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**About this book :- **
**Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus ** written by
** Michael Spivak **

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.

The approach taken here uses elementary versions of modern methods found i n 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.

The first half of the book covers that simple part of advanced calculus which generalizes elementary calculus to higher dimensions. Chapter 1 contains preliminaries, and Chapters 2 and 3 treat differentiation and integration. The remainder of the book is devoted to the study of curves, surfaces, and higher-dimensional analogues. Here the modern and classical treatments pursue quite different routes ; there are, of course, many points of contact, and a significant encounter occurs in the last section. The very classical equation reproduced on the cover appears also as the last theorem of the book. This theorem (Stokes' Theorem) has had a curious history and has undergone a striking metamorphosis.

**Book Detail :- **
** Title: ** Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus
** Edition: ** 5th
** Author(s): ** Westview Press
** Publisher: **
** Series: **
** Year: ** 1998
** Pages: ** 159
** Type: ** PDF
** Language: ** English
** ISBN: ** 0805390219,9780805390216
** Country: ** US

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**About Author :- **

Author ** Michael David Spivak ** was born in May 25, 1940 at Queens, New York. He is an American mathematician specializing in differential geometry, an expositor of mathematics.

In 1964 Spivak received a Ph.D. from Princeton University under the supervision of John Milnor. In 1985 Spivak received the Leroy P. Steele Prize. Spivak is the author of the five-volume A Comprehensive Introduction to Differential Geometry.

Spivak has also 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) and the creator of Science International.

Spivak has also authored several well-known undergraduate textbooks. Among them, his text Calculus (W. A. Benjamin Inc., 1967; Publish or Perish, 4th ed., 2008) takes a rigorous and theoretical approach to introductory calculus and includes proofs of many theorems taken on faith in most other introductory textbooks.[8] Spivak acknowledges in the preface that others might consider this textbook to be an introduction to mathematical analysis rather than a calculus book. Another of his well-known textbooks is Calculus on Manifolds (W. A. Benjamin Inc., 1965; Addison-Wesley, revised edition, 1968), a concise (146 pp.) but rigorous and modern treatment of multivariable calculus accessible to advanced undergraduates. Spivak has also written The Joy of TeX: A Gourmet Guide to Typesetting With the AMS-TeX Macro Package and The Hitchhiker's Guide to Calculus. The book Morse Theory, by John Milnor, was based on lecture notes by Spivak and Robert Wells (as mentioned on the cover page of the booklet).

**All Famous Books of this Author :- **

Here is list all books, text books, editions, versions or solution manuals avaliable of this author, We recomended you to download all.

** • Download PDF Calculus (3E) by Michael Spivak **

** • Download PDF Calculus (3E) Solution by Michael Spivak **

** • Download PDF Calculus on Manifolds: Advanced Calculus (5E) by Michael Spivak **

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**Book Contents :- **
**Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus ** written by
** Michael Spivak **
cover the following topics.

Preface
**1. Functions on Euclidean Space**

NORM AND INNER PRODUCT

SUBSETS OF EUCLIDEAN SPACE

FUNCTIONS AND CONTINUITY
**2. Differentiation**

BASIC DEFINITIONS

BASIC THEOREMS

PARTIAL DERIVATIVES

DERIVATIVES

INVERSE FUNCTIONS

IMPLICIT FUNCTIONS

NOTATION
**3. Integration**

BASIC DEFINITIONS

MEASURE ZERO AND CONTENT ZERO

INTEGRABLE FUNCTIONS

FUBINI

S THEOREM

PARTITIONS OF UNITY

QHANGE OF VARIABLE
**4. Integration on Chains**

ALGJEBRAIC PRELlMlNARms

FIELDS AND FORMS

GJWMETRIC PRFELIMINARIES

THE FUNDAMENTAL THEOREM OF CALCULUS

>5. Integration on Manifolds, 86

MANIFOLDS

FmLDS AND FORMS ON MANIFOLDS

STOKES THEOREM ON MANIFOLDS

THE VOLUME ELEMENT

THE CLASSICAL THEOREMS

Bibliography

Index

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