Mathematics

Articles

General Analysis. A guide to the frameworks of calculus, complex analysis and functional analysis consisting of standard theorems and proofs as well as some original ideas.

Differentiation in Banach spaces. An introduction to the Fréchet derivative and its most useful properties.

Line integrals in Banach spaces. A general formulation of the line integral, and an introduction to complex analysis from a more advanced point of view.

Misc.

Books

I’ve studied some mathematics on my own; on this page are books that I have read along with some comments. Please note that I cannot guarantee the mathematical validity/correctness/accuracy of the content below.

John M. Lee’s Introduction to Smooth Manifolds. Click here for my (very incomplete) solutions.
Topics: Smooth manifolds.
Prerequisites: Algebra, basic analysis in Rn, general topology, basic algebraic topology.

Great writing as usual, with plenty of examples and diagrams where appropriate. Chapters 6 (Sard’s Theorem) and 9 (Integral Curves and Flows) are a bit technical, and some of the exercises are quite hard. There are more than enough computational exercises to ensure that you are capable of applying the theory that you have learned.

John M. Lee’s Introduction to Topological Manifolds. Click here for my solutions.
Topics: General topology, algebraic topology.
Prerequisites: Metric spaces and basic group theory, but no general topology.

The exercises are excellent and vary in difficulty. The book introduces some basic category theory at the end of Chapter 7 (Homotopy and the Fundamental Group), and the exercises slowly add examples of categorical concepts all the way up to Chapter 13 (Homology). Places that need extra concentration: the section on paracompactness and partitions of unity (Chapter 4), CW complexes (Chapter 5).

James Munkres’s Analysis on Manifolds.
Topics: Analysis in Rn, vector calculus, smooth submanifolds of Rn.
Prerequisites: Basic real analysis, linear algebra.

The exposition in Chapters 2 (Differentiation), 3 (Integration) and 4 (Change of Variables) is great, but the proofs are too long/bloated for my tastes. As for the rest of the book – skip (or skim through) it and go straight to a smooth manifolds book after learning some general topology. Places that need extra concentration: Section 8 (The Inverse Function Theorem) – read Rudin’s proof instead, Section 19 (Proof of the Change of Variables Theorem), Section 32 (The Action of a Differentiable Map).

Serge Lang’s Complex Analysis.
Topic: Complex analysis.
Prerequisites: Basic real analysis.

I’m not sure why I picked this book. There’s nothing “GTM” about this GTM book, at least in Part One (Basic Theory). There are very few exercises beyond basic computations. However, the explanations are great and unlike with most other books, Lang starts off with power series. Places that need extra concentration: Section 2.3 (Relations Between Formal and Convergent Series).

Steven Roman’s Advanced Linear Algebra. Click here for my solutions.
Topic: Linear algebra.
Prerequisites: (Abstract) algebra, basic real analysis in some places.

This book contains some of the best writing I’ve ever seen in a mathematics book, plus great exercises. For best results, do not learn linear algebra before using this book (or unlearn it if you have). In the first part of the book, Roman proves the essential theorems about finitely-generated modules over PIDs in order to carefully structure your knowledge of eigenvectors, normal operators, and other topics that are usually badly handled elsewhere. The second part of the book explores some very interesting topics including tensor products, convexity, affine geometry, and the umbral calculus. Places that need extra concentration: Chapter 6 (Modules over a Principal Ideal Domain).

Walter Rudin’s Principles of Mathematical Analysis. Click here for my solutions.
Topic: Real analysis.
Prerequisites: None.

Do not use any other book for real analysis. You will be wasting your time.

Frederick M. Goodman’s Algebra: Abstract and Concrete. (Free!)
Topic: (Abstract) algebra.
Prerequisites: None.

A great selection of topics (groups, rings, modules, field extensions) and very readable proofs. However, there are too many typos to count (although that’s not necessarily a bad thing), and Chapters 9 (Field Extensions – Second Look) and 10 (Solvability) are somewhat hard to read.

Here are some other great books:

11 Comments

  1. I too enjoy studying mathematics, and I just wanted to say I found your articles and notes to be a great resource. Thanks!

    Reply

  2. In a former lifetime, I earned a Ph.D. in Mathematics, which to this day (as a software engineer) I hold in deepest regard and love. My father was a mathematician, and my son is just entering math grad school — perhaps something rubbed off, or he simply discovered the subject’s beauties and pleasures out of earlier forays into physics. Currently, I’m enjoying new walks in some of the territories he’s exploring, particularly John Lee’s fine books on manifolds. I think you provide a nicely tailored outline of mathematical self-study. Thanks for recommending Steven Roman’s writing in Advanced Linear Algebra, an interesting sounding book. I’ll check it out once I get out from under John Lee’s thrall. 🙂 Kudos and appreciation for providing your site (which I found in a search for Lee’s books) — indeed a great resource!

    Reply

  3. All this math u writing here where do u use in everyday life? Give some examples to give me more hints to start working math again 😀

    Reply

    1. Hi, all this maths is the basis of all applied maths in day-to-day situations in engineering computers etc. Up to you where you decide to draw your level of entry from leaving all the algorithm to the computer (lowest level of abstraction) to understanding theoretical basis (listed here) or somewhere in-between – where many engineering-oriented mathbooks will usually operate. I hope this helps.

      Reply

  4. What an excellent resource you have provided for those working through texts without any guidance from tutors and professors …

    Thank you!

    Peter

    Reply

  7. Quick question have you used any of the mathematics you’ve studied/learned in process hacker and if so what was it and how was it integrated in process hacker also was it something that would normally not be an applied math

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  9. I think Rudin has two pre-requisites: a very strong ability in algebraic manipulations, and intellectual fortitude. It’s pretty clear, but he leaves out a lot of steps and you really have to work through it very carefully.

    Reply

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