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Learn college-level mathematics

A guide by a reader

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A good book always helps in learning. When it comes to mathematics in college, by the time you get out, you will need to know all of the Basic Level (described below), some of the Intermediate Level, and hopefully at least one topic from the Advanced Level.

Disclaimer: I am not the author of any of these books, but I do know many of the authors in the list.

Basic Level:
1) Calculus.
2) Linear Algebra.

Intermediate level:
3) Analysis.
4) Algebra (Rings, Groups).
5) Probability.
6) Ordinary Differential Equations.
7) Number Theory.
8) Discrete Mathematics.
9) Advanced Calculus.
10) Differential Geometry.

Advanced Level:
11) Algebraic Topology.
12) Algebra (Fields, Galois Theory).
13) Partial Differential Equations.
14) Functional Analysis.
15) Complex Variable (could be intermediate).

If you are a student in college, your instructor will be choosing the book for the course, leaving you no choice, but you may still be able to borrow the following suggested reading from the library.

1) Calculus books these days are almost all equivalent to each other. They all will have plenty of worked-out examples, and plenty of exercises for you to do. My suggestion is that you read Spivak's Calculus book on the side. Calculus, 4th edition

2) This is a subject that needs to be learned at many levels. The book by Lay, Linear Algebra and Its Applications, 3rd Updated Edition (Book & CD-ROM), will give you the basics. For a slightly more abstract view, try Strang's Linear Algebra and Its Applications, 4th Edition. For a more abstract treatment, I recommend Linear Algebra (2nd Edition), Linear Algebra and Its Applications, and/or Finite-Dimensional Vector Spaces (Undergraduate Texts in Mathematics), but these may be tough to read on your own. (Still, check them out.)

For the intermediate level, some of the texts become more abstract, perhaps necessarily.

3) The classic text is Rudin's Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics), but it is not an easy read. If your instructor did not select it, perhaps you shouldn't try reading it on your own. (But do give it a go.) Are there good alternatives? You could try Fomin & Kolmogorov Introductory Real Analysis (Dover Books on Mathematics), Marsden Elementary Classical Analysis, Beals Analysis: An Introduction, or Reed Fundamental Ideas of Analysis.

4) I recommend Gallian's Contemporary Abstract Algebra, but also Herstein Abstract Algebra or Topics in Algebra, 2nd Edition, and Dummit & Foote Abstract Algebra, 3rd Edition. With these last texts we are entering graduate school material.

5) For probability I recommend Ross' A First Course in Probability (8th Edition). If you really want something special, then Feller An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition is your book, but it is a harder read.

6) I like Borrelli & Coleman Differential Equations: A Modeling Perspective, and also Boyce & DiPrima Elementary Differential Equations and Boundary Value Problems. For a slightly more abstract treatment, Hirsch & Smale Differential Equations, Dynamical Systems, and Linear Algebra (Pure and Applied Mathematics (Academic Press), 60.) is good.

7) I like the little book Multiplicative Number Theory (Graduate Texts in Mathematics) (v. 74) by Davenport (but please buy the cheaper Dover edition!), but perhaps you will be better served by Silverman's Friendly Introduction to Number Theory, A (3rd Edition).

8) I strongly suggest Concrete Mathematics: A Foundation for Computer Science (2nd Edition). A more standard text would be Scheinerman's Mathematics: A Discrete Introduction.

9) Do people still teach Advanced Calculus? The books by Buck Advanced Calculus, Third Edition, Widder Advanced Calculus, Kaplan Advanced Calculus (5th Edition) are books I can recommend, but I am sure there are good options.

10) The cheap Lectures on Classical Differential Geometry: Second Edition (Dover Books on Mathematics) by Struik is good, the expensive Differential Geometry of Curves and Surfaces by do Carmo is excellent.

For the advanced level, the books are necessarily harder to read, being graduate level stuff.

11) Massey's A Basic Course in Algebraic Topology is my suggestion.

12) Galois Theory, Third Edition (Chapman Hall/Crc Mathematics) is a good option for Galois Theory.

13) I suggest Partial Differential Equations: An Introduction by Strauss.

14) Functional Analysis by Lax.

15) Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics (3rd Edition) by Saff & Snider. There are many good books around, and this book is at an intermediate level.

A book I want to suggest outside these narrow categories is Hardy's A Course of Pure Mathematics Centenary edition (Cambridge Mathematical Library), because I like it. These other two books of which he is also an author are also excellent: Inequalities (Cambridge Mathematical Library), An Introduction to the Theory of Numbers.

Finally, a suggestion that may surprise some. The Schaum's Outlines collection has some excellent titles that you should consult, including their volumes in calculus, differential equations, complex variables (really good), and probability. Don't turn your nose down at these!

Good luck!

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Last updated: 7/11/10
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