## 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.

- Differentiation done correctly: 1. The derivative
- Differentiation done correctly: 2. Higher derivatives
- Differentiation done correctly: 3. Partial derivatives
- Differentiation done correctly: 4. Inverse and implicit functions
- Differentiation done correctly: 5. Maxima and minima
- Convex functions, second derivatives and Hessian matrices
- Fréchet derivative of the (matrix) exponential function
- First-order ODEs, matrix exponentials, and det(exp)

**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.

- Line integrals: 1. Exact, conservative and closed forms
- Line integrals: 2. Locally exact forms and singular homology
- Line integrals: 3. Applications to complex analysis

**Misc.**

- Horn’s inequality for singular values via exterior algebra
- Every continuous open mapping of R into R is monotonic
- Formula for the circumference of an ellipse
- Power series of tan, cot and csc

## 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 * R^{n}*, general topology, basic algebraic topology.

**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.

**James Munkres’s Analysis on Manifolds.**

Topics: Analysis in * R^{n}*, vector calculus, smooth submanifolds of

*.*

**R**^{n}Prerequisites: Basic real analysis, linear algebra.

**Serge Lang’s Complex Analysis.**

Topic: Complex analysis.

Prerequisites: Basic real analysis.

**Steven Roman’s Advanced Linear Algebra.** Click here for my solutions.

Topic: Linear algebra.

Prerequisites: (Abstract) algebra, basic real analysis in some places.

**Walter Rudin’s Principles of Mathematical Analysis.** Click here for my solutions.

Topic: Real analysis.

Prerequisites: None.

**Frederick M. Goodman’s Algebra: Abstract and Concrete.** (Free!)

Topic: (Abstract) algebra.

Prerequisites: None.

Here are some other great books:

**Concrete Mathematics.**Essential for developing summation and binomial coefficient manipulation skills.**Kenneth P. Bogart’s Combinatorics Through Guided Discovery.**The entire book consists of very entertaining exercises.**Combinatorial Species and Tree-like Structures.**A framework for counting.

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

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!

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 😀

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.

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

Thank you!

Peter

Muy buena la pagina me ayudo mucho en mi seminario I

me podrian facilitar el solucionario de las preguntas del libro de Jhon lee Introduction manifolds el capitlo 3 (vectpres tangentes ) …..rojoanti@gmail.com

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

good yes

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.