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Math at its core is about establishing truths separate from sensual qualities, seeking patterns based upon these truths, systematically removing contradictions/inconsistencies from the patterns, and formulating conjectures with all of the above in mind. It is the one true language apart from reality which ironically makes it so useful.

The application of math in any field is considered a specialization which requires a significant amount of skill to which contemporary cultures try to measure. While math is commonly used in scientific fields, it is not a subset of them. In fact, the application of math to scientific fields is generally considered THE advancement of science. Unfortunately, due to this schism between what math is predicated on (formal logic) and what common language is predicated on (reality), the study of math for any scientific field is considered distinct and being less emphasized in education.

Here we wish to provide resources which will help you to develop mathematical skills for where ever you choose to apply them.


Mathematics First Year Undergraduate Education (by an Anon) (Reiterating that this is for math majors, and is actually more like a full undergraduate education than one year)

    Books in the maths first year undergraduate guide:
  • James Stewart's Single Variable Calculus [d]
    • Spivak's Calculus (3rd edition is best) is preferred for learning mathematics rigorously (that is, with proofs) [d]
  • Intro to proofs book- Daniel J. Velleman's How to Prove It: A Structured Approach and Joseph J. Rotman's Journey Into Mathemathics: An Introduction to Proofs are good, but most proofs books will do
  • ???
  • Walter Rudin, Principles of Mathematical Analysis [d] (Baby Rudin)
  • Walter Rudin, Real and Complex Analysis [a] [d] [s] (Papa Rudin is not even close to core undergrad material, and isn't even the best book for its subject)
  • Apostol, Calculus
    • Volume 1 [d]
      • Vol. 1 Solutions [d]
    • Volume 2 [d]
      • Vol. 2 Solutions [?]
Guide (more like notes) for everyone else:
  • Do in order: single variable calculus > linear algebra and multivariable calculus > differential equations
  • Abstract algebra and complex analysis (and analysis in general) you don't need unless you know what you are doing.
  • Mathematical proofs is a middle ground. Maybe do it for a mental exercise.
  • Axiomatic Euclidian geometry is extremely useful in lower-division physics/engineering classes.


Lists or Collections

The MAA Basic Library List HIGHLY recommended "The MAA's Basic Library List is a list of books recommended by the Mathematical Association of America for purchase by college and university libraries." The full list isn't available without membership or subscription, just search if you can't get or don't want to do either.

Advanced Mathematics Book List HIGHLY recommended Amazon book list with reviews. Assumes high-school level knowledge of math.

How to Become a Pure Mathematician (or Statistician) HIGHLY recommended a List of Undergraduate and Basic Graduate Textbooks and Lecture Notes - the blog

Chicago Undergraduate Mathematics Bibliography

Recommended Books in the Mathematical Sciences Worth looking into notes on books within areas of mathematics and advice on general study, edited by maths professor James M. Cargal.

Mathematics (DOE-HDBK-1014-92) Vol I : Vol II [Department of Energy Fundamentals Handbooks]

PhysicsForums - A Cumulative List of Math Textbooks A listing of the best math textbooks grouped by categories.

MathOverflow - Undergraduate Level Math Books a good thread

Undergraduate Math Textbooks for UC Berkeley (torrent) A torrent with all textbooks used in undergraduate mathematics courses at the University of California (Berkeley) from 2009 to 2011.


Edwin E. Moise, Elementary Geometry from an Advanced Standpoint [a]


Analytic Combinatorics A suggestion for further reading in an "Intro to Combinatorics" course


Precalculus: Mathematics for Calculus; James Stewart, RedlinWatson
    5th Edition

Calculus: Early Transcendentals; James Stewart
    4th Edition; Partial Solutions
    5th Edition; Full Solutions
    6th Edition; Partial Solutions
    7th Edition; Partial Solutions

Calculus: Early Transcendentals; Briggs/Cochran
    International Edition

Calculus Made Easy[f]; Silvanus P. Thompson (1914) [Dated, but still very useful for gaining an intuitive grasp on certain concepts]

Calculus: Graphical Numerical Algebraic; Ross L. Finney, Franklin Demana, Bert K. Waits, Daniel Kennedy [Some book I still have from high school]

Schaum's Outline of Mathematical Handbook of Formulas and Tables [Formulas and table for Analysis, Calculus, Differential Equations, Probability and Statistics]

Schaum's Outline of Advanced Calculus (3rd Edition) [a] [Summary Multivariable Calculus with a review of the final portions of Single Variable. Possible typos?]


Principles of Mathematical Analysis; Walter Rudin [Real Analysis; colloquially called "Baby Rudin"]

Real and Complex Analysis; Walter Rudin ["Papa Rudin"]


Elementary Mathematics
    I - Single variable, limits and differentiation
    II - Single variable, integration techniques
    III - Multivariable
Linear Algebra
Abstract Algebra
Analytic Geometry
Differential Geometry

Comparative charts:


Free Math Books Math books

The Assayer Math Book Compilation

Amazon Mathematics Guide Undergraduate mathematics textbook recommendations Advanced Math Books, Lectures, Videos Compilation

MathOnline Undergraduate to Research Level Books (Searchable)  [Down]

17thCenturyMaths Bunch of famous historical translated maths books mainly from the 17th & 18th century

Mathematics books (9GB) torrent At your own risk

Web Text, Sources, References, and Articles

Betterexplained Some core math concepts explained intuitively

MathReference Primers on about every mathematical topic

Wikibooks/Mental Maths Mental Math

Purplemath [e] Basic-Intermediate Algebra

Maths for Morons Like Us[e] Basic - Intermediate Algebra

Virtual Math Lab [e] Basic-Intermediate Algebra

Dave's Short Trig Course Basic-Intermediate Trigonometry

Trig without Tears Basic-Intermediate Trigonometry (With emphasis on understanding)

Themathpage [e] High School Math

S.O.S. Math [e] Alegebra to Calculus

Harvey Mudd Math tutorials Precalculus through Linear Algebra

Paul's Online Math Notes Math Notes

AoPS/Alcumus [e] Learning system aimed at Gifted High Schoolers

Introduction to Logic [e] Beginner’s Logic

Introduction to Group Theory Also goes into Rubix Cubes

MathPages Some advanced topics (Ancient website layout)

Analysis Webnotes Undergrad Math (Analysis)

MathOnline [b] Undergraduate to Research Level Lecture Notes Analysis

Book of Proof [b] Intro to how to mathematically prove

Blogic Online Logic Textbook

Statsoft [b] Statistics Online Book

The Little Handbook of Statistical Practice

SOCR Online Statistics Book

Numerical Analysis University of South Florida

Multivariable Calculus and Vector Analysis

Common math errors

Approximation theory

Wolfram Math World Mathematics Encyclopedic 

Cambridge Tripos example sheets 

Basic Calculus  ( Mathematics IA) - short ultimate basic calculus script

Linear equations (Mathematics IA)

Nonlinear equations  (M IA)

Higher order equations (M IA)

Differential equations summary

Vector calculus

Numerical analysis

WebMath Helpful links for hands-on math

Math Insight Includes narrative and interactive applets to teach topics in mathematics. Some topics include Multivariable Calculus and Vector Algebra


Khan Academy Arithmetic to Advanced (More emphasis on understanding)

PatrickJMT Arithmetic to Advanced (Less emphasis on understanding)

Brightstorm Algebra through Calculus  [Paywalled]

Introduction to Higher Mathematics 19 video youtube playlist introducing higher maths

Youtube/derekowens Algebra to Calculus

Youtube/IntuitiveMath Algebra to Calculus

Youtube/njwildberger History, framework of mathematics & other topics

Midnight Tutors  Calculus

The New Boston Math

Youtube/ MathTV Algebra through Calculus help [b] Advanced Math Books, Lectures, Videos Compilation

A-Level Maths Revision (UK) videos detailing exam solutions (also past papers from Edexcel)

Elementary Statistics

Math In Real Life Organized collection of videos and webtext for high school mathematics

Meaure-Based Probability Series

Game Theory 101 Series describing introductory game theory

An Introduction to Number Theory By Edward Burger, 24 lectures

Calculus Lecture video series Does what it says on the tin!

Numerical Analysis Includes Euler's Method, Trapezoidal Rule, Numerical Differentiation of Continuous Functions, and more.

Study Stove: Calculus Shows step by step how to do problems

It's So Blatant This YouTube channel has lectures on Abstract Algebra, Lie Algebra, Set Theory and Probability Theory from Harvard University, as well as other lectures from other institutions on Chaos Theory, Dimension Theory and Algebraic Quantum Field Theory.


BBC FOUR - Marcus du Sautoy - The Story of Maths

Tools & Apps

Wolfram Alpha All math homework threads should be looked up here before posting

SOCR Shitload of Statistics apps, games & tools

Triangle Solver

Falstad Math Simulations/Applets

Multivariable calculus and vector analysis Readings on calc + Java simulations

PhET Math Java apps hosted by University of Colorado

Fractal Explorer With fractal explorer you can plot fractals and even make zoom videos from the mandelbrot set.


Getting to grips with LaTeX

Getting Started with LaTeX

MathURL Live LaTeX equation editing (can create links to each equation)

Detextify Draw characters in freehand to find its LaTeX character codes Examples

TeX Users Group

MathIM Chat in LaTeX

AoPS/LaTeX LaTeX Reference

MathBin Pastebin with LaTeX

LaTeX video tutorials for complete beginners, quite nicely presented.

Minimal TikZ A minimal (and very understandable) introduction to plotting/graphs in LaTeX with the TikZ package.

Kile Open source LaTeX editor

Problems & Challenges

Khan Academy/Exercises Basic - Intermediate Problem Generator (Click “Practice”)

nrich/Problem Generator Intermediate Math Problem Generator

Math Fun Facts Math paradoxes, theorems, puzzles, etc. explained such that a layman could understand

Senior Mathematical Challenge Papers [b] ‘88 - ‘09 Questions (Aimed at 16-18)

Cut the Knot Math Puzzles & Problems

Mathschallenge Math Puzzles & Problems

nrich/Maths Math Puzzles & Problems

Wikipedia/ Unsolved Problems in Mathematics Good luck, son

British Mathematical Olympiad Download the papers separately or most of them in this pack (Pack contains all BMO1 & BMO2 up to 2011; no IMO papers)

Advice / Articles

Lockhart's Lament Essay on the state of Math education

AoPS Math Advice

Mathematical Articles by Stan Brown

Terry Tao's Blog / Advice Writing in Mathematics

Tim Gowers/ Advice About using examples in writing

Wikipedia/ Mathematical Maturity

Elements of Math by Steven Strogatz Series of articles on various topics in math

Xamuel Logic and Math blog

Tips for Calculus Tips on doing well in a calculus course


Wolfram MathWorld Wiki-like articles


Geometry Formulas & Facts

E-Calc Math Help Math reference & cheat sheets

MacTutor History of Mathematics

Earliest Uses of Mathematical Words

Earliest Uses of Mathematical Symbols

Tricki Store of Problem Solving techniques

Map of Mathematics Visualise the huge discipline of mathematics

The Online Encyclopedia of Integer Sequences

Subpages (1): Calculus III
Oh Waker,
Feb 19, 2013, 12:15 AM
Oh Waker,
Feb 19, 2013, 12:15 AM