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Table of contents

If the infinitesimal approach intrigues you but you've already done a course in calculus or currently reading through another page book on calculus, Infinitesimal Calculus Dover Books by Henle and Kleinberg is a nice short 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. 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.

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. 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. II" by Richard Courant and Fritz John the paperback is split into 2 parts to continue on with the standard rigorous approach. The following texts take a slightly more rigorous 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.

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.

Mathematics Textbooks for Self Study A Guide for the Autodidactic

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. 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. The necessary prerequisite knowledge is just precalculus but some calculus knowledge is useful and may appear in a few examples. 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 2 '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.

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 "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 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 2 aforementioned books above are far cheaper thanks to them being published by Dover and the last 2 are free.

Bonus is they are all free. 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. After reading one of them, you'll be more than ready to move onto advanced Numerical Linear Algebra and Matrix Analysis textbooks.

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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:. Of course there's also "Linear Algebra done Right" by Axler and on the one hand, the stuff he does is great 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 an unique development of the subject. It is good for learning linear algebra for the first time if you're a hot shot freshman, 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". The term Advanced Calculus has come to mean different things over the course of the past century.

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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.

A good supplement to any of the above is Visual Complex Analysis by Needham. 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.

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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:. 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.

The 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:.

For more mathematical detailed books see the Fourier Analysis books below. 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. 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 [1] [2] [3] trick and using it to his advantage over and over again.

Mathematics Textbooks for Self Study -- A Guide for the Autodidact

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 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 and its applications were examined by the electrical engineer Oliver Heaviside in the s but the first textbook on the subject was only published in 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. 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.

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Fuller undergraduate treatments can be found with:. Once you have the required background in analysis, you can really study the meat of PDEs in detail with the following:. 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 wrong or at the very least hold most of the meat of the proof in proving it. 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:. 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 :. 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.

Combinatorics, graph theory , linear algebra involving vector spaces , and 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 LaTeX pronounced "lay-tech" to typeset your proofs and future papers in. This is the formal study of the Foundations of Mathematics using mathematics, particularly on Set Theory which much of mathematics is built on and 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.

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. Starting with the unique factorization property of the integers, the theme of factorization is revisited several times throughout the book to illustrate how the ideas handed down from Euclid continue to reverberate through the subject. In particular, the book shows how the Fundamental Theorem of Arithmetic, handed down from antiquity, informs much of the teaching of modern number theory.

The result is that number theory will be understood, not as a collection of tricks and isolated results, but as a coherent and interconnected theory.

Abstract Algebra

A number of different approaches to number theory are presented, and the different streams in the book are brought together in a chapter that describes the class number formula for quadratic fields and the famous conjectures of Birch and Swinnerton-Dyer. The final chapter introduces some of the main ideas behind modern computational number theory and its applications in cryptography.

Written for graduate and advanced undergraduate students of mathematics, this text will also appeal to students in cognate subjects who wish to be introduced to some of the main themes in number theory. This is helped along by a good sized bibliography plus many problems ….