**Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus by Michael Spivak**

[PDF eBook eTextbook]

- Publisher: CRC Press (January 22, 1971)
- Language: English
- 161 pages
- ISBN-10: 9780805390216
- ISBN-13: 978-0805390216

A supplementary text for undergraduate courses in the calculus of variations which provides an introduction to modern techniques in the field based on measure theoretic geometry. * Varifold geometry* is presented through and appraisal of Plateau’s problem.

This little book is especially concerned with those portions of “advanced calculus” in which the subtlety of the concepts and methods makes rigor difficult to attain at an elementary level. The approach taken here uses elementary versions of modern methods found in sophisticated * mathematics*. The formal prerequisites include only a term of linear algebra, a nodding acquaintance with the notation of set theory, and a respectable first-year calculus course (one which at least mentions the least upper bound (sup) and greatest lower bound (inf) of a set of real numbers). Beyond this a certain (perhaps latent) rapport with abstract mathematics will be found almost essential.

**Table of Contents:**

Editors’ Foreword, v

Preface, vii

1. Functions on Euclidean Space

NORM AND INNER PRODUCT, 1

SUBSETS OF EUCLIDEAN SPACE, 5

FUNCTIONS AND CONTINUITY, 11

2. Differentiation

BASIC DEFINITIONS, 15

BASIC THEOREMS, 19

PARTIAL DERIVATIVES, 25

DERIVATIVES, 30

INVERSE FUNCTIONS, 34

IMPLICIT FUNCTIONS, 40

NOTATION, 44

3. Integration

BASIC DEFINITIONS, 46

MEASURE ZERO AND CONTENT ZERO, 50

INTEGRABLE FUNCTIONS, 52

FUBINI1S THEOREM, 56

PARTITIONS OF UNITY, 63

<lHANGE OF VARIABLE, 67

4. Integration on Chains

ALGEBRAIC PRELIMINARIES, 75

FIELDS AND FORMS, 86

G~;OMETRIC PRELIMINARIES, 97

THE FUNDAMENTAL THEOREM OF CALCULUS, 100

5. Integration on Manifolds

MANIFOLDS, 109

FIELDS AND FORMS ON MANIFOLDS, 115

STOKES’ THEOREM ON MANIFOLDS, 122

THE VOLUME ELEMENT, 126

THE CLASSICAL THEOREMS, 134

Bibliography, 139

Index, 141

* Michael David Spivak* (born May 25, 1940) is an American mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Perish Press. Spivak is the author of the five-volume A Comprehensive Introduction to Differential Geometry. In 1964 Spivak received a Ph.D. from

*under the supervision of John Milnor. In 1985 he received the Leroy P. Steele Prize.*

**Princeton University**Spivak was born in Queens, New York.

Spivak has lectured on elementary physics. Spivak’s most recent book, Physics for Mathematicians: Mechanics I, which contains the material that these lectures stemmed from and more, was published on December 6, 2010. Spivak is also the designer of the MathTime Professional 2 fonts which are widely used in academic publishing.

Spivak coined Spivak pronouns, a set of English gender-neutral pronouns.

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