Highly Recommended Math Books for University Self Study

Recently, a viewer of my website asked if I was able to suggest any undergraduate level university textbooks for self study that follows the university curriculum.

Self-study is challenging but not impossible. Choosing a good and appropriate book of the right level is of crucial importance. For instance, for beginners to Calculus, I wouldn’t recommend Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) by Rudin. It is simply too difficult for beginners or even intermediate students. Any book by Bourbaki is also not suitable for beginners, for instance.

Update: I recently found a book that is a better alternative to Rudin: Mathematical Analysis, Second Edition by Apostol! Many online sources have very positive reviews on Apostol’s Analysis book. I have read it and found it much more readable than Rudin.

I would like to suggest the following books (mainly for Pure Mathematics). Ideally, the motivated student is able to self study and obtain the knowledge equivalent to a 4 Year course at a university.

The recommendations are divided into Year 1, Year 2, Year 3 and Year 4.

If you have any other recommendations, please feel free to comment below!

Year 1

Introduction to Pure Math and Proofs:

How to Prove It: A Structured Approach

Calculus:

Thomas’ Calculus (12th Edition)

Linear Algebra:

Linear Algebra and Its Applications, 4th Edition

Multivariable Calculus:

Thomas’ Calculus, Multivariable (13th Edition)

Year 2

Linear Algebra II (Second Year Course):

Linear Algebra, 4th Edition

Analysis I: 

Introduction to Real Analysis

Abstract Algebra I:

A First Course in Abstract Algebra (3rd Edition)

This book will be an introduction to Group Theory.

Probability:

Introduction to Probability, 2nd Edition

Analysis II:

Calculus, 4th edition

(Note: Despite the title “Calculus”, this book is actually a rather rigorous book on Analysis, suitable as a second course textbook)

Complex Analysis I:

Complex Variables and Applications (Brown and Churchill)

Year 3

Analysis III:

Introductory Real Analysis (Dover Books on Mathematics)

ODE (Ordinary Differential Equations):

Ordinary Differential Equations (Dover Books on Mathematics)

Graph Theory:

A First Course in Graph Theory (Dover Books on Mathematics)

Algebra II:

Abstract Algebra, 3rd Edition

Algebra II will usually be a course on Rings, Modules.

(Note: You can use this book for learning Galois Theory too)

Differential Geometry:

Differential Geometry of Curves and Surfaces

Year 4 

Number Theory:

An Introduction to the Theory of Numbers

Galois Theory:

Abstract Algebra, 3rd Edition

(Note: Same textbook as for Algebra II)

PDE (Partial Differential Equations):

A First Course in Partial Differential Equations: with Complex Variables and Transform Methods (Dover Books on Mathematics)

Logic:

A Beginner’s Guide to Mathematical Logic (Dover Books on Mathematics)

Functional Analysis:

Introductory Functional Analysis with Applications

Topology:

Topology (2nd Edition)

(Note: See also my book review on Topology by Munkres)

Measure and Integration:

The Elements of Integration and Lebesgue Measure

Congratulations for reaching the bottom of this long list!

All the best for your studies in Mathematics. 🙂