How to learn Mathematics from the ground up

A guide to the intrepid adventurer

Description

Imagine the following scenario, you are a middle age engineer that studied applied mathematics in the context of a certain field of engineering and would like to remember everything again, or better to learn everything again from the ground up! With your knowledge of having done the path once, and experience to know what is a good book, what would be the best path to revisit everything again, or to structure the best a path to a friends children or a family member life journey? This will be a journey from the most basics mathematics, all the way to pure mathematics, a real adventure along 50 books :-D

Important and free resource

One of the most valuable resources in math from kinder garden to college math.

• Khan Academy
  https://www.khanacademy.org/

Motivational books on Mathematics

1. Video - The Map of Mathematics
   https://www.youtube.com/watch?v=OmJ-4B-mS-Y

2. The Math Book - Big Ideas Simply Explained
   by DK

3. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth
   by Paul Hoffman

4. Logicomix: An epic search for truth
   by Apostolos Doxiadis, Christos Papadimitriou

First years in math

This is a series that focus on giving you a problem solving mentality, with this book series the intent is to teach you how you to solve problems with mathematics as a tool. The solution books have worked out problems, so you can rely on them for self study.

1. The Art of Problem Solving: Prealgebra
   by Richard Rusczyk, David Patrick, Ravi Boppana
   Text: 608 pages. Solutions: 224 pages.

2. The Art of Problem Solving: Introduction to Algebra, 2nd Ed
   by Richard Rusczyk
   Text: 656 pages. Solutions: 312 pages.

3. The Art of Problem Solving: Introduction to Counting & Probability, 2nd Ed
   by David Patrick
   Text: 256 pages. Solutions: 120 pages.

4. The Art of Problem Solving: Introduction to Geometry, 2nd Ed
   by Richard Rusczyk
   Text: 557 pages. Solutions: 226 pages.

5. The Art of Problem Solving: Introduction to Number Theory
   by Mathew Crawford
   Text: 336 pages. Solutions: 144 pages.

Intermediate years in math

1. The Art of Problem Solving: Intermediate Algebra
   by Richard Rusczyk and Mathew Crawford
   Text: 720 pages. Solutions: 336 pages.

2. The Art of Problem Solving: Intermediate Counting & Probability
   by David Patrick
   Text: 400 pages. Solutions: 208 pages.

Pre-Calculus

At this point you have three good options see what adjusts better to you.

1. The Art of Problem Solving: Precalculus, 2nd Ed
   by Richard Rusczyk
   Text: 528 pages. Solutions: 272 pages.

2. Precalculus: Mathematics for Calculus 7th ed
   by James Stewart, Lothar Redlin, Saleem Watson

3. Precalculus
   by Jay Abramson
   https://openstax.org/details/books/precalculus

University or college mathematics

Linear Algebra

You will learn Linear Algebra with examples in code (Python and Matlab) without calculus.

1. Linear Algebra: Theory, Intuition, Code
   by Mike X Cohen

Calculus and friends

This is a book to motivate you to go further in the most beautiful way!

1. Calculus Made Easy
   by Silvanus P. Thompson, Martin Gardner

Then to my knowledge there are 3 similar good paths that you can follow, but with increasing depth in mathematics and Calculus.

First path

2. Engineering Mathematics, 5th Ed
   by Prof Anthony Croft, Dr Robert Davison, et al.

Second path

3. Modern Engineering Mathematics, 6th Ed
   by Glyn James, Phil Dyke

4. Advanced Modern Engineering Mathematics, 5th Ed
   by Glyn James, David Burley, Dick Clements, et al.

Third path

5. Mathematical Methods for Physics and Engineering: A Comprehensive Guide 3rd Ed
   by K. F. Riley

6. Student Solution Manual 1st Ed for Mathematical Methods for Physics and Engineering 3th Ed
   by K. F. Riley

Fourier Series and Transforms

1. Fourier Analysis: An Introduction
   by Elias M. Stein and Rami Shakarchi

2. Fast Fourier Transform and Its Applications 2th Ed
   by E. Brigham

Probability and Statistics

Then you need to learn about Probability and Statistics the following are two nice books with a companion book with the solutions.

1. Probability: For the Enthusiastic Beginner
   by David J. Morin

2. Introduction to Probability, Statistics, and Random Processes
   by Hossein Pishro-Nik
   http://www.probabilitycourse.com/preface.php

3. Student's Solutions Guide for Introduction to Probability, Statistics, and Random Processes
   by Hossein Pishro-Nik
   http://www.probabilitycourse.com/preface.php

4. All of Statistics: A Concise Course in Statistical Inference
   by Larry Wasserman

Optimization

Then you will need to learn about Optimization, two good books, the first with code in Julia.

1. Algorithms for Optimization
   by Mykel J. Kochenderfer, Tim A. Wheeler
   Note: See the book PDF site link on the authors page.
   https://mykel.kochenderfer.com/textbooks/

2. Convex Optimization
   by Boyd, Vandenberghe
   https://web.stanford.edu/~boyd/cvxbook/

Discrete Mathematics

1. Discrete Mathematics with Applications 5th Ed
   by Susanna S. Epp

Numerical Analysis and Computational Mathematics

1. Numerical Methods for Engineers 8th Ed
   by Steven Chapra, Raymond Canale

2. Numerical Recipes 3rd Edition: The Art of Scientific Computing
   by William H. Press

3. Numerical Methods in Physics with Python
   by Alex Gezerlis

4. Computational Physics: Problem Solving with Python 3rd Ed
   by Rubin H. Landau, Manuel J Páez, Cristian C. Bordeianu

5. Applied Computational Physics
   by Joseph F. Boudreau, Eric S. Swanson

6. Hans Petter Langtangen - Various writings
   http://hplgit.github.io/

7. Hans Petter Langtangen - Last versions
   https://library.oapen.org/discover?rpp=10&etal=0&query=Langtangen%2C+Hans+Petter&scope=&group_by=none&page=1

Information Theory

1. A Mind at Play: How Claude Shannon Invented the Information Age
   by Jimmy Soni, Rob Goodman
   

2. Information Theory, Inference and Learning Algorithms
   by David J. C. MacKay
   Note: In the author site you have de book and the video lectures.
   http://www.inference.org.uk/mackay/itila/

Error Correction Codes

1. Error-Correction Coding and Decoding: Bounds, Codes, Decoders, Analysis and Applications
   by Martin Tomlinson, Cen Jung Tjhai, Marcel A. Ambroze, Mohammed Ahmed, Mubarak Jibril
   Note: Book on open access.
   https://www.springer.com/gp/book/9783319511023

2. Error Correction Coding: Mathematical Methods and Algorithms 2nd Ed
   by Todd K. Moon

Problems with answers

1. Schaum's 3,000 Solved Problems in Calculus
   by Elliott Mendelson

2. Schaum's Outline of Calculus, 6th Ed
   by Frank Ayres, Elliott Mendelson

3. Schaum's Outline of Advanced Calculus, 3rd Ed
   by Robert Wrede, Murray Spiegel

4. Schaum's Outline of Advanced Mathematics for Engineers and Scientists
   by Murray Spiegel

5. Schaum's Outline of Probability and Statistics, 4th Ed
   by John Schiller, R. Alu Srinivasan, Murray Spiegel

6. Schaum's Outline of Discrete Mathematics, 3rd Ed
   by Seymour Lipschutz, Marc Lipson

7. Schaum's Outline of Complex Variables, 2th Ed
   by Murray Spiegel, Seymour Lipschutz, John Schiller, Dennis Spellman

8. Schaum's Outline of Differential Equations, 4th Ed
   by Richard Bronson, Gabriel B. Costa

9. Schaum's Outline of Partial Differential Equations
   by Paul DuChateau, D. Zachmann

10. Vector Analysis, 2nd Ed
    by Murray Spiegel, Seymour Lipschutz, Dennis Spellman

11. Schaums Outline of Tensor Calculus
    by David Kay

Very good synthesis of all around Mathematics

1. The Princeton Companion to Mathematics
   by Timothy Gowers, June Barrow-Green, Imre Leader

2. The Princeton Companion to Applied Mathematics
   by Nicholas J. Higham, Mark R. Dennis, Paul Glendinning, Paul A. Martin, Fadil Santosa, Jared Tanner

To go further into Pure Mathematics

This book section specific recommendation come from the wonderful video about learning pure mathematics, see the video and the video description for more details.

• Aleph 0 - How to learn pure mathematics on your own: a complete self-study guide
  https://www.youtube.com/watch?v=fo-alw2q-BU

Real Analysis

1. Calculus, 4th Ed
   by Michael Spivak, Michael Spivak

2. Combined Answer Book For Calculus Third and Fourth Editions, 1th Ed
   by Michael Spivak

3. Understanding Analysis
   by Stephen Abbott.

Linear Algebra

2. Linear Algebra Done Right
   by Sheldon Axler

And for the problems.

3. Linear Algebra
   by Insel, Freidberg, and Spence

Topology

1. Topology through Inquiry
   by Su and Starbird

Differential Equations

1. Differential Equations with Boundary Value Problems
   by Zill and Cullen

Complex Analysis

1. A Friendly Approach to Complex Analysis
   by Sara Maad and Amol Sasane

2. Visual Complex Analysis
   by Tristan Needham

Abstract Algebra

1. Contemporary Abstract Algebra
   by Gallian

Differential Geometry

1. A Geometric Approach to Differential Forms
   by David Bachman

2. Introduction to Manifolds
   by Loring Tu

The great men and women behind mathematics

1. Men of Mathematics
   by E.T. Bell

All my other guides

• The links to all my guides are in:
  Guides on Linux - Programming - Embedded - Electronics - Aeronautics
  https://github.com/joaocarvalhoopen/Guides_Linux-Programming-Electronics-Aeronautics

Have fun!

Best regards,
João Nuno Carvalho