Name: Vector calculus, linear algebra, and differential forms
ISBN: 9780971576650

Vector calculus, linear algebra, and differential forms

This text covers most of the standard topics in multivariate calculus and part of a standard first course in linear algebra. It focuses on underlying ideas, integrates theory and applications, offers a host of pedagogical aids, and features coverage of differential forms and an emphasis on numerical methods to prepare students for modern applications of mathematics. *Covers important material that is usually omitted. *Presents more difficult and longer proofs (e.g. Proofs of the Kantorovitch theorem, the implicit function theorem) in an appendix. *Makes a careful distinction between vectors and points. *Features an innovative approach to the implicit function theorem and inverse function theorem using Newton's method. *Always emphasizes the underlying meaning - what is really going on (generally, with a geometric interpretation) - eg. The chain rule is a composition of linear transformations; the point of the implicit function theorem is to guarantee that under certain circumstances, non-linear equations have solutions. *Integrates theory and applications. *Begins most chapters with a treatment of a linear problem and then shows how the 7 methods apply to corresponding non-linear p

What Hacker News says

debanjan16Mar 2023

For mathematics:

1. Spivak's Calculus

2. Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by Hubbard and Hubbard

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zrkrlcJan 2023

Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach

It's a rigorous but chatty textbook in the style of Spivak but written by someone who is sensitive to applied maths. I would not have survived my astrophysics classes without it.

(Not to mention it's where I first saw this really intuitive way of doing matrix multiplication: https://blogs.ams.org/mathgradblog/2015/10/19/matrix-multipl...)

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nextosJan 2023

These are good recommendations, but I think beginners tend to burn out due to the lack of a structured program and/or exercise solutions if they are trying to study on their own. The simplest structured program I can think of that satisfies both is:

* Basic Mathematics by Lang. Covers basic algebra and geometry at high school level.

Then one of these two, depending on your interests, or both:

* Vector Calculus, Linear Algebra and Differential Forms by Hubbard and Hubbard. Takes you through linear algebra, single-variable calculus and multiple variable calculus. Analysis is discussed in an appendix. All proofs have a constructive bias, so it's very algorithmic and natural for a CS-minded student. Solutions are in a separate volume.

* Program = Proof by Mimram. Discusses logic and computation, and takes you from the basics to depedent type theory and beyond. Uses OCaml and Agda. Freely available at: https://www.lix.polytechnique.fr/Labo/Samuel.Mimram/teaching...

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debanjan16Jun 2022

Vector Calculus, Linear Algebra, and Differential Forms by Barbara Burke Hubbard and John H. Hubbard is a good example I think.

https://www.amazon.com/Vector-Calculus-Linear-Algebra-Differ...

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jpfrMay 2022

The best book for me was "Vector Calculus, Linear Algebra, and Differential Forms" by Hubbard and Hubbard.

It is the only book (I know of) that brings you from absolute basics to an integrated development of the subjects from its title. And that integrated developments actually leads to a good didactic method.

The search function will show you many comments on HN recommending it.

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blaisehorvathFeb 2022

Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach - This is my all time favourite math book. It teaches Linear algebra in very intuitive manner. Along the pure math side it also gives examples of the introduced concepts from Physics, Machine learning etc... You can buy new one on [author's own publishing company](https://matrixeditions.com/5thUnifiedApproach.html).

I own Spivak's Calculus (which is a great book), but if you are not interested in the pure math side or you just don't want to grind a 700 page tome I recommend Calculus made easy. Contrary to the title and the small size of this book, it's an amazing and intuitive. When I read Calculus Made Easy I also read Infinite powers by Steven Strogratz they complete each other very well.

I have a copy of "The Art of Probability: For Scientists and Engineers" by Hamming it's a really good book.

If you are interested in Bayesian probability, machine learning or information theory I recommend "Information Theory, Inference, and Learning Algorithms" by MacKay it's an amazing book. It's available as in PDF format on the author's website.

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