Calculus on manifolds

What Hacker News says

1. Y_YDec 2024
   If you liked the "functional" style of calculus in SICM, or want a calculus only book in this vein I recommend Baby Spivak: https://en.wikipedia.org/wiki/Calculus_on_Manifolds_(book)

   (And obviously Functional Differential Geometry by the authors of SICM)

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2. okaleniukMay 2023
   Calculus on Manifolds by Michael Spivak. http://www.strangebeautiful.com/other-texts/spivak-calc-mani...

   For me, the eye opening thing was that any n-dimensional body is a transformation of the n-dimensional standard cube. This brings some mathematical consequences I don't understand, but what's important for me is that equations get much easier for the standard cube transformations because your corners are just ones and zeros.

   E. g. I wrote an app that "unbends" a book page from a photo so it looks flat. It can be done with a single but rather large transformation, or with a pair of transformations: to and from the standard cube, and each transformation is then much simpler than their composition. 15 years after I learned that trick and 10 years after I wrote an app, I wrote a book called Geometry for Programmers.

   If not for Spivak, I would have drowned in equations, never written the app and would have had nothing to write a book about.

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3. okaleniukJan 2023
   Calculus on Manifolds by Spivak. Brilliant. And relatively thin too.
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4. 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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5. jordibcJan 2023
   For sure I got significantly(?) better with classics like Spivak, Apostol, Rudin.

   "Real and Complex Analysis" by Rudin, and the two books both named "Calculus" from Spivak and Apostol. But also from Apostol his more concise and far-reaching "Mathematical Analysis". And from Spivak his small gem "Calculus On Manifolds" made quite a dent on me.

   Other than more "classic math" books, I also wanted to mention two outliers that I found eye-opening and generally awesome:

   * Street-Fighting Mathematics, by Mahajan (http://streetfightingmath.com/). Intuitive, useful and fun.

   * Geometric Algebra for Physicists, by Doran and Lasenby. I found the power and elegance of geometric algebra mesmerizing, and even if this book is also about physics and there may be more appropriate math-only books about geometric algebra, this is the one that made it for me.

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6. enriqutoJan 2021
   While Halmos' book is lovely, I still prefer the geometric definition of determinant to the algebraic one: The determinant of a matrix is the signed volume (or area) of the parallellepiped spanned by its columns. Equivalently, the determinant of a linear map is the volume of the image of a unit cube by that map (or any arbitrary shape of volume one, not necessarily a cube). All the algebraic properties of the determinant follow easily from the geometric definition (multi-linearity, anti-symmetry, etc).

   Really, I don't see what you like about Halmos definition of the determinant... I have just read it (page 99 of my copy) and he admits that it is a "somewhat roundabout procedure", just after giving the definition! There's other references that seem much cleaner (e.g. Spivak's calculus on manifolds, using exterior algebra).

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