View source for Mathematics

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== Intro ==
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 makes it ironic that it is so useful.

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

== Precalculus ==
Note that any autodidactic education requires a minimum amount of fundamentals, and to grasp the higher levels of math you absolutely need to understand the basic concepts known as precalculus, which is generally the math you will see up to high school. If you lack any of these fundamentals, you should refresh your knowledge at pages like [https://www.khanacademy.org Khan Academy] or [http://patrickjmt.com/ PatrickJMT]. If you think you are fit, you can also directly start with calculus, although I would advise to skim a Precalculus book before you do so.

If you totally forgot everything or are a beginner, it is recommended you do the interactive exercises on Khan Academy because they are really helpful tools to quickly refresh your school knowledge up until calculus. You should do all the chapters up to Precalculus, that is: Early Math, Arithmetic, Pre-Algebra, Basic Geometry, Algebra I, Geometry, Algebra II, Trigonometry, Probability and Statistics. You don't need to listen to every video, but you should cover each exercise once to check if you understand it. Once you finish the Precalculus module, you can continue with your first book.

If you are still fit regarding math, you should at least do the precalculus module on Khan Academy to be sure you have grasped everything necessary. For a general overview on the topics to come, choose any book on precalculus, though I recommend one of the following:
* Simmons' ''Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry'' (Concise refresher)
* Stewart's ''Precalculus: Mathematics for Calculus''
* Axler's ''Precalculus: A Prelude to Calculus''
* Cohen's ''Precalculus with Unit Circle Trigonometry''
* [http://www.stitz-zeager.com/ Stitz & Zeager's ''Precalculus''] (Free but might have a bit too much extra material)
* Lang's ''Basic Mathematics''
You can finish Stewart's book in a few months. It is already structured in a way that you can do 1-2 sections per day for 6 days and do a review day on the 7th. You will be familiar with most concepts in this book, but especially if you just come out of high school or have just finished Khan Academy from zero, it will be a good exercise for you.

Those of you who've never done precalculus, trigonometry, or even algebra 2 in high school or feel like you've forgotten all of it might be thinking that you should start with an algebra 2/3, college/intermediate algebra, or trigonometry book before starting a precalculus book since that's how it worked in high school, but don't be. If you take a look at Stewart's ''Algebra and Trigonometry'' book, all the content is mostly identical to his precalculus book. Similarly, Stewart's ''College Algebra'' and ''Trigonometry'' books also just recycle the corresponding material in his precalculus book.

If you're already at an advanced level and want to revisit basic mathematics just for fun, consider reading Linderholm's ''Mathematics Made Difficult: Handbook for the Perplexed''.

=== Basic Euclidean geometry ===
Disclaimer: You don't really need to learn this stuff anymore unless you want to. Synthetic geometry (aka axiomatic or pure geometry) books use isn't how people usually go about solving geometry problems. They use analytic geometry (better named coordinate geometry) and solve problems by reducing them into algebra, trigonometry, and calculus problems. The axiomatic approach stuck around beyond the invention of coordinate geometry and calculus as a means to teach proofs and rigorous reasoning but that too has been replaced by [[Mathematics#Proofs and Mathematical Reasoning|Intro to Proofs]] (or Discrete Math) classes based on naive set theory and logic. With that said, classical Euclidean geometry becomes somewhat more interesting in the context of non-Euclidean or Klein geometries, which you can learn about [https://4chan-science.fandom.com/wiki/Mathematics#Classical_Geometries later].
* Kiselev's ''Geometry: Books I and II''
*# ''Planimetry & Geometry''
*# ''Stereometry (Euclid's Elements distilled)''
* ''Basic Geometry'' by Birkhoff and Beatley (Uses Birkhoff's axioms rather than the ones Euclid chose)
* ''Geometry: A High School Course'' by Lang and Murrow
** A modern approach to rigorous geometry that covers somewhat different topics than other books do and isn't afraid to use coordinate
Some advanced 2nd course material:
* ''Geometry Revisited'' by Coxeter and Greitzer
* ''Advanced Euclidean Geometry'' by Posamentier
* ''Advanced Euclidean Geometry'' (Dover Books) by Johnson
* ''College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle'' (Dover Books) by Altshiller-Court

=== Elementary Algebra ===
* ''Elements of Algebra'' by Euler (Don't let its age discourage you, it's one of the best books ever written on elementary algebra even 250~ years later. Minor errata: Euler makes a small mistake when defining division and multiplication by √-1. [https://plus.maths.org/content/os/issue45/features/sangwin/index See this review]. He writes that 1/√-1 = √(1/-1) = √-1 but this is incorrect as 1 = √-1/√-1 = √-1√-1 = (√-1)<sup>2</sup> = -1. Instead multiply by 1=√-1/√-1 to simplify it, 1/√-1 = 1/√-1 * √-1/√-1 = √-1/(√-1)<sup>2</sup> = -√-1. This may or may not be pointed out depending on your translation.)
* ''Elementary Algebra for Schools'' by Knight and Hall <sup>[[https://archive.org/details/elementaryalgebr00hall Archive.org] for the 1896 edition, [https://www.amazon.com/Elementary-Algebra-Schools-Containing-Chapter/dp/1402159064/ Amazon] for the 1906 edition]</sup>
Some advanced "honors" books that cover extra material that hasn't been covered in algebra/public school courses for a century:
* ''Algebra: An Elementary Text-Book - For the Higher classes of Secondary Schools and Colleges'' Volumes 1 & 2 by Chrystal
* ''Higher Algebra: a Sequel to Elementary Algebra for Schools'' by Hall and Knight
*''College Algebra'' (AMS Chelsea Publishing) by Henry Burchard Fine

=== Vector Geometry ===
These books are the linear algebra approach to geometry. Many of the topics covered here are scattered throughout other courses so they're not required reading to progress. The material is collected & organized together here for beginners who want to see the modern approach geometric problems or for someone who wants to revise his geometric skills. 
* ''A Vector Space Approach to Geometry'' (Dover Books) by Melvin Hausner
* ''Elementary Vector Geometry'' (Dover Books) by Seymour Schuster
* ''Vector Geometry'' (Dover Books) by Gilbert de B. Robinson

== Problem books ==
These elementary problem books are meant for additional non-routine practice, challenges & puzzles, killing time, preparing for school competitions and exams, or to steal interesting questions from when you're teaching or tutoring students.
* ''Challenging Problems in Algebra'' (Dover Books) by Posamentier
* ''Challenging Problems in Geometry'' (Dover Books) by Posamentier
* ''The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics'' (Dover Books) by Shklarsky, Chentzov, and Yaglom
* ''103 Trigonometry Problems: From the Training of the USA IMO Team'' by Andreescu and Feng
* ''104 Number Theory Problems: From the Training of the USA IMO Team'' by Andreescu
* ''105 Algebra Problems from the AwesomeMath Summer Program'' by Andreescu
* ''106 Geometry Problems from the AwesomeMath Summer Program'' by Andreescu
* ''107 Geometry Problems from the Awesomemath Year-Round Program'' by Andreescu
* ''108 Algebra Problems from the Awesomemath Year-Round Program'' by Andreescu
* ''Problems From the Book'' and ''Straight From the Book'' by Andreescu
* ''The Stanford Mathematics Problem Book'' (Dover Books) by Polya and Kilpatrick
* ''Sequences, Combinations, Limits'' (Dover Books) by Gelfand, Gerver, Kirillov, Konstantinov, and Kushnirenko
* ''Challenging Mathematical Problems With Elementary Solutions'' (Dover Books) by A.M. Yaglom and I.M. Yaglom
* ''Hungarian Problem Book I-IV containing the Eötvös Mathematical Competitions from 1894–1963''
The following have more advanced problems at the university level up to preparing for qualifying exams during graduate school
* ''The Green Book of Mathematical Problems'' (Dover Books) by Hardy and Williams
* ''The Red Book of Mathematical Problems'' (Dover Books) by Williams and Hardy
* ''William Lowell Putnam Mathematical Competition: Problems & Solutions: 1938-1964''
* ''The William Lowell Putnam Mathematical Competition: Problems and Solutions 1965–1984''
* ''The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions and Commentary''
* ''Contests in Higher Mathematics: Miklos Schweitzer Competitions, 1962-1991'' by Szekely
* ''Problems in Mathematical Analysis I: Real Numbers, Sequences and Series'' by Kaczor and Nowak
* ''Problems in Mathematical Analysis II: Continuity and Differentiation'' by Kaczor and Nowak
* ''Problems in Mathematical Analysis III Integration'' by Kaczor and Nowak
* ''A Collection of Problems on Complex Analysis'' (Dover Books) by Volkovyskii, Lunts, and Aramanovich
* ''Problems in Group Theory'' (Dover Books) by Dixon
* ''Problems and Theorems in Analysis I: Series, Integral Calculus, Theory of Functions'' by George Polya and Gabor Szegö
* ''Problems and Theorems in Analysis II: Theory of Functions, Zeros, Polynomials, Determinants, Number Theory, Geometry'' by George Polya and Gabor Szegö
* ''A Hilbert Space Problem Book'' by Halmos
* ''Berkeley Problems in Mathematics'' by Paulo Ney de Souza and Jorge-Nuno Silva
* ''Problems and Solutions in Mathematics'' (Major American Universities PH.D. Qualifying Questions and Solutions)
* ''[https://arxiv.org/abs/0906.0290 Puzzles in geometry that I know and love] ''by Petrunin

== Problem Solving and Heuristics ==
Some strategies on how to approach difficult problems to solve them exactly or heuristically and dealing with Fermi problems:
* ''How to Solve It: A New Aspect of Mathematical Method'' by Polya
* ''Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin'' by Weinstein and Adam
* ''Guesstimation 2.0: Solving Today's Problems on the Back of a Napkin'' by Weinstein and Edwards
* ''Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving'' by Mahajan
* [http://ocw.mit.edu/resources/res-6-011-the-art-of-insight-in-science-and-engineering-mastering-complexity-fall-2014/online-textbook/ ''The Art of Insight in Science and Engineering: Mastering Complexity'' by Mahajan]
* ''How to Solve It: Modern Heuristics by Michalewicz and Fogel''
** Heuristics focusing on CS/optimization problems
* ''Problem-Solving Through Problems'' by Larson
* ''Solving Mathematical Problems: A Personal Perspective'' by Terence Tao
* ''Putnam and Beyond'' by Gelca and Andreescu

== Overview of Mathematics ==
* ''What Is Mathematics? An Elementary Approach to Ideas and Methods'' by Richard Courant and Herbert Robbins
* ''Prelude to Mathematics'' (Dover Books) by Sawyer
* ''Concepts of Modern Mathematics'' (Dover Books) by Ian Stewart
* ''Mathematics: Its Content, Methods and Meaning'' (Dover Books) by Aleksandrov, [https://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], [http://www-history.mcs.st-and.ac.uk/Biographies/Lavrentev.html Lavrentev], [https://en.wikipedia.org/wiki/Sergei_Sobolev Sobolev], [https://en.wikipedia.org/wiki/Israel_Gelfand Gel'fand], et al.
* ''A Mathematical Gift: The Interplay Between Topology, Functions, Geometry, and Algebra I, II, III'' by Ueno et al.

== Calculus ==
[https://en.wikipedia.org/wiki/Calculus Calculus] is the study of change (derivatives) and accumulation (integrals) of functions. These topics are linked by the [https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus Fundamental Theorem of Calculus] which states informally that "the accumulation of the changes of a function" and "the change of the accumulation of a function" result in the original function. The rest of calculus is just spamming these ideas on various applications and situations.

Some students struggle with calculus but honestly it is a really straight forward subject, especially compared to other advanced subjects in math. As long as you pay attention and go in trying to learn, you should quickly end up joining the rest of the math students in calling it 'trivial' in retrospect. Don't set yourself up to failure by thinking you're not capable of learning it because it will all click once you look at it right.

=== Single Variable Calculus ===
The standards texts (Stewart, Rogawski, et al.) you see required for college classes are, in all honesty, quite terrible since they are not written with self-study in mind but just as a collection of exercises and a review of the basic methods. ''Do ''use them to practice calculus but not as a means to learn it.

For a well done intuitive approach using infinitesimals, which is the way everyone ends up thinking about calculus which is also technically ''nonstandard ''but by no means mathematically incorrect, [https://www.math.wisc.edu/~keisler/calc.html ''Elementary Calculus: An Infinitesimal Approach'' by Jerome Keisler] is a fantastic and free public domain book (also available in an inexpensive Dover paperback edition). If the infinitesimal approach intrigues you but you've already done a course in calculus or currently reading through another 1000 page book on calculus, ''Infinitesimal Calculus'' (Dover Books) by Henle and Kleinberg is a nice short 144 page book that develops the theory of infinitesimals in calculus in an accessible and clear manner. As you can probably infer from the page count, Henle's book doesn't have any material on the applications of calculus so don't use it as a standalone book to learn all of calculus from but as a supplement to see a different approach in understanding the subject the way it was originally invented.

For a rigorous ''standard'' (δ-ε) approach to the subject, your options are ''Calculus'' by Spivak or the classic ''Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra'' by Apostol. Spivak's writing certainly has its fans but it sadly lacks much of the applications and motivation (related rates, optimization) that are standard in calculus making it hard to use on its own. Apostol's ''Calculus'' doesn't have this oversight and it's probably the best one to learn the material from on your own. Another rigorous option that also has copious amounts of physics applications, motivation, and intuition presented at the same time is ''Introduction to Calculus and Analysis, Volume I'' by Richard Courant and Fritz John. The book is a modern rewrite of the classic ''Differential and Integral Calculus'' by Richard Courant and includes the most material of the three and its exercise are the most difficult (perhaps a bit too difficult in places).

Other calculus books leave out the involved proofs, which you'll see later in analysis, and focus on conceptual understanding and applying calculus to accommodate weaker students or students that are less prepared in rigorous/abstract mathematics. This isn't a completely bad way of learning calculus but you might be annoyed by the occasional lack of explanation/justification in some isolated places. Some good books in this category are ''Calculus: An Intuitive and Physical Approach'' (Dover Books) by Kline, C''alculus With Analytic Geometry'' by Simmons (contains a lot on history of the subject and its applications in physics and science), and ''A First Course in Calculus'' by Lang.

To just learn the methods of calculus, there are plenty of lectures on calculus for you to choose from on YouTube, such as 3Blue1Brown's intuitive [https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr ''Essence of Calculus''] series, and the videos of [https://www.youtube.com/user/ilectureonline/playlists?view=50&sort=dd&shelf_id=11 Martin Van Biezen], [https://www.youtube.com/watch?v=EKvHQc3QEow&list=PL19E79A0638C8D449 Sal Khan], [https://www.youtube.com/results?search_query=patrickjmt+calculus+ PatrickJMT], and [https://www.youtube.com/user/blackpenredpen/playlists?disable_polymer=1 blackpenredpen]

=== Multivariable and Vector Calculus ===
Again, the usual suspects you'll find assigned in college courses tend to make good exercise books but terrible introductions to the subject. Your options are the latter part of Keisler's book above for an infinitesimals approach; Lang's ''Calculus of Several Variables'' or the latter part of Simmons' book to continue with their approach for weaker and less prepared students; and ''Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability'' by Apostol, or ''Introduction to Calculus and Analysis, Vol. II'' by Richard Courant and Fritz John (the paperback is split into 2 parts) to continue on with the standard rigorous approach. For another text that covers this material in a more geometric way, consider Callahan's ''Advanced Calculus: A Geometric View<sup>[[http://www.math.smith.edu/~callahan/ website]]</sup>.''

The following texts take a slightly more abstract approach than Apostol or Courant and go a bit deeper into the subject by covering differential forms and manifolds. Most single semester courses on vector calculus do not have time to reasonably cover this material, and consequently is usually skipped until later, but this advanced perspective can greatly aid one's understanding of the subject. You could study this material either when you first learn multivariable calculus or when you want a second pass on the subject, after just learning the basic methods, to improve your understanding while deepening your knowledge by generalizing what you've seen before. They can also be used as supplements or stepping stone to an advanced multivariable analysis course.
* C. H. Edwards Jr.'s ''Advanced Calculus of Several Variables'' (Dover Books)
* Hubbard and Hubbard's ''Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach''
*Shifrin's ''Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds'' (a bit expensive if you decide to purchase the book)
* Harold M. Edwards' ''Advanced Calculus: A Differential Forms Approach''
* Bressoud's ''Second Year Calculus: From Celestial Mechanics To Special Relativity''

=== Curves and Surfaces in ℝ² and ℝ³ ===
This subject is the study of Geometry using the tools that you learned in Vector Calculus and serves as a preparation to more abstract approaches to Differential Geometry you'll see in the future. Most schools only quickly pass through the subject during multivariable calculus but it will help in the long run if you study the material early on.
* Tapp's ''Differential Geometry of Curves and Surfaces''
* Pressley's ''Elementary Differential Geometry''
* Christian Bär's ''Elementary Differential Geometry''
* do Carmo's ''Differential Geometry of Curves and Surfaces'' (Dover Books) (standard book)

=== Ordinary Differential Equations ===
The standard text used in college courses is ''Elementary Differential Equations'' by Boyce and DiPrima, which many people do seem to like (not me however). A cheap and very good alternative is ''Ordinary Differential Equations'' by Tenenbaum & Pollard (Dover Books) which is perfect for self study. Other well written books are ''Differential Equations with Applications and Historical Notes'' by Simmons and ''Differential Equations'' by Ross.
* Coddington's ''An Introduction to Ordinary Differential Equations'' (Dover Books)
*Hirsch and Smale's ''Differential Equations, Dynamical Systems, and Linear Algebra''
**A unified approach that combines linear algebra and ODE. The newer editions, featuring the author Devaney, ''does'' offer more applications plus an intro to chaos theory, but the older editions without Devaney is more rigorous. In either case, PDFs of both can be found rather easily.
* Arnold's ''Ordinary Differential Equations'' (advanced)
** Requires some analysis
** Has a lot of intuition on smooth manifolds

== Linear Algebra ==
When speaking of linear algebra, people refer to one of 2 complementary but different subjects: matrix algebra/computational linear algebra and theoretical linear algebra/finite vector space theory. Oddly enough, you could study them in any order but canonically you're typically expected to learn some matrix algebra first and then transition to vector spaces and/or more applied/numerical topics second. The necessary prerequisite knowledge is just precalculus but some calculus knowledge is useful and may appear in a few examples.

=== Matrix Algebra ===
For a first exposure to the subject there really isn't that much to learn. You typically cover systems of equations, matrix operations, Gaussian elimination (also known as row reduction), LU decomposition, determinants, eigenvectors and eigenvalues, and diagonalization possibly with a few additional fluff subjects to round out a whole course. Many times you can pick up this material while studying calculus or ODEs (like with Apostol or Hubbard<sup>2</sup>'s book) so you can just skip to more advanced material. Also, the introductory material in first few chapters of advanced textbooks are often good enough to learn matrix algebra from if you're in a rush. 

But while some students seem to inhale these topics and quickly move on, others will need to take their time before operating with matrices becomes natural to them. A gentle introduction for learners with weaker math skills can be found in ''Matrices and Linear Algebra'' by Schneider and Barker. Learners with slightly better math abilities can benefit more from ''Matrices and Linear Transformations'' by Cullen which is aimed at STEM students and contains extra material at the end on advanced material. A free book for students seeking a honors introduction to linear algebra (and basic proofs) is [http://www.math.brown.edu/~treil/papers/LADW/LADW.html ''Linear Algebra Done Wrong'' by Treil] (Don't worry, the title is a pun on Axler's "done right" book below). Another popular free book is [http://joshua.smcvt.edu/linearalgebra/ Hefferon's ''Linear Algebra'']. There's also a whole host of vulgarly over expensive textbooks used by college courses at this level (like Strang's ''Introduction to Linear Algebra'', Lay's ''Linear Algebra and Its Applications'', Friedberg's ''Elementary Linear Algebra'', etc) but most of them aren't very good and even if they were, the first two aforementioned books above are far cheaper thanks to them being published by Dover and the last two are free. If you prefer learning from videos, you can get a very solid intro using the linear algebra playlists on [https://www.khanacademy.org/math/linear-algebra Khan Academy], [http://patrickjmt.com/#linear-algebra PatrickJMT], or [https://www.engineer4free.com/linear-algebra.html Engineer4Free]. Bonus is they are all free. 

=== Applied Linear Algebra ===
For a first book in applied linear algebra, ''Linear Algebra and Its Applications'' by Strang is the standard text used but it is one of those love-it-or-hate-it texts. If you fall into the hate-it camp, then Meyer's ''Matrix Analysis and Applied Linear Algebra'' is a good alternative. After reading one of them, you'll be more than ready to move onto advanced [[Computer Science and Engineering#Numerical Linear Algebra|numerical linear algebra]] and matrix analysis textbooks.

=== Finite-Dimensional Vector Spaces ===
To get started on the theoretical side of linear algebra you obviously should be familiar with the basics of proofs. Once you are, theory side has a lot of classic and well loved textbook to choose from:
* ''Linear Algebra'' by Shilov (Dover Books)
* ''Finite-Dimensional Vector Spaces'' by Halmos (Dover Books)
* ''Linear Algebra'' by Friedberg, Insel, and Spence
* ''Linear Algebra'' by Hoffman and Kunze 
Of course there's also ''Linear Algebra done Right'' by Axler and on the one hand, the stuff he does is great... but on the other hand, he fucking <u>HATES</u> determinants and goes crazy avoiding them. Because of that you shouldn't use his book alone to learn from and you really should read Shilov alongside of it. But Axler certainly gives a unique development to the subject.

=== Refreshers and Advanced Books in LA ===
Now if you want a challenge, start off with ''Linear Algebra and Its Applications'' by Lax. It is good for learning linear algebra for the first time if you're a hot shot freshman and already know the basics of calculus and analysis, using it as a second book on linear algebra, or as a 3rd refresher book for those who are entering graduate school. Another good 3rd book for deeper linear algebra study, and if you have the abstract algebra background for it, is Roman's ''Advanced Linear Algebra''. ''Abstract Linear Algebra'' by Curtis is a concise 162 page book that builds up the core of linear algebra from the beginning in a general abstract way and ends with Hurwitz's Theorem as a finale.

''Linear Algebra and Geometry'' by Kostrikin & Manin is an excellent advanced book bridging Linear Algebra to many advanced topics including Lie algebras, category theory, Clifford algebras, affine and projective geometry, tensors and multilinear algebra, and many applications in quantum mechanics and physics.

== Advanced Calculus ==
The term ''Advanced Calculus'' has come to mean different things over the course of the past century. During the first half of the 20th century, ''Advanced Calculus'' courses consisted of what's now commonly found in Multivariable and Vector Calculus possibly with some Differential Equations topics thrown in. Lately, it has been fashionable to call very watered down "Real Analysis" courses ''Advanced Calculus'' even though it's not ''advanced'' nor ''calculus ''and goes no deeper into analysis than a good rigorous calculus book does. Here Advanced Calculus means what the name implies, advanced topics in calculus (and tools from analysis) typically not found in the usual calculus sequence but still very useful for solving difficult problems in science, engineering, and mathematics.

=== Complex Variables ===
Complex variables (also known as ''complex calculus'' or ''applied complex analysis'') is the generalization of calculus over the complex field and shares many parallels with multivariable/vector calculus.
* ''A First Course in Complex Analysis with Applications'' by Zill and Shanahan
* ''Fundamentals of Complex Analysis: With Applications to Engineering and Science'' by Saff and Snider
* ''Complex Variables: Introduction and Applications'' by Ablowitz and Fokas <sup>[[https://sites.google.com/site/ablowitz//publications/books/errata Errata]]</sup>
* ''Functions of a Complex Variable: Theory and Technique'' by Carrier, Krook, and Pearson
A good supplement to any of the above is ''Visual Complex Analysis'' by Needham. Once you've finished a book on complex variables, ''Conformal Mapping: Methods and Applications'' (Dover Books) by Schinzinger and Laura makes a nice supplement with many applications of conformal mapping in MechE/AeroE and EE.

=== Special Functions ===
Special functions used to be the subject of a second semester complex variables course until it was sucked into mathematical physics, advanced engineering mathematics and other similar courses. The problem with such courses is that they spend far too little time developing subject as they try to cover complex variables, PDEs, differential geometry, topology, variations, algebra, and numerical methods among other subjects at the same time. The following books give a more focused and fuller development of special functions:
* ''Special Functions & Their Applications'' (Dover Books) by Lebedev
* ''Special Functions for Scientists and Engineers'' (Dover Books) by Bell
* ''The Functions of Mathematical Physics'' (Dover Books) by Hochstadt
For books with more mathematical theory:
* ''Special Functions'' by Andrews, Askey, and Roy
* [https://archive.org/details/courseofmodernan00whit ''A Course of Modern Analysis''] by Whittaker and Watson
* ''Special Functions'' by X. Z. Wang and Guo (Great complement to Whittaker and Watson)
Whittaker and Watson has been the bible for special functions for over a century now. Part 1 contains a review of the essential real and complex analysis needed for Part 2 which details the major special functions.

=== Fourier Transforms ===
The [https://en.wikipedia.org/wiki/Fourier_transform Fourier transform] and related transforms are powerful techniques used throughout STEM that convert a function into its frequency components. Tragically, many science and engineering programs can't find room for such a course in their curricula and try to get away with throwing in brief discussion of how to use them into the courses that require them. This in the end fails to create any conceptual understanding of what's going on beyond the mindless crank turning. These books will help you see the Fourier transform beyond just a 'trick' and be better equipped to apply them:
* ''The Fourier Transform & Its Applications'' by Bracewell (Great for conceptual understanding)
* ''Fourier Transforms: An Introduction for Engineers'' by Gray and Goodman
* ''Linear Systems, Fourier Transforms, and Optics'' by Gaskill
* ''A First Course in Fourier Analysis'' by Kammler
The subject matter overlaps considerably with [[Electrical and Electronics Engineering#Signals .26 Systems|EEE's ''Systems and Signals'']] books. For more mathematical detailed books see the [[Mathematics#Fourier Analysis|fourier analysis]] books below.

=== Calculus of Variations ===
[https://en.wikipedia.org/wiki/Calculus_of_variations Calculus of variations] is the subject of finding ''functions'' that maximize or minimize some equation. For example, finding a path that minimizes the distance traveled from point a to b.
* ''Calculus of Variations'' (Dover Books) by Gelfand and Fomin
* ''Calculus of Variations: with Applications to Physics and Engineering'' (Dover Books) by Weinstock
* ''Calculus of Variations'' (Dover Books) by Elsgolc

=== Integral Equations ===
* ''Introduction to Integral Equations with Applications'' by Jerri
* ''The Classical Theory of Integral Equations: A Concise Treatment'' by Zemyan
* ''Integral Equations'' (Dover Books) by Tricomi
* ''Linear Integral Equations'' by Kress
* ''Singular Integral Equations: Boundary Problems of Function Theory and Their Application to Mathematical Physics (Dover Books)'' by Muskhelishvili (more advanced)

=== Asymptotics ===
* ''Asymptotic Methods in Analysis'' (Dover Books on Mathematics) by de Bruijn
* ''Asymptotic Expansions'' (Dover Books on Mathematics) by Erdelyi
* ''Asymptotic Expansions of Integrals'' (Dover Books on Mathematics) by Bleistein and Handelsman
* ''Asymptotics and Special Functions'' by Olver

=== Perturbation Theory ===
* ''A First Look at Perturbation Theory (Dover Books on Physics)'' by Simmonds and Mann Jr (Primmer)
* ''Introduction to Perturbation Methods'' by Holmes
* ''Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory'' by Bender and Orszag
* ''Perturbations: Theory and Methods'' by Murdock
* ''Perturbation Theory for Linear Operators'' by Tosio Kato

=== Nonlinear Dynamics and Chaos ===
* ''Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering'' by Strogatz <sup>[[https://www.youtube.com/playlist?list=PLbN57C5Zdl6j_qJA-pARJnKsmROzPnO9V Author's online course]]</sup> 
* ''Dynamical Systems'' (Dover Books on Mathematics) by Shlomo Sternberg <sup>[[http://www.math.harvard.edu/~shlomo/ Free]]</sup>
* ''Introduction to Applied Nonlinear Dynamical Systems and Chaos'' by Wiggins

=== Obscure Topics ===

==== Integration Techniques ====
If you have read about Feynman, you may have heard his story of coming across ''Advanced Calculus'' by Woods and discovering the differentiating parameters under the integral sign[https://web.williams.edu/Mathematics/lg5/Feynman.pdf <sup><nowiki>[1]</nowiki></sup>]<sup>[http://fy.chalmers.se/~tfkhj/FeynmanIntegration.pdf <nowiki>[2]</nowiki>][https://en.wikipedia.org/wiki/Leibniz_integral_rule <nowiki>[3]</nowiki>]</sup> trick and using it to his advantage over and over again. These books are in a similar vein, they contain many integration techniques like [https://en.wikipedia.org/wiki/Integral_of_inverse_functions ∫f<sup>-1</sup>(x) dx = xf<sup>-1</sup>(x) - F(f<sup>-1</sup>(x))] that aren't covered in a standard single variable calculus course but are accessible to anyone who's completed the course.
* ''Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals'' by Boros and Moll
* ''Inside Interesting Integrals: (with an introduction to contour integration) A Collection of Sneaky Tricks, Sly Substitutions, and Numerous Other Stupendously Clever, Awesomely Wicked, and Devilishly Seductive Maneuvers for Computing Nearly 200 Perplexing Definite Integrals From Physics, Engineering, and Mathematics (Plus 60 Challenge Problems with Complete, Detailed Solutions)'' by Nahin
* ''(Almost) Impossible Integrals, Sums, and Series ''by Valean

==== Fractional Calculus ====
Just as you can iterate to get second derivatives and triple integrals, it's possible to extend the order of these operators from integers to fractions or to any real or complex number. For example, you can define a half derivative operator where if you apply it twice to a function, you get the usual derivative of that function. This is the domain of [https://en.wikipedia.org/wiki/Fractional_calculus fractional calculus] which has a wide variety of applications in many branches of physics and engineering. The idea of fractional calculus is an old one dating back to Leibniz in 1695 and its applications were examined by the electrical engineer Oliver Heaviside in the 1890s but the first textbook on the subject was only published in 1974 by Oldham and Spanier. Since then fractional calculus has steadily been gaining more attention but it still remains relatively unknown to many in the STEM field.
* ''The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order (Dover Books)'' by Oldham and Spanier
* ''An Introduction to the Fractional Calculus and Fractional Differential Equations'' by Miller and Ross

== Partial Differential Equations ==
Historically, the study of PDEs was a major impetus for the development of many results of analysis. Without this advanced math knowledge, the study of PDE is destined to be somewhat more trickier than what you've seen before in your studies. Be prepared to do some real work.

For a quick primer on PDEs, ''Partial Differential Equations for Scientists and Engineers'' (Dover Books) by Farlow is pretty good albeit somewhat shallow. Fuller undergraduate treatments can be found with:
* ''Applied Partial Differential Equations: With Fourier Series and Boundary Value Problems'' by Haberman
** The best applied text at this level 
* ''Partial Differential Equation: An Introduction'' by Strauss
* ''Partial Differential Equations'' by Fritz John
* ''Partial Differential Equations: Theory and Technique by Carrier and Pearson''
** Excellent source of insightful challenging problems

=== Graduate Partial Differential Equations ===
Once you have the required background in analysis, you can really study the meat of PDEs in detail with the following:
* ''Partial Differential Equations'' by Jost (Strong bias for elliptic equations)
* ''Partial Differential Equations'' by Evans (The standard introduction text for graduate PDEs)
* ''Introduction to Partial Differential Equations'' by Folland (An more intermediate graduate level PDEs book than Evans)
* ''Partial Differential Equations I-III'' by Taylor
*# Basic Theory
*# Qualitative Studies of Linear Equations
*# Nonlinear Equations

== Numerical Analysis ==
Survey of Numerical Analysis
* ''A Theoretical Introduction to Numerical Analysis'' by Ryaben'kii and Tsynkov
* ''A First Course in Numerical Analysis'' (Dover Books) by Ralston and Rabinowitz
* ''Functional Analysis and Numerical Mathematics ''by Collatz
See also the CS&E pages on
* [http://4chan-science.wikia.com/wiki/Computer_Science_and_Engineering#Overviews_of_Numerical_Analysis Overviews of Numerical Analysis]
* [http://4chan-science.wikia.com/wiki/Computer_Science_and_Engineering#Numerical_Linear_Algebra Numerical Linear Algebra]
* [http://4chan-science.wikia.com/wiki/Computer_Science_and_Engineering#Approximation_Theory Approximation Theory]
* [http://4chan-science.wikia.com/wiki/Computer_Science_and_Engineering#Numerical_Ordinary_Differential_Equations Numerical Ordinary Differential Equations]
* [http://4chan-science.wikia.com/wiki/Computer_Science_and_Engineering#Finite_Difference_Methods Finite Difference Methods]
* [http://4chan-science.wikia.com/wiki/Computer_Science_and_Engineering#Finite_Element_Methods Finite Element Methods]
* [http://4chan-science.wikia.com/wiki/Computer_Science_and_Engineering#Spectral_Methods Spectral Methods]

== Proofs and Mathematical Reasoning ==
True mathematics involves proofs, lots and lots of proofs (cry more physicists). The importance of mastering the art of writing valid proofs that do not make careless unstated assumptions or unproven assertions can not be understated. Oftentimes when you view some statement as initially obvious, it will turn out to be either dead <u>wrong</u> or at the very least hold most of the meat of the proof in proving it. Another aspect in learning proofs is following along when reading a proof in mathematical texts which requires diligently filling in ''all'' the skipped steps and checking which assumptions could be removed/weakened or what fails when they are removed/weakened. At their core, basic proofs are really easy and frequently just a matter of unwrapping the definition and following your nose, but getting into the right mindset for them might take the neophyte some practice in order to see them that way. Therefore you should work through a book or two on proofs before moving onto advanced mathematics and then blaming those books for being written badly because you lacked the prerequisite mathematical maturity from skipping this step. Some good books to learn proofs are:
* ''A Transition to Advanced Mathematics'' by Smith, Eggen, and St. Andre
* ''A Primer of Abstract Mathematics'' by Ash
*''The Foundations of Mathematics'' by Sibley
*''Mathematical Proofs: A Transition to Advanced Mathematics'' by Chartrand, Polimeni, and Zhang
*''Conjecture and Proof'' by Laczkovich
** An excellent supplement to any of the above books that shows a larger variety of proofs in mathematics
* ''Proofs from THE BOOK'' by Aigner and Ziegler
** Not a textbook on proofs but it is an excellent collection of well done and elegant proofs to appreciate and draw inspiration from
If you find yourself struggling with proofs, then the following books provide more hand holding on the subject (but at the cost of excluding some additional material):
* ''How to Prove It: A Structured Approach'' by Velleman
* ''How to Read and Do Proofs: An Introduction to Mathematical Thought Processes'' by Solow
* [http://www.people.vcu.edu/~rhammack/BookOfProof/ ''Book of Proof''] by Hammack
After this, set theory and mathematical logic are the logical continuation of this material and reading books on them will deepen your understand of what sets and proofs ''really are'' as well as mathematics as a whole with meta-mathematics. They also make excellent next steps in getting better at proofs and abstract mathematics in general before moving on to the much more difficult subjects like algebra and analysis. [[Mathematics#Combinatorics and Graph Theory|combinatorics, graph theory]], [[Mathematics#Finite-Dimensional Vector Spaces|linear algebra involving vector spaces]], and [[Mathematics#Number Theory|number theory]] textbooks would then be the next level to practice on and are fairly easy to read at this stage of mathematical maturity.

Since you will likely find yourself revising your proofs quite often, now would be an ideal time to finally learn [[Universal Material#LaTeX|LaTeX]] (pronounced "lay-tech") to typeset your proofs and future papers in.

== Set Theory, Mathematical Logic, and MetaMathematics ==
This is the formal study of the [https://en.wikipedia.org/wiki/Foundations_of_mathematics foundations of mathematics] using mathematics, particularly on [https://en.wikipedia.org/wiki/Set_theory set theory] which much of mathematics is built on and [https://en.wikipedia.org/wiki/Mathematical_logic mathematical logic] which studies what proofs are and the limits of what can be done. When starting in this subject the question of where to start pops up. Ideally, you would want to know a some logic while studying set theory and know some set theory while studying logic leading to a bit of a dilemma. This is solved by most introductory books giving just enough material on the other subject so you don't get lost but once you move on to intermediate and beyond books, you are assumed to have already studied both set theory and logic at least at the introductory level.

=== Introductory Set Theory ===
* ''Elements of Set Theory'' by Enderton
Enderton is a gentle, clear, and easy to read textbook that's perfect for someone just finishing a book or course on proofs and looking for the next step to improve their math skills further. He will construct the real numbers from ZF axioms in the first five chapters.
* ''The Joy of Sets: Fundamentals of Contemporary Set Theory'' by Devlin
* ''Introduction to Set Theory'' by Hrbacek and Jech (baby Jech)
These books would be better for someone who already has a few proof based math courses under their belt. They're a notch harder than Enderton and go into a few more advanced topics too.

=== Introductory Logic ===
* ''Introduction to Logic: and to the Methodology of Deductive Sciences'' (Dover Books) by Alfred Tarski
* ''Introduction to Metamathematics'' by Kleene
* ''A Mathematical Introduction to Logic'' by Enderton (gold standard)

=== Intermediate Set Theory and Logic ===
* ''Set Theory: An Introduction to Independence Proofs'' by Kunen
** The newer edition is just called "Set Theory" but still is focused on independence proofs
* ''Model Theory: An Introduction'' by Marker (have had a good course in algebra)
* ''Fundamentals of Mathematical Logic'' by Hinman
* ''Computability Theory'' by Cooper (overlaps with chapter 3 of Enderton Logic)

=== Advanced Computability (Recursion) Theory ===
* ''Recursively Enumerable Sets and Degrees'' by Soare
* ''Computable Structures and the Hyperarithmetical Hierarchy'' by Ash and Knight 
* ''Higher Recursion Theory'' by Sacks 

=== Advanced Set Theory ===
* ''Set Theory'' by Jech (More of a reference book) 
* ''The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings'' by Kanamori
*''Constructibility by Devlin'' (there are errors be careful look up on SE first)<sup>[[https://projecteuclid.org/euclid.pl/1235419477 Free] ] </sup>
*''Descriptive Set Theory'' by Moschovakis (getting dated)
For an alternative guide through logic, consider the [https://www.logicmatters.net/tyl/ Teach Yourself Logic] guide.

== Number Theory ==

=== Elementary Number Theory ===
* ''An Introduction to the Theory of Numbers'' by Niven, Zuckerman, and Montgomery
** [https://math.stackexchange.com/a/342/606034 This book will make you into a number theorist]
* ''An Introduction to the Theory of Numbers'' by Hardy and Wright
* ''Elementary Number Theory'' (Dover Books) by Dudley
* ''Number Theory'' (Dover Books) by Andrews
** The first half is a combinatorial approach to basic number theory followed by an introduction to combinatorial number theory by a pioneer in the field

=== Algebraic Number Theory ===
* ''A Classical Introduction to Modern Number Theory'' by Ireland and Rosen 
** A bit harder than the above, the go-to book after you've taken undergrad abstract algebra
** Starts off with elementary number theory
* ''A Course in Arithmetic'' by Serre (first half; much harder than the above)
* ''Primes of the Form x<sup>2</sup>+ny<sup>2</sup>: Fermat, Class Field Theory, and Complex Multiplication'' by Cox
* ''Algebraic Number Theory and Fermat's Last Theorem'' by Ian Stewart and Tall
* ''Number Theory 1-3'' by Kazuya Kato, Nobushige Kurokawa, and Takeshi Saito
*# Fermat's Dream
*# Introduction to Class Field Theory
*# Iwasawa Theory and Modular Forms
* ''Algebraic Number Theory'' by Cassels and Fröhlich
* ''Algebraic Number Theory'' by Neukirch and Schappacher

=== Analytic Number Theory ===
* ''Introduction to Analytic Number Theory'' by Apostol
* ''A Course in Arithmetic'' by Serre (second half)
* ''Modular Functions and Dirichlet Series in Number Theory'' by Apostol
* ''Multiplicative Number Theory'' by Davenport 
* ''Multiplicative Number Theory I: Classical Theory'' by Montgomery and Vaughan
* ''Analytic Number Theory'' by Iwaniec and Kowalski (The reference)

=== Computational Number Theory ===
* [http://www.shoup.net/ntb/ ''A Computational Introduction to Number Theory and Algebra''] by Shoup
* ''Prime Numbers: A Computational Perspective'' by Crandall and Pomerance
* ''Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography'' by Buhler and Stevenhagen
* ''A Course in Computational Algebraic Number Theory'' by Cohen
* ''Advanced Topics in Computational Number Theory'' by Cohen

=== Elliptic Curves ===
* ''Rational Points on Elliptic Curves'' by Silverman and Tate
* ''The Arithmetic of Elliptic Curves'' by Silverman
* ''Advanced Topics in the Arithmetic of Elliptic Curves'' by Silverman

== Probability and Randomness ==

=== Probability (Multivariable Calculus based) ===
* ''Introduction to Probability'' by Bertsekas and Tsitsiklis
* ''Introduction to Probability'' by Blitzstein and Hwang
* ''First Course in Probability'' by Ross
* ''Introduction to Probability Theory'' by Hoel, Port, and Stone
* ''The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)'' by Ash

=== Stochastic Processes ===
* ''Introduction to Stochastic Processes'' by Hoel, Port, and Stone
* ''Introduction to Probability Models'' by Ross
* ''Stochastic Processes'' by Ross
* ''A First Course in Stochastic Processes'' by Karlin and Taylor
* ''A Second Course in Stochastic Processes'' by Karlin and Taylor
* ''Probability and Random Processes'' by Grimmett and Stirzaker
* ''Stochastic Processes in Physics and Chemistry'' by Van Kampen

=== Mathematical Statistics ===
See the universal recommendations on [[Universal Material#Statistics|Statistics]].

==== Design of Experiments ====
See the universal recommendations on [[Universal Material#Design of Experiments|Design of Experiments]].

=== Measure Theoretic Probability Theory ===
* ''Probability with Martingales'' by David Williams
* ''Probability and Measure'' by Billingsley
* ''An Introduction to Probability Theory and Its Applications Vol. 2'' by Feller
* ''Probability Theory: A Comprehensive Course'' by Klenke
* ''Probability-1'' by Shiryaev

=== Stochastic Calculus ===
* ''Stochastic Differential Equations: An Introduction with Applications'' by Bernt Øksendal
* ''Diffusions, Markov Processes and Martingales: Volumes 1 and 2'' by Rogers and Williams
*# ''Foundations''
*# ''Itô Calculus''
* ''Brownian Motion and Stochastic Calculus'' by Ioannis Karatzas and Steven Shreve

== Combinatorics and Graph Theory ==

=== Primers in Combinatorics and Graph Theory ===
These 2 books are aimed at high school students with knowledge of elementary algebra to give them a taste of pure mathematics.
* ''Mathematics of Choice: Or, How to Count Without Counting'' by Ivan Niven
* ''Introduction to Graph Theory'' (Dover Books) by Trudeau

=== Introduction to Combinatorics ===
* ''Combinatorics: Topics, Techniques, Algorithms by Cameron'' [http://www.maths.qmul.ac.uk/~pjc/comb/ <sup><nowiki>[Website]</nowiki></sup>]
* ''A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory'' by Miklós Bóna
* ''A Course in Combinatorics'' by van Lint and Wilson

=== Introduction to Graph Theory ===
* [http://diestel-graph-theory.com/index.html ''Graph Theory''] by Diestel
* ''Modern Graph Theory'' by Bollobas

=== Enumerative Combinatorics ===
* ''A Course in Enumeration'' by Aigner
* ''Enumerative Combinatorics: Volumes 1 & 2'' by Stanley <sup>[http://www-math.mit.edu/~rstan/ec/ <nowiki>[Website]</nowiki>]</sup>

=== Extremal Combinatorics and Graph Theory ===
* ''Extremal Combinatorics: With Applications in Computer Science'' by Jukna [http://www.thi.informatik.uni-frankfurt.de/~jukna/EC_Book/ <sup><nowiki>[Website]</nowiki></sup>]
* ''Extremal Graph Theory'' (Dover Books) by Bollobas
* ''The Probabilistic Method'' by Alon and Spencer

=== Algebraic Graph Theory ===
* ''Algebraic Graph Theory'' by Biggs
* ''Algebraic Graph Theory'' by Godsil and Royle [http://www.math.uwaterloo.ca/~cgodsil/agt2/index.html <sup><nowiki>[Website]</nowiki></sup>]

==== Linear Algebraic Graph Theory ====
* ''Graphs and Matrices'' by Bapat
* ''Graph Spectra'' by Brouwer and Haemers [http://homepages.cwi.nl/~aeb/math/ipm/ <sup><nowiki>[Draft]</nowiki></sup>]
* ''An Introduction to the Theory of Graph Spectra'' by Cvetković, Rowlinson, and Simić
See also [[Computer Science and Engineering#Combinatorial Optimization and Network Flows|Combinatorial Optimization &amp; Network Flows]] and [[Computer Science and Engineering#Combinatorial Game Theory|Combinatorial Game Theory]] on the CS&E page.

== Abstract Algebra ==
Abstract algebra (also called modern algebra or just algebra) is the study of mathematical structures that consist of a set with algebraic rules defined on the set's elements. This enables us to prove general results that depend only on the particular rules the structures have and not a particular example structure (like the rationals, reals, quaternions, polynomials, matrices, or integers modulo n) we have in mind that satisfies those rules.

Abstract algebra is not to be mistaken with "college algebra" as that refers to the elementary algebra which is typically done in grade school. It's called "college algebra" because, well, nobody would pay for a course called ''The Algebra You Should Have Learned in High School But Were Too Much Of A Fuck Up To Do So''. If you're looking for resources on that, see the [[Mathematics#Precalculus|Precalculus]] section above.

=== Group Theory Teaser ===
These books are accessible enough to give freshmen or high school students a digestible taste of abstract math and build intuition for when they later get to algebra
* ''Groups and Their Graphs'' by Grossman and Magnus (sadly out of print)
* ''Visual Group Theory'' by Carter
* ''Groups and Symmetry'' by Armstrong

=== Undergraduate Algebra ===
* ''Algebra'' by Artin
* ''Topics in Algebra'' by Herstein
** Herstein's ''Abstract Algebra'' is an abbreviated version of his ''Topics'' for a one semester course
** Terminology is slightly old-fashioned
* ''Abstract Algebra'' by Dummit and Foote (more of an encyclopedic reference than a book)
* ''Algebra'' by Mac Lane and Birkhoff
* ''Abstract Algebra'' by Grillet
* ''A Course in Algebra'' by Vinberg
** Similar to Artin but friendlier overall
** Technically a graduate book but its level is closer to undergrad
* ''Undergraduate Algebra: A Unified Approach'' by Brešar
Herstein's ''Topics'' takes a fairly conventional approach to the subject while Artin's book does things in a rather unique and geometrical way. While both are well written texts on their own, but pairing them is very useful.

==== Survey of Abstract Algebra Applications ====
* ''[https://judsonbooks.org/aata/ Abstract Algebra: Theory and Application]'' by Tom Judson
** Also covers the theory; a good general introduction
* ''Applied Abstract Algebra'' by Lidl and Pilz
* ''Applications of Abstract Algebra with Maple and Matlab'' by Klima, Sigmon, and Stitzinger
* ''Topics in Applied Abstract Algebra'' by Nagpaul and Jain

See also [[Computer Science and Engineering#Finite Fields|Finite Fields]]

==== Matrix Groups ====
* ''Matrix Groups for Undergraduates'' by Tapp
* ''Naive Lie Theory'' by Stillwell

=== Graduate Algebra ===
* ''Basic Algebra I & II'' (Dover Books) by Jacobson
** Vol I is closer to 1st year books in terms of level
* Algebra: Chapter 0 by Aluffi
** Can be used as a first introduction as well
* Algebra by Hungerford
** Don't mistake it for his ''Abstract Algebra'' which is at a lower level
* ''Algebra'' by Lang <sup>[[https://math.berkeley.edu/~gbergman/.C.to.L/ A Companion to Lang's Algebra]]</sup>

==== Commutative Algebra ====
* ''Introduction to Commutative Algebra'' by Atiyah and Macdonald
* ''Basic Commutative Algebra ''by Singh
* ''[https://www.centerofmath.org/textbooks/commalgebra/index.html A Term of Commutative Algebra] ''by Altman and Kleiman
* ''Commutative Algebra with a View Towards Algebraic Geometry'' by Eisenbud

==== Group Theory ====
* ''An Introduction to the Theory of Groups'' by Rotman
* ''Finite Group Theory'' by Isaacs
* ''A Course in the Theory of Groups'' by Robinson

==== Noncommutative Rings ====

* ''Noncommutative Rings'' by Herstein
* ''A First Course in Noncommutative Rings'' by Tsit-Yuen Lam 
* ''Exercises in Classical Ring Theory'' by Tsit-Yuen Lam 
** Solutions and commentary to the above
* ''Lectures on Modules and Rings'' by Tsit-Yuen Lam
** Sequel to ''First Course in Noncommutative Rings'' but independent

* ''Exercises in Modules and Rings'' by Tsit-Yuen Lam 
** Solutions to the above

<br />
== Analysis ==
The origins of [https://en.wikipedia.org/wiki/Mathematical_analysis mathematical analysis] are found in the age old struggle of mathematicians to deal with the infinite and [https://en.wikipedia.org/wiki/Infinitesimal infinitesimal] dating all the way back to antiquity with the works of [https://en.wikipedia.org/wiki/Eudoxus_of_Cnidus Eudoxus of Cnidus] and [https://en.wikipedia.org/wiki/Archimedes Archimedes of Syracuse]. With the piecemeal development of calculus by [https://en.wikipedia.org/wiki/Bonaventura_Cavalieri Cavalieri], [https://en.wikipedia.org/wiki/Blaise_Pascal Pascal], [https://en.wikipedia.org/wiki/Pierre_de_Fermat Fermat], [https://en.wikipedia.org/wiki/Ren%C3%A9_Descartes Descarte], [https://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz Leibniz], [https://en.wikipedia.org/wiki/Leonhard_Euler Euler], [https://en.wikipedia.org/wiki/Joseph-Louis_Lagrange Lagrange], [https://en.wikipedia.org/wiki/Joseph_Fourier Fourier] and many others, calculus gradually showed itself to be a powerful yet deeply troubled tool. As much as mathematicians tried, they struggled with clearly defining key stumbling points: the concept of an infinitesimal number smaller than 1/n for all integers n yet nonzero in a logically consistent manner, the concept of infinite approach, division by 0, and the rules in which an infinite series may be manipulated and examined. These were not just pointless trifles that could be brushed off as more philosophy than math but of increasing practical importance. As time went on, many counterexamples (and not just [https://en.wikipedia.org/wiki/Pathological_(mathematics) pathological] ones) where the naive application of the methods of calculus would produce erroneous results cast a shadow on the validity of all other results of calculus. Many critics wanted to end its study altogether and relentlessly mocked the concept of infinitesimals as "the ghosts of departed quantities". Since the triumphs for calculus were both numerous and far reaching, mathematicians strongly sought to make the results of calculus proven rather than discarding the subject all together. This situation was finally resolved only in the early 19th century with the work of [https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy Cauchy] and [https://en.wikipedia.org/wiki/Karl_Weierstrass Weierstrass] and the [https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit ε-δ definition of a limit] (which would ironically kill off infinitesimals until the 1960s and the advent [https://en.wikipedia.org/wiki/Non-standard_analysis nonstandard analysis]) that birthed the new field of analysis. This sparked off a massive revolution in mathematics and the field of analysis quickly exploded into various distinct but interconnected directions.

Students just finishing the study of calculus and basic proofs often fail to realize the sheer importance of careful work in analysis and scoff the whole subject off as merely "intellectual or autistic masturbation". This mentality comes from being coddled with the toy-problems you see in calculus that are selected to hide any possible nastiness that comes from complicated situations that frequently arise in science and engineering. Even if the student is aware of the importance of being careful, they are often insulted when forced to work through "obvious" theorems. The problem here is that many results in analysis that seem obvious are frankly very difficult to prove (for example see the [https://en.wikipedia.org/wiki/Jordan_curve_theorem Jordan curve theorem]) or even dead wrong. In order to gain the ability to prove important and powerful theorems hidden away in analysis, students need plenty of practice working through basic problems to gain familiarity and mathematical maturity to move on to difficult work even if this means you need to spend time proving that "every open ball is open".

A good reference to keep with you and refer to often is ''Counterexamples in Analysis'' by Gelbaum and Olmsted published by Dover Books.

=== Inequalities ===
A lot of the exercises in analysis often boil down to spamming the triangle inequality until you get the result you want. If you haven't done much work with inequalities since grade school, practicing them can make the subject seem vastly easier.
* ''The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities'' by Steele
* ''Inequalities'' by Hardy, Littlewood, and Polya

=== Real Analysis (Metric Space based) ===
This is where the results of single variable calculus are finally made both rigorous and generalized.

The gold standard for the subject is the first 8 chapters of Rudin's ''Principles of Mathematical Analysis'' whose slick proofs and challenging exercises can't be beat. Chapters 9 and 10 of Rudin on multivariable analysis are bit sparse to learn from so you're better off moving fuller treatment on analysis on manifolds (see below) to learn from. The final chapter 11 is completely skippable as other books are far better in their treatment of Lebesgue integrals/measure theory than Rudin's brief survey. Apostol's ''Mathematical Analysis'' goes through a bit more material than Rudin, gives more worked out proofs, and has relatively easier problems. If you're struggling with Rudin, give Abbott or Apostol a try. Zorich's ''Mathematical Analysis I & II'' starts lower level than Rudin but ends on much higher level covering many additional topics including manifolds. The price paid is that his books is quite longer than Rudin and larger time investment. 

A more classical treatment of analysis, but starting from the Lebesgue integral is ''An Introduction to Classical Real Analysis'' by Hewitt and Stromberg. This book is like how analysis was done a hundred years ago but with a more sophisticated integral. 

Another alternative treatment that you could read afterwards is [http://www.math.harvard.edu/~shlomo/ Sternberg and Loomis' ''Advanced Calculus''] . Differentiation is done on normed vector spaces, and integration is done using the theory of content; multivariable stuff is done via manifolds. It also covers some physics. Its abstract approach makes it more challenging than the other books.

=== Analysis on Manifolds ===
This is the study of analysis on multidimensional spaces making multivariate and vector calculus rigorous and pushing the subject further. A good grounding in linear algebra is required. Going to a book like Tu's Manifolds instead of one of the below can also be viable.
* Munkres' ''Analysis on Manifolds''
* Spivak's ''Calculus on Manifolds''
* do Carmo's ''Differential Forms and Applications''

=== Fourier Analysis ===
* ''Fourier Series'' (Dover) by Tolstov
* ''Fourier Analysis: An Introduction'' by Stein & Shakarchi
* ''Fourier Analysis and its Applications'' by Folland
* ''Fourier Analysis'' by Körner
* ''Fourier Series and Integrals'' by Dym and McKean

=== Complex Analysis ===
* ''Complex Analysis'' by Stein & Shakarchi
* ''Complex Analysis''<sup>[[https://www.math.ucla.edu/~twg/CA.book.html website]][[https://www.math.ucla.edu/~twg/CA.book.html guide]]</sup>'''' by Gamelin
* ''Functions of One Complex Variable ''by Conway
* ''Complex Analysis'' by Ahlfors
* ''Visual Complex Functions: An Introduction With Phase Portraits'' by Wegert
* ''Complex Analysis'' by Serge Lang
** Also check out the solutions guide by Shakarchi
* ''Complex Analysis in One Variable'' by Narasimhan (advanced)

=== Graduate Real Analysis ===
* ''Real Analysis: Measure Theory, Integration, and Hilbert Spaces'' by Stein & Shakarchi
* ''Real Analysis'' by Royden
* ''Real Analysis: Modern Techniques and Their Applications'' by Folland
* ''Real and Complex Analysis'' by Rudin
* ''Real Analysis'' by Loeb (also includes some nonstandard analysis) 
* ''Introduction to Modern Analysis'' by Kantorovitz (very broad)

=== Functional Analysis ===
* ''Introduction to Topology and Modern Analysis'' by George Simmons (very few prerequisites)
* ''Functional Analysis: Introduction to Further Topics in Analysis'' by Stein & Shakarchi
* ''Functional Analysis by Lax''
** Includes several historic notes of the subject and the fate of its researchers during WWII
* ''Functional Analysis'' by Rudin
* ''Linear Operators'', 3 Volume Set by Nelson Dunford & Jacob T. Schwartz

=== Abstract Harmonic Analysis ===
* ''Introduction to Abstract Harmonic Analysis'' (Dover Books) Loomis
* Fourier Analysis on Groups (Dover Books) by Walter Rudin
* A Course in Abstract Harmonic Analysis by Folland

=== Nonstandard Analysis ===
* ''Nonstandard Analysis (Dover Books)'' by Robert
* ''Applied Nonstandard Analysis (Dover Books) ''by Martin Davis
* ''Nonstandard Analysis: Theory and Applications'' by Arkeryd, Cutland and Henson
* ''Real Analysis Through Modern Infinitesimals'' by Nader Vakil (suitable for undergraduates)
* ''Lectures on the Hyperreals: An Introduction to Nonstandard Analysis ''by Goldblatt
* ''Nonstandard Analysis for the Working Mathematician ''by Loeb and Wolff
* ''Nonstandard Methods in Stochastic Analysis and Mathematical Physics (Dover Books) ''by Albeverio, Hoegh-Krohn, Fenstad, and Lindstrom
* ''Non-standard Analysis ''by Abraham Robinson (The original book on the subject)

=== Sets of Books on Analysis ===
Because of how broad and how essential the subject of analysis is, there are quite a few series of volumes that attempt to guide the reader through the landscape. Not all of these books necessary form a series, but they do form a logical grouping by the same authors.

As mentioned before: Baby, Papa/Big, and Grandpa Rudin.
# ''Principles of Mathematical Analysis''
# ''Real and Complex Analysis''
# ''Functional Analysis''
The most well-known series in analysis is the Princeton Lectures in Analysis by Stein and Shakarchi.
# ''Fourier Analysis: An Introduction''
# ''Complex Analysis''
# ''Real Analysis: Measure Theory, Integration, and Hilbert Spaces''
# ''Functional Analysis: Introduction to Further Topics in Analysis''
Terence Tao has also written quite extensively on introductory analysis.
# ''Analysis I''
# ''Analysis II''
# ''An Introduction to Measure Theory''
As mentioned, Zorich's books only come in two volumes, but they cover a lot.
# ''Mathematical Analysis I''
#* Sets, real numbers, limits, continuous functions
#* Differential calculus in one and several variables
#* Riemann integration
# ''Mathematical Analysis II''
#* Metric and topological Spaces
#* Differential calculus on normed vector spaces
#* Multiple integrals
#* Differential Forms and integration
#* Classical vector analysis and physics
#* Fourier Analysis
Amann and Escher treat introductory analysis from a highly abstract point of view. Commutative diagrams galore. Only for the daring or the prepared.
# ''Analysis I''
#* Sets and logic, abstract algebra, linear algebra, real and complex numbers
#* Convergence, Banach spaces, series
#* Topology
#* Differential calculus
#* and more
# ''Analysis II''
#* Integration with Banach spaces
#* Differential calculus and an introduction to manifolds
#* Complex analysis with Pfaff forms
# ''Analysis III''
#* Measure theory and integration
#** includes Bochner-Lebesgue integral
#* Differentiation on manifolds
#** with some Riemannian geometry
#* Integration on manifolds
[http://www.math.stonybrook.edu/~aknapp/ Knapp (free)]
# ''Basic Real Analysis''
# ''Advanced Real Analysis''
Comprehensive Course in Analysis by Barry Simon
# ''Real Analysis''
# ''Complex Analysis 2A and 2B''
## ''Basic Complex Analysis''
## ''Advanced Complex Analysis''
# ''Harmonic Analysis''
# ''Operator Theory''

== Topology ==

=== Point-set Topology ===
* ''Topology ''by James Munkres (The standard for most topology courses)
* ''Introduction to Topological Manifolds ''by John Lee
* ''Elementary Topology (Dover Books)'' by Michael Gemignani
* ''General Topology (Dover Books) by ''Stephen Willard (A bit more difficult than the above)
A great supplement to any General Topology book is ''Counterexamples in Topology'' (Dover Books) by Steen and Seebach. See also [https://topology.jdabbs.com/ Pi-Base], which is a digital, searchable version with more counterexamples.

=== Algebraic Topology ===
* [http://pi.math.cornell.edu/~hatcher/AT/ATpage.html ''Algebraic Topology'']'' ''by Hatcher
* ''Differential Forms in Algebraic Topology ''by Bott and Tu
* ''An Introduction to Algebraic Topology ''by Rotman
* ''A Concise Course in Algebraic Topology ''by May (Advanced)

=== Differential Topology ===
* ''Topology from the Differential Viewpoint'' by Milnor (very short)
* ''Introduction to Differential Topology ''by Bröcker and Jänich
* ''Differential Topology'' by Victor Guillemin and Alan Pollack
* ''Differential Topology'' by Hirsch (More advanced)

=== Geometric Topology ===

==== Low-Dimensional Topology ====
* ''Low-Dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots'' by Francis Bonahon (elementary presentation)
* ''Three-Dimensional Geometry and Topology'' by Thurston and Levy

==== Knot Theory ====
* ''Knot Theory'' by Livingston (Primer)
* ''Knots and Physics''; ''On Knots''; Formal Knot Theory (Dover Books) by Kauffman
* ''An Introduction to Knot Theory'' by Lickorish
* ''Knots and Links'' by Rolfsen
* ''Knot Theory and Its Applications''; ''A Study of Braids'' by Kunio Murasugi
* ''Knots, Links, Braids and 3-Manifolds: An Introduction to the New Invariants in Low-Dimensional Topology'' by Prasolov and Sossinsky

== Geometry ==

=== Classical Geometries ===
* ''Geometry by Its History'' by Ostermann and Wanner
** From the Babylonians up to the 20th century
* ''Geometry: Euclid and Beyond'' by Hartshorne
** Supplementary text to Euclid
* ''Non-Euclidean Geometry'' by Coxeter
* ''Geometry'' by Audin
** Transformation geometry
** Projective geometry
** Envelopes and evolutes 
** Touches on curves and surfaces
* [https://mphitchman.com/ ''Geometry with an Introduction to Cosmic Topology'' by Hitchman]
* ''Geometries'' by A. B. Sossinsky
* ''Introduction to Classical Geometries'' by Galarza and Seade

* ''Geometry From a Differentiable Viewpoint'' by McCleary
** Also covers geometry of curves and surfaces and touches on manifolds. The book of P.M.H. Wilson is similar

* Gibson's Trilogy
*# ''Elementary Euclidean Geometry: An Undergraduate Introduction''
*# ''Elementary Geometry of Differentiable Curves: An Undergraduate Introduction''
*# ''Elementary Geometry of Algebraic Curves: An Undergraduate Introduction''
* Borceux's Trilogy
*# ''An Axiomatic Approach to Geometry''
*# ''An Algebraic Approach to Geometry''
*# ''A Differential Approach to Geometry''
* ''Geometry I and II'' by Berger (more advanced)

=== Modern Differential Geometry ===

==== Smooth Manifolds ====
* ''Introduction to Smooth Manifolds'' by John Lee
* ''An Introduction to Manifolds'' by Loring W. Tu 
* ''A Short Course in Differential Geometry and Topology'' by Fomenko and Mishchenko 
* ''A Comprehensive Introduction to Differential Geometry, Volume 1'' by Spivak

==== Riemannian Geometry ====
* ''Riemannian Geometry'' by do Carmo
* ''Riemannian Manifolds: An Introduction to Curvature'' by John Lee
* ''Introduction to Riemannian Manifolds'' by John Lee
* ''Differential Geometry: Connections, Curvature, and Characteristic Classes'' by Loring Tu
* ''Riemannian Geometry and Geometric Analysis'' by Jost 

==== Synthetic Differential Geometry ====
This is a reformulation of differential geometry, but with a foundation in category and topos theory. Do note that the law of the excluded middle does not hold in this theory.
* ''Synthetic Geometry of Manifolds'' by Anders Kock
* ''Basic Concepts of Synthetic Differential Geometry'' by René Lavendhomme
* ''Synthetic Differential Geometry'' by Anders Kock
Smooth infinitesimal analysis is analysis with a similar foundation.
* ''A Primer of Infinitesimal Analysis'' by John L. Bell
* ''Models for Smooth Infinitesimal Analysis'' by Ieke Moerdijk and Gonzalo E. Reyes

=== Algebraic Geometry ===

==== Overview ====
* ''An Invitation to Algebraic Geometry'' by Karen Smith, Lauri Kahanpää, Pekka Kekäläinen, and William Traves

==== Introductory Algebraic Geometry without Schemes ====
* ''Lectures on Curves, Surfaces and Projective Varieties'' by Beltrametti et al.
* ''Algebraic Geometry: A Problem-Solving Approach'' by Garrity et al.
* ''Basic Algebraic Geometry 1: Varieties in Projective Space'' by Shafarevich
* ''Algebraic Geometry, A First Course'' by Harris
* ''Algebraic Geometry I: Complex Projective Varieties'' by Mumford

==== With Schemes ====
* ''A Royal Road to Algebraic Geometry ''by Holme
* ''Algebraic Geometry I: Schemes, With Examples and Exercises ''by Görtz and Wedhorn
* ''[http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf Foundations of Algebraic Geometry ]''[http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf by Vakil]
* ''The Geometry of Schemes (Graduate Texts in Mathematics) ''by Harris and Eisenbud
* ''Algebraic Geometry and Commutative Algebra'' by Bosch (CA+AG in one book!)
* ''Introduction to the Theory of Schemes'' by Manin and Leitis
* ''Basic Algebraic Geometry 2: Schemes and Complex Manifolds'' by Shafarevich
* ''3264 and All That: A Second Course in Algebraic Geometry ''by Harris and Eisenbud
* ''Algebraic Geometry II ''by Mumford

==== More Serious Algebraic Geometry ====
* ''Algebraic Geometry ''by Robin Hartshorne (The standard)
* ''Algebraic Geometry I-III'' by Kenji Ueno
* ''Principles of Algebraic Geometry'' by Griffiths and Harris (Complex Geometry)

* ''Intersection Theory ''by Fulton

==External Links==

=== Book Recommendations ===
[[Math Textbook Recommendations]]

[https://www.maa.org/press/maa-reviews/browse MAA Book Reviews]

[https://www.maa.org/press/maa-reviews/browse?field_tags_tid=All&field_bll_rating_tid=any_bll&field_maa_review_value=All The Basic Library List: Mathematical Association of America's Recommendations for Undergraduate Libraries]

[https://www.ocf.berkeley.edu/~abhishek/chicmath.htm Chicago undergraduate mathematics bibliography]

* Note: The original author now also hosts their bibliography on [https://github.com/ystael/chicago-ug-math-bib GitHub]. Anyone may contribute to it as they see fit.

[https://web.archive.org/web/20150423201942/http://www.amazon.com/gp/richpub/syltguides/fullview/r1ge1p236k3ysv Amazon's "So you'd like to... Learn Advanced Mathematics on Your Own"]

[http://www.topology.org/tex/conc/differential_geometry_books.html Differential geometry textbook recommendations and historically interesting works]

[http://people.math.gatech.edu/~cain/textbooks/onlinebooks.html A list of free online math textbooks]

[https://sites.google.com/site/scienceandmathguide/subjects/mathematics Old /sci/ guide] ( https://sites.google.com/site/scienceandmathguide/ )

[https://math.stackexchange.com/questions/tagged/book-recommendation Stack Exchange Math Book Recommendation]

[https://www.physicsforums.com/forums/science-and-math-textbooks.21/ Physics Forums - Science and Math Textbooks Forum]

[https://www.quantstart.com/articles/How-to-Learn-Advanced-Mathematics-Without-Heading-to-University-Part-1 QuantStart - How to Learn Advanced Mathematics part1] , [https://www.quantstart.com/articles/How-to-Learn-Advanced-Mathematics-Without-Heading-to-University-Part-2 part 2], [https://www.quantstart.com/articles/How-to-Learn-Advanced-Mathematics-Without-Heading-to-University-Part-3 part 3]

[https://www.springer.com/series/3423?detailsPage=titles Springer Undergraduate Mathematics Series]

[https://www.springer.com/series/136?detailsPage=titles Springer Graduate Texts in Mathematics]

[https://www.reddit.com/r/math/comments/8ewuzv/a_compilation_of_useful_free_online_math_resources/ Compilation of Useful, Free, Online Math Resources]

[http://web.evanchen.cc/coursework.html Evan Chen's Mathematics Coursework and Lecture Notes]

[https://bookstore.ams.org/ American Mathematical Society Bookstore]

[https://hbpms.blogspot.com/ How to Become a Pure Mathematician (or Statistician)]

[http://www.tuloomath.com/tuloomath-mathematics-site/tuloomath-toc/mathemagicians-comprehensive-online-mathematics-library/ TULOO (an unreadable but somewhat useful blog)]

[https://www.logicmatters.net/tyl/ Teach Yourself Logic (2020)]

[http://www.geometry.org/ Alan U. Kennington (differential geometry, logic, and advanced physics books)]

=== Reference ===
[http://mathworld.wolfram.com/ Wolfram MathWorld]

[https://proofwiki.org/wiki/Main_Page ProofWiki]

=== Tools and Apps ===
[http://www.wolframalpha.com/ Wolfram Alpha] (use this before making threads on /sci/ asking for help with integrals, matrix computations etc.)

[https://wolfreealpha.gitlab.io/input Wolfree Alpha] - Provides free access to Wolfram Alpha's step by step solutions feature.

[http://mathinsight.org/thread/multivar#s15 Readings and java simulations for multivariable calculus]

[[Category:/sci/]]
[[Category:Generals]]