Under construction

Introduction

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.

Guides

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 [?]

Books

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.

Geometry

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

Combinatronics

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

Calculus

Precalculus: Mathematics for Calculus; James Stewart, Redlin, Watson
    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?]

Analysis

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

• Solutions
  

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

Courses

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

Comparative charts:

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E-Books

Free Math Books Math books

The Assayer Math Book Compilation

Amazon Mathematics Guide Undergraduate mathematics textbook recommendations 

homepages.nyu.edu 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

• Abstract Algebra: The Basic Graduate Year by Robert B. Ash, University of Illinois
• Abstract Algebra: Theory and Applications by Thomas W. Judson, Stephen F. Austin State University
• Advanced Calculus by Sternberg Shlomo and Lynn Loomis, Harvard University
• Algebra 1 (Grades 9 & 10) by Andrew and Anne Gloag, High Tech High in San Diego, California
• Algebra and Analysis for Computer Science by Jean Gallier, University of Pennsylvania.
• Applied Probability by Paul Pfeiffer, Rice University
• Basic Analysis by Jiří Lebl, University of Illinois at Urbana-Champaign
• Basic Concepts of Mathematics by Elias Zakon, The University of Windsor
• Basic Probability Theory by by Robert B. Ash, University of Illinois
• Basic Probability and Statistics by Brenda Meery, high school mathematics teacher in Glace Bay, Nova Scotia
• Book of Proof by Dr. Richard Hammack, Virginia Commonwealth University
• Calculus (PDF) by David Guichard, Whitman College.
• Calculus (PDF) by Gilbert Strang, MIT
• Calculus 1 by Paul Dawkins, Lamar University
• Calculus 2 by Paul Dawkins, Lamar University
• Calculus 3 by Paul Dawkins, Lamar University
• CK-12 Geometry (Grades 10-12) by multiple authors. Covers California curriculum.
• CK-12 Single Variable Calculus (Grades 11-12) by Multiple authors. Covers California curriculum.
• CK-12 Trigonometry – Second Edition (Grades 10-12) by multiple authors. Covers California curriculum.
• Collaborative Statistics by Barbara Illowsky, Ph.D and Susan Dean, De Anza College
• College Algebra by Paul Dawkins, Lamar University
• Complex Analysis by George Cain, Georgia Institute of Technology
• Complex Variables by Robert B. Ash, University of Illinois
• A Computational Introduction to Number Theory and Algebra by Victor Shoup, NYU
• A Course In Algebraic Number Theory by Robert B. Ash, University of Illinois
• A Course In Commutative Algebra by Robert B. Ash, University of Illinois
• Design of Comparative Experiments by R. A. Bailey, University of London
• Difference Equations to Differential Equations: An Introduction to Calculus by Dan Sloughter, Furman University
• Differential Equations by Paul Dawkins, Lamar University
• Dynamical Systems by S. Sternberg, Harvard University
• Electronic Statistics Textbook by T. Hill & P. Lewicki.
• Elementary Abstract Algebra by W. Edwin Clark, University of Southern Florida
• Elementary Calculus: An Infinitesimal Approach (PDF) by H. Jerome Keisler, University of Wisconsin-Madison
• Elementary Linear Algebra by Keith Matthews, University of Queensland
• Elementary Number Theory by W. Edwin Clark, University of Southern Florida
• Elements of Abstract and Linear Algebra by Edwin H. Connell, University of Miami
• First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, and Dennis Pixton, SUNY – Binghamton
• First Course in Linear Algebra by Dr. Robert Beezer, University of Puget Sound
• GeneratingFunctionology by Herbert Wilf, University of Pennsylvania
• Introduction to Matrix Algebra by Autar K Kaw, University of South Florida
• Introduction to Probability (PDF) by Charles Grinstead & J. Laurie Snell, Swarthmore & Dartmouth Colleges
• Introduction to Social Network Methods by Robert A. Hanneman and Mark Riddle, UC Riverside
• Introduction to Statistical Thought by Michael Lavine, UMASS
• Introduction to the Theory of Numbers by Leo Moser, McGill
• Introductory Statistics: Concepts, Models, and Applications by David W. Stockburger, Missouri State
• Lectures on Probability, Statistics and Econometrics by Marco Taboga, Economist, Bank of Italy
• Lectures on Statistics by Robert B. Ash, University of Illinois
• Linear Algebra (PDF) by Jim Hefferon, Saint Michael’s College
• Linear Algebra by Paul Dawkins, Lamar University
• Math Alive by Ingrid Daubechies, Princeton University
• Math in Society by David Lipmann, Pierce College
• Mathematical Methods of Engineering Analysis (PDF) by Erhan Çinlar and Robert J. Vanderbei, Princeton University
• Multivariable Calculus by George Cain & James Herod, Georgia Tech
• Proofs and Concepts: The Fundamentals of Abstract Mathematics by Dave Witte Morris and Joy Morris, University of Lethbridge
• Real Variables with Basic Metric Space Topology by Robert B. Ash, University of Illinois
• Statistics by Philip B. Stark, University of California – UC Berkeley
• A Summary of Calculus by Karl Heinz Dovermann, University of Hawaii
• The Book “A=B” by Marko Petkovsek, Herbert Wilf and Doron Zeilberger, Univ. of Pennsylvania and Rutgers
• The Calculus of Functions of Several Variables by Dan Sloughter, Furman University
• Yet Another Calculus Text by Dan Sloughter, Furman University

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

MathCS.org 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