# Calculus and Analysis in Euclidean Space

• Jerry Shurman

Part of the Undergraduate Texts in Mathematics book series (UTM)

1. Front Matter
Pages i-xiii
2. Jerry Shurman
Pages 1-20
3. ### Multivariable Differential Calculus

1. Front Matter
Pages 21-21
2. Jerry Shurman
Pages 23-58
3. Jerry Shurman
Pages 59-130
4. Jerry Shurman
Pages 131-197
5. Jerry Shurman
Pages 199-250
4. ### Multivariable Integral Calculus

1. Front Matter
Pages 251-251
2. Jerry Shurman
Pages 253-348
3. Jerry Shurman
Pages 349-374
4. Jerry Shurman
Pages 375-408
5. Jerry Shurman
Pages 409-501
5. Back Matter
Pages 503-507

### Introduction

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.

### Keywords

derivative implicit functions integration of differential forms linear mappings multivariable differentiable calculus multivariable integral calculus parameterized curves textbook adoption calculus textbook adoption analysis single-variable calculus Euclidean space inverse functions approximation smooth functions parametrized curves

#### Authors and affiliations

• Jerry Shurman
• 1
1. 1.Reed College Dept. MathematicsPortlandUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-319-49314-5
• Copyright Information Springer International Publishing AG 2016
• Publisher Name Springer, Cham
• eBook Packages Mathematics and Statistics
• Print ISBN 978-3-319-49312-1
• Online ISBN 978-3-319-49314-5
• Series Print ISSN 0172-6056
• Series Online ISSN 2197-5604