Calculus on Manifolds by Spivak

[micromass]

• Author: Michael Spivak
• Title: Calculus on Manifolds
• Amazon link: https://www.amazon.com/dp/0805390219/?tag=pfamazon01-20
• Prerequisities: Rigorous Calculus
• Level: Undergrad

Table of Contents:

[LIST]
[*] Foreword
[*] Preface
[*] Functions on Euclidean Space
[LIST]
[*] Norm and Inner Product
[*] Subsets of Euclidean Space
[*] Functions and Continuity
[/LIST]
[*] Differentiation
[LIST]
[*] Basic Definitions
[*] Basic Theorems
[*] Partial Derivatives
[*] Derivatives
[*] Inverse Functions
[*] Implicit Functions
[*] Notation
[/LIST]
[*] Integration
[LIST]
[*] Basic Definitions
[*] Measure Zero and Content Zero
[*] Integrable Functions
[*] Fubini's Theorem
[*] Partitions of Unity
[*] Change of Variable
[/LIST]
[*] Integration on Chains
[LIST]
[*] Algebraic Preliminaries
[*] Fields and Forms
[*] Geometric Preliminaries
[*] The Fundamental Theorem of Calculus
[/LIST]
[*] Integration on Manifolds
[LIST]
[*] Manifolds
[*] Fields and Forms on Manifolds
[*] Stokes' Theorem on Manifolds
[*] The Volume Element
[*] The Classical Theorems
[/LIST]
[*] Bibliography
[*] Index
[/LIST]

 

[mathwonk]

I learned calculus rigorously from Michael Spivak. First I graded a course from his beginning Calculus book, then a little later I taught a course from this book and read most of it carefully and worked as many problems as possible.

It is short and clear and covers only the most important concepts and theorems, differentiation, integration, partitions of unity, and then the inverse/implicit function theorem, change of variables and Fubini theorem for integration, and Stokes theorem. He also explains clearly the concept of differential forms, and integration of forms over chains for integration theory.

Knowing this stuff sets you apart from the herd who have studied only from ordinary calculus texts, and are always afterwards trying to master this material.

 

[Robert1986]

I agree with mathwonk. I didn't really understand much of what I did in multi-variable calculus (aside from surface-level stuff and knowing how to carry out change of variables, etc) until I read this book.

 

[mathwonk]

the only topic not treated in spivak is differential equations, e.g. the solution of first order ordinary d.e.'s. After reading Spivak I walked into the Univ of Washington 2 hour Phd prelim exam on advanced calc and walked out with almost a perfect score after only 30 minutes. The one question I did not nail was on diff eq. I suggest Lang's Analysis I for that.

 

[Petek]

I used this book twice. The first time was as an undergrad around 1967 or 1968. The course ran for two quarters (approx. 24 weeks) and was taught by someone (a probability theorist, IIRC) who was learning the material for the first time. I recall that we didn't even get to Stokes' Theorem (which is the primary result). The second time was several years ago. After a long hiatus from math, I started studying on my own. After working though several other texts, I decided to tackle Spivak again. This time I made it all the way through the book. I found it rough going because of typos, editing problems and a few outright errors. In my opinion, the exercises are the best part of the book. They provide examples (and counter examples) for the main text.

 

[raopeng]

I feel this pamphlet is a great supplement to any Mathematical Analysis courses, with focus on some introduction to differential geometry and a geometric insight, not a good textbook for self-studies, however, in my opinion due to its conciseness. The book at my hand is a 1960s edition, for a more recent work Mathematical Analysis by Zorich whose Volume II covers and extends beyond the materials in this book has referred to this book several times and successfully integrated seamlessly this marvellous "modern approach" to a first year undergraduate Analysis course.

 

[mathwonk]

I just counted the books in my math library and found over 350 volumes, ranging from Euclid's Elements to abstract texts on sheaf theory and homological algebra. Of all those books, this is probably the one I benefited most from, followed perhaps by Mike's Calculus.