Name: Comprehensive Introduction to Differential Geometry
ISBN: 9780914098850

What Hacker News says

enriqutoJan 2023

The classical stuff is great:

* Geometry and the imagination by Hilbert and Cohn-Vossen

* Methods of mathematical physics by Courant and Hilbert

* A comprehensive introduction to differential geometry by Spivak (and its little brothers Calculus and Calculus on manifolds)

* Fourier Analysis by Körner

* Arnold's books on ODE, PDE and mathematical physics are breathtakingly beautiful.

* The shape of space by Weeks

* Solid Shape by Koenderink

* Analyse fonctionnelle by Brézis

* Tristan Needhams "visual" books about complex analysis and differential forms

* Information theory, inference, and learning algorithms by MacKay (great book about probability, plus you can download the .tex source and read the funny comments of the author)

And finally, a very old website which is full of mathematical jewels with an incredibly fresh and clear treatment: https://mathpages.com/ ...I'm in love with the tone of these articles, serious and playful at the same time.

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jasomillNov 2021

While certainly not the usual approach, calculus in terms of infinitesimals has been placed of formal foundations[1]. To be fair to Courant, however, the approach to infinitesimals in nonstandard analysis is pretty far removed from the views of Leibniz, et al., and "mystical associations" is a fair, if not necessarily impartial, criticism of the latter.

IIRC, even Newton was at least a bit skeptical of the validity of his method of fluxions, even going so far as to formulate his Principia in terms of (nearly impenetrable IMO) arguments from classical geometry instead, and the non-rigorous application of related ideas led even the best mathematicians to questionable conclusions at times (e.g., Euler's argument that 1 + 2 + 4 + ⋯ = -1, which may or may not have influenced computer science in terms of two's complement arithmetic).

As for abstraction as a source of seemingly artificial difficulty in modern mathematics, from the introduction to Courant's Differential and Integral Calculus, the first edition of Introduction to Calculus and Analysis[2],

The presentation of analysis as a closed system of truths without reference to their origin and purpose has, it is true, an aesthetic charm and satisfies a deep philosophical need. But the attitude of those who consider analysis solely as an abstractly logical, introverted science is not only highly unsuitable for beginners but endangers the future of the subject; for to pursue mathematical analysis while at the same time turning one's back on its applications and on intuition is to condemn it to hopeless atrophy.

For similar arguments that over-reliance on formalism begins far earlier in the modern mathematical curriculum, see Feynman's New Textbooks for the "New" Mathematics [3], in which he discusses his frustrations as a working (theoretical!) physicist reviewing grade school textbooks.

As a partial counterpoint, I personally often have an easier time understanding the more abstract treatments. For example, I struggled with the traditional presentation of multivariable calculus in terms of "physically meaningful" differential operators until working through the first couple books of Spivak's Comprehensive Introduction to Differential Geometry after-hours, at which point everything sort of clicked. Had my intro (college) calculus course not coincidentally been taught by a differential geometer with a habit of presenting some of the basic ideas as asides, I may have never made it any further in the field (in terms of learning; I work in software, not maths).

Nevertheless, I'd never propose Bourbaki as a good model for elementary education!

Revised (Introduction to…) editions of Courant's calc textbooks are particularly noteworthy in this respect; to me, at least, they strike a very good balance between classical and modern formalisms, and between applications to pure and applied mathematics.

As for the application of "general abstract nonsense" to computer science and engineering, I have no doubt Courant would be pleased, with the applications if not in their presentation, as the interplay between pure mathematics and its applications was always an important subject to him (see also the introduction to his [and, nominally, Hilbert's] Methods of Mathematical Physics [4], the address he gave on variational methods in PDEs [5], often cited as a foundational work in finite element analysis, and, well, pretty much the entirety of What is Mathematics?[6]).

[1] https://en.wikipedia.org/wiki/Nonstandard_analysis

[2] https://onlinelibrary.wiley.com/doi/pdf/10.1002/978111803324...

[3] http://calteches.library.caltech.edu/2362/1/feynman.pdf

[4] https://onlinelibrary.wiley.com/doi/pdf/10.1002/978352761721...

[5] https://www.ams.org/journals/bull/1943-49-01/S0002-9904-1943...

[6] https://archive.org/details/WhatIsMathematics

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arvidFeb 2008

You really need to learn Math by doing it there really is no other way. Books are really just references and guides and can give you good problems to work from. Either find a friend who will study with you or get direction from a math professor.

I do not know Spivak's Calculus but his advanced books (by Publish or Perish) are excellent, So I assume his calculus book is also. Especially Calculus on Manifolds and A Comprehensive Introduction to Differential Geometry. Anyone who wants to understand calculus on higher dimensions should read Calculus on Manifolds.

If you want to learn from the masters and you have the confidence, audacity and intelligence. I would suggest Fundamentals of Abstract Analysis by Andrew Gleason and Geometry and the Imagination by David Hilbert.

Just a warning. These books are for people adept at mathematics and are willing to spend hours on a page or two. If you are not, then avoid these books.

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