My Notes on Mathematics Textbooks

Created in August 15, 2023

2023   ·   math   ·   study

I present my reviews of mathematics textbooks with personal prefernces on gaining complete understanding and intuitive insights. Note that I did not complete reading all of the books listed below, so please only serve this note as an auxiliary resource for you to find your ideal self-studying material instead of formal critique.

Linear Algrbra

  1. Linear Algebra, 4th edition by Stephen H. Friedberg et al.
    • Level: Introductory - Intermediate
    • Readability: 3/5
    • Personal Preference: 4/5
    • Friedberg’s Linear Algebra is one of the most widely used textbooks at the undergraduate level. It features a rigorous writing style, which can be somewhat challenging for beginners. However, it is worth investing time in reading, as it provides a solid foundation in linear algebra within a single book.
  2. Linear Algebra and Its Applications, 4/e by Gilbert Strang
    • Level: Introductory
    • Readability: 3.5/5
    • Personal Preference: 2.5/5
    • I used Strang’s textbook for an applied mathematics course. Some find it easy to follow, and I agree, although the notations can be confusing at times. The writing style is suitable for those who want to apply linear algebra in fields like physics and engineering, though it leans towards practical applications and may lack some mathematical intuition. Fortunately, there are ample online resources that complement Strang’s textbook.

Analysis

  1. Real Mathematical Analysis by Charles C. Pugh
    • Level: Introductory - Intermediate
    • Readability: 4/5
    • Personal Preference: 4.5/5
    • Pugh’s textbook is my favorite for analysis. I discovered it while looking for an alternative to Rudin and preferred it over Goursat within the first few pages. Some criticize it for its ambiguity in definitions, but I appreciate its smooth flow and ample examples. The contexts and exercises are insightful, nurturing a strong sense of mathematical intuition. Highly recommended for self-study.
  2. Principles of Mathematical Analysis by Walter Rudin
    • Level: Introductory - Advance
    • Readability: 2/5
    • Personal Preference: 3/5
    • A classic and rigorous textbook. However, after covering the entire material, it can be challenging to construct an overall understanding of the field of analysis. A better appreciation of the textbook often comes after working through the extensive and demanding exercises.

Geometry and Differential Equations

  1. Introduction to Tensor Analysis and the Calculus of Moving Surfaces by Pavel Grinfeld
    • Level: Introductory
    • Readability: 4/5
    • Personal Preference: 2/5
    • If one is seeking a book to understand general relativity, I would recommend going directly to Sean Carroll’s Spacetime and Geometry.
  2. Green’s Functions with Applications by Dean G. Duffy
    • Level: Advance
    • Readability: 4/5
    • Personal Preference: 4.5/5
    • It is well-suited for those already familiar with both ordinary and partial differential equations. Instead of delving into higher-dimensional problems with Bessel and Legendre functions, the Green’s functions approach emphasizes the concept of “propagators,” which is conceptually intertwined with the path integral formulation in quantum mechanics. Highly recommended for those who are interested in dynamical systems.

Statistics

  1. Probability, Statistics and Econometrics by Oliver Linton
    • Level: Introductory - Intermediate
    • Readability: 4/5
    • Personal Preference: 5/5
    • It covers foundational knowledge and essential content for econometrics. While the explanations are relatively rigorous compared to other introductory statistics textbooks, the book remains highly readable. However, I would recommend supplementing your learning with additional course materials to build a more complete understanding of the subject.
  2. Theoretical Statistics: Topics for a Core Course by Robert W. Keener
    • Level: Intermediate - Advance
    • Readability: 5/5
    • Personal Preference: 4.5/5
    • A concise textbook on theoretical statistics. The writing pace is fast, beginning with basic measure theory and quickly progressing to exponential family and sufficient statistics. I recommend this book for individuals who are already familiar with statistics and are seeking an advanced textbook to strengthen their theoretical background.
  3. Statistical Inference by G. Casella and R. L. Berger
    • Level: Intermediate - Advance
    • Readability: 3/5
    • Personal Preference: 3/5
    • This is a classic textbook with rigorous mathematical content. However, it is heavily focused on mathematical proofs and statements, making it challenging to follow and apply to statistical problems. It may be most suitable for mathematics majors, as it builds on probability theory, which forms the foundation for their approach to statistics. It’s also a good choice for those seeking a comprehensive and solid understanding of all aspects of statistical inference.
  4. A First Course in Bayesian Statistical Methods by Peter D. Hoff
    • Level: Introductory
    • Readability: 5/5
    • Personal Preference: 5/5
    • One of my most enjoyable textbooks for self-study. It is suitable for those who have a grasp of conditional probability and some statistical background. It illuminates the thought processes behind why one would use and apply Bayesian methods to analytical and practical problems, providing the necessary techniques to develop a taste for Bayesian statistics.
  5. The Bayesian Choice by Christian P. Robert
    • Level: Advance
    • Readability: 2.5/5
    • Personal Preference: 3/5
    • An advanced textbook on Bayesian statistics, requiring a strong background in probability theory. The first half of the book presents theoretical foundations that connect information theory with decision theory for prior distribution, making it a valuable read for those with a specific interest in justifying the use of Bayesian methods.
  6. Bayesian Data Analysis, 3rd edition by Andrew Gelman et al.
    • Level: Intermediate
    • Readability: 3.5/5
    • Personal Preference: 4/5
    • The book covers both basic and advanced techniques in Bayesian data analysis, including meta-analysis, hierarchical models, multi-parameter estimation, and posterior MCMC methods. The exercises, encompassing both derivation and numerical problems, are robust and offer a deeper understanding, yet they are not overly challenging for those who are relatively new to Bayesian statistics.