaling the skills of geometric intuition, algebraic manipulation, and incisive use of natural language

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.

Table of contents

Front Matter
Pages i-xiii

Results from One-Variable Calculus
Pages 1-20

Multivariable Differential Calculus
Front Matter
Pages 21-21
Euclidean Space
Pages 23-58
Linear Mappings and Their Matrices
Pages 59-130
The Derivative
Pages 131-197
Inverse and Implicit Functions
Pages 199-250

Multivariable Integral Calculus
Front Matter
Pages 251-251
Integration
Pages 253-348
Approximation by Smooth Functions
Pages 349-374
Parametrized Curves
Pages 375-408
Integration of Differential Forms
Pages 409-501

Back Matter
Pages 503-507