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# Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus

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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
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Paperback, 160 pages

Published
January 22nd 1971
by Westview Press
(first published 1965)

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(las hana ekki á spænsku, útgáfan sem ég las er ekki á goodreads og kápan á nýju útgáfunni þykir mér svo ljót)

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.

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. ...more

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 ...more

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 ...more

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Michael 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 ...more

His book Calculus takes a very rigorous and theoretical approach to ...more

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