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Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus
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.
162 pages, Paperback
First published January 1, 1965
40 people are currently reading
1067 people want to read
About the author
Michael Spivak
46 books111 followersMichael David Spivak is a mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Perish Press. He is the author of the five-volume Comprehensive Introduction to Differential Geometry. He received a Ph.D. from Princeton University under the supervision of John Milnor in 1964.
His book Calculus takes a very rigorous and theoretical approach to introductory calculus. It is used in calculus courses, particularly those with a pure mathematics emphasis, at many universities.
Spivak's book Calculus on Manifolds (often referred to as little Spivak) is also rather infamous as being one of the most difficult undergraduate mathematics textbooks.
His book Calculus takes a very rigorous and theoretical approach to introductory calculus. It is used in calculus courses, particularly those with a pure mathematics emphasis, at many universities.
Spivak's book Calculus on Manifolds (often referred to as little Spivak) is also rather infamous as being one of the most difficult undergraduate mathematics textbooks.
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Displaying 1 - 15 of 15 reviews
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September 1, 2010What yo mamma never told you you could do with calculus!
October 23, 2019
Ehhhhh... reading it now it's good. It's a succinct exposition of everything leading up to Stokes-Cartan, basically.
But I already know most of the material. When I originally tried reading it without already knowing differential geometry it was totally opaque.
So: good for review, bad for self-study.
But I already know most of the material. When I originally tried reading it without already knowing differential geometry it was totally opaque.
So: good for review, bad for self-study.
Read
April 27, 2017Mjög falleg bók. Lestur minn var engan veginn fullnægjandi enda er hún oft sögð erfiðasta kennslubókin í stærðfræði ætluð grunnnemum en ég naut hennar þó. Kem vonandi aftur að henni eftir ár eða tvö.
(las hana ekki á spænsku, útgáfan sem ég las er ekki á goodreads og kápan á nýju útgáfunni þykir mér svo ljót)
(las hana ekki á spænsku, útgáfan sem ég las er ekki á goodreads og kápan á nýju útgáfunni þykir mér svo ljót)
Want to read
February 5, 2011recommended by one of the Columbia theory applicants as a "must-read" in physics/math
August 25, 2020
მესამე გვერდს ვერ გავცდი. ყველაზე მეტად მესამე გვერდი მომეწონა. გირჩევთ ყველას - საუკეთესო ტრაგედიას რაც კი შექმნილა.
December 6, 2017
A timeless classic.
December 31, 2018
a good 5th / 10th review of calculus, for people who want that kind of thing.
I like his explanation of where √π comes from in the normal distribution, but in the same breath Spivak lets us know that he's an asshole.
It's mercifully short. Major points for that in a field where nearly everything you could buy explaining the material is a doorstop or wordy/obtuse in some other way.
I like his explanation of where √π comes from in the normal distribution, but in the same breath Spivak lets us know that he's an asshole.
It's mercifully short. Major points for that in a field where nearly everything you could buy explaining the material is a doorstop or wordy/obtuse in some other way.
August 8, 2024
I only read this book as 'bedtime reading' long after my formal courses in analysis in order to gain some familiarity with forms and analysis on manifolds. I skimmed through the first three chapters just to get used to Spivak's notation since I already knew almost all of the material. Those three chapters contain the standard results from differential and integral calculus. I really like the fact that Spivak aims to prepare undergraduate students for more advanced courses such as general topology, functional analysis and measure theory. I thoroughly enjoyed the third chapter on integration where Spivak introduces the very basics of measures and contents and his examples are one step away from elucidating the main differences between the Riemann and the Lebesgue integral. Additionally, he also introduces the theorem on a smooth partition of unity, which will prove vital in further courses in functional analysis and PDEs. I had to focus more in the remaining two chapters on tensors, forms and integration on manifolds. I found the exposition quite clear and succinct. Some concepts were probably explained too briefly and without a prior course in differential topology I would have been left wondering why certain definitions actually make sense. That being said, I find this text on integration on manifolds much clearer than other (more advanced) texts on the subject (the notation is also slightly different since most of the books on the subject employ Einstein summation convention while this book does not). However, this is only an introductory text, so only the necessary minimum on tensors and manifolds (only subsets of Euclidean spaces are considered so one does not need to be familiar with topological spaces) is covered here. Still, the book succeeds in stating and proving the generalized Stokes theorem as well as demonstrating why it is the fundamental theorem of calculus (the book concludes by showing how the theorem relates to the trinity of theorems from classical vector calculus, namely Green's, Stokes' and the divergence theorem). I am not sure if I would have been able to grasp the concepts from the last two chapters as a first-year undergrad student self-studying from this book though. It would probably have taken a lot of effort. Therefore, I recommend this book either to people who are already familiar with more abstract math or to undergrad students with a knack for abstract math.
July 1, 2019
Fairly good exposition, shows its age in parts but strong on problem sets. Among the better treatments of the topic for more able students.
January 21, 2023
Very concise, but very on point as well.
June 15, 2007
This is one the best instructional books for analysis. It was probably the first real math book I ever read and probably what first made me appreciate the difference between calculations and pure mathematics and the power and beauty of the later.
The book is extremely well structured and works towards a definite objective: to derive Stoke's theorem on Euclidean spaces and manifolds. It starts from the very basics - linear algebra and topology - and works up to the goal by deriving multi-variate calculus, Grassman algebras, and integration on chains. As a bonus, the last chapter introduces manifolds and shows how the result can be generalized to curved spaces.
Although it may sound complicated, this is actually an introductory text. The only prerequisites are linear algebra and multi-variate calculus. Some knowledge of topology also helps, although Spivak introduces all the topological concepts necessary for his subject.
I highly recommend this book to anyone interested in mathematics.
The book is extremely well structured and works towards a definite objective: to derive Stoke's theorem on Euclidean spaces and manifolds. It starts from the very basics - linear algebra and topology - and works up to the goal by deriving multi-variate calculus, Grassman algebras, and integration on chains. As a bonus, the last chapter introduces manifolds and shows how the result can be generalized to curved spaces.
Although it may sound complicated, this is actually an introductory text. The only prerequisites are linear algebra and multi-variate calculus. Some knowledge of topology also helps, although Spivak introduces all the topological concepts necessary for his subject.
I highly recommend this book to anyone interested in mathematics.
April 7, 2016
Good introduction to the topic.
Excellent chapters on basic R^n topology and differentiable calculus, including inverse function and implicit function theorems. The notation is non-classical (but standard) and exceedingly clear. Sadly, the proofs are fairly unmotivated, and one has to work hard to do more than just check their validity. Inverse function, for example, is proved in a way that does not generalize to infinite-dimensional spaces, and one is at a loss to find an alternate proof.
The chapter on differential forms and integration is technically flawless, but also lacks motivation and intuition. The final chapter, on generalizations to manifolds, suffers a similar drawback, but slightly less so.
The exercises range from trivial to interesting, but all are very "doable" --- and important! E.g. C^infinity partitions of unity are constructed in the exercises.
Excellent chapters on basic R^n topology and differentiable calculus, including inverse function and implicit function theorems. The notation is non-classical (but standard) and exceedingly clear. Sadly, the proofs are fairly unmotivated, and one has to work hard to do more than just check their validity. Inverse function, for example, is proved in a way that does not generalize to infinite-dimensional spaces, and one is at a loss to find an alternate proof.
The chapter on differential forms and integration is technically flawless, but also lacks motivation and intuition. The final chapter, on generalizations to manifolds, suffers a similar drawback, but slightly less so.
The exercises range from trivial to interesting, but all are very "doable" --- and important! E.g. C^infinity partitions of unity are constructed in the exercises.
April 5, 2010
I just read this again because another book was obfuscating exterior calculus for me, where it had formerly been clear. It's Spivak's usual clear (and very brief style) which I love. Stokes' Theorem in half a page!
August 9, 2016
My bible <3
October 31, 2014
Quite good. Terse.
Displaying 1 - 15 of 15 reviews