What is a highly recommended book for studying abstract algebra?

[micromass]

Welcome to the Physicsforums bibliography thread. The purpose of this thread is to make a list of science books of various different topics. What follows is the list of science books sorted by their different categories. Clicking on the name of a book brings up all the relevant information including comments from other users.

Note: in order to add books or comments to this thread, please see the discussion thread: https://www.physicsforums.com/showthread.php?t=638548

Mathematics

• Algebra
  Roman - Advanced Linear Algebra
• Analysis
  Rudin - Principles of Mathematical Analysis
• Applied Mathematics and Mathematical Methods
• Calculus
  Spivak - Calculus
  Apostol - Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra
• Discrete Mathematics
• Foundations
• Geometry
• High-School Mathematics
  Lang - Basic Mathematics
• Number Theory
• Probability and Statistics
• Topology
  Lee - Introduction to Topological Manifolds
• Other Topics

Physics

• Astronomy and Cosmology
• Classical Mechanics
• Classical Optics
  Hecht (or Hecht and Zajak) - Optics
• Condensed Matter
• Electromagnetism
  Franklin - Classical Electromagnetism
• Experimental Methods
• Nuclear and Particle Physics
• Quantum Physics
• Relativity
• Statistical Mechanics and Thermodynamics
• Other Topics

Engineering

• Mechanical Engineering
• Electrical Engineering
• Chemical Engineering
• Nuclear Engineering

Other sciences

• Chemistry
  March, Smith - March's Advanced Organic Chemistry: Reactions, Mechanisms, and Structure
  Morrison, Boyd - Organic Chemistry
• Computer Science
• Biology
• Geology and geography

 

[micromass]

• Author: Spivak
• Title: Calculus
• Amazon Link: https://www.amazon.com/dp/0914098918/?tag=pfamazon01-20
• Prerequisities: High-school mathematics
• Contents: Continuity, differentiation, integration, sequences, series. Does not contain multi-variable calculus.

User comments:

• micromass:
  This is a wonderful and exciting book. I feel that this is one of the few books that any math major should read. It mainly covers single-variable calculus and it does so very rigorously. Make no mistake about it, the book is rigorous and quite hard. The exercises tend to be very challenging. As such, I would consider the book more an introduction to real analysis than an actual calculus book. If you wish to read this book, I would recommend that you had some experience with calculus already and preferably an experience with proofs too.

 

[Redbelly98]

• Author: Hecht, or Hecht & Zajak, depending on the edition
• Title: Optics
• Amazon Link: https://www.amazon.com/dp/0805385665/?tag=pfamazon01-20
  Cheaper, used, older edition: https://www.amazon.com/dp/B004HX9WXM/?tag=pfamazon01-20
• Prerequisities: Electricity and magnetism; multivariable calculus
• Contents:

User comments:

• Redbelly98:
  Over the last 30 or so years, this has become a standard textbook for an introductory optics course typically taken by college juniors or seniors. I often refer to it to refresh my background since using it for a course in 1987.

 

[micromass]

• Author: Robert T. Morrison, Robert N. Boyd
• Title: Organic Chemistry
• Amazon Link: https://www.amazon.com/dp/0136436692/?tag=pfamazon01-20
• Prerequisities: No prerequisites
• Contents:

User comments:

• https://www.physicsforums.com/member.php?u=331656:
  I can't really say much about who needs it but i am in High School and i love this book. I guess there are no prerequisites for this book.

 

[micromass]

• Author: Jerry March, Michael B. Smith
• Title: March's Advanced Organic Chemistry: Reactions, Mechanisms, and Structure
• Amazon Link: https://www.amazon.com/dp/0471720917/?tag=pfamazon01-20
• Prerequisities: High-School Chemistry
• Contents:

User comments:

• https://www.physicsforums.com/member.php?u=331656:
  I suppose that this book is not really made for high school students like me but i have bought it because it has almost all the reactions which i encounter while solving the problems. You should check out the reviews on amazon.

 

[jbunniii]

• Author: Rudin
• Title: Principles of Mathematical Analysis
• Amazon Link: https://www.amazon.com/dp/007054235X/?tag=pfamazon01-20
• Prerequisities: Rigorous calculus, including epsilon-delta proofs. Spivak's "Calculus" would be more than sufficient preparation.
• Contents: Metric space topology, series and sequences, differentiation, Riemann-Stieltjes integration, uniform convergence, functions of several variables, differential forms, basics of Lebesgue integration

User comments:

• jbunniii
  For the well prepared reader, this is a beautifully clear treatment of the main topics of undergraduate real analysis. Yes, it is terse. Yes, the proofs are often slick and require the reader to fill in some nontrivial gaps. No, it doesn't spend much time motivating the concepts. It is not the best book for a first exposure to real analysis - that honor belongs to Spivak's "Calculus." But don't kid yourself that you have really mastered undergraduate analysis if you can't read Rudin and appreciate its elegance. It also serves as a nice, clean, uncluttered reference which few graduate students would regret having on their shelves.
• micromass
  This is a wonderful book iff you can handle it. Do not use Rudin as your first exposure to analysis, it will be a horrible experience. However, if you already completed a Spivak level text, then Rudin will be a wonderful experience. It contains many gems and many challenging problems. Personally, I find his approach to differential forms and Lebesgue integration quite weird though. I think there are many books that cover it better than him. But the rest of the book is extremely elegant and nice.

 

[Meir Achuz]

• Author: Jerrold Franklin
• Name: Classical Electromagnetism
• Amazon url: https://www.amazon.com/dp/0805387331/?tag=pfamazon01-20
• Level: First year graduate course text.

User comments:

• Meir Achuz
  This book is on the level of Jackson, but with the readability of Griffiths.
  It has good Amazon reviews.

 

[espen180]

• Author: Steven Roman
• Title: Advanced Linear Algebra
• Amazon link https://www.amazon.com/dp/0387728287/?tag=pfamazon01-20
• Contents: Vector spaces, linear maps, module theory, structure theory of linear operators, metric spaces, normed & inner product spaces, Hilbert spaces, tensor products, linear programming, affine geometry, algebras, umbral calculus.
• Prerequisities: Having completed at least one year of proof based linear algebra. Basic abstract algebra, in particular group and ring theory, is also assumed.
• Level: Grad

User comments:

• espen180
  This is the most comprehensive and the best written linear algebra book I have seen. The exposition is clear, thorough, and rigorous. It is a great textbook and is also a good reference book.
• micromass
  This is a very nice book on linear algebra. If you're looking for an advanced text on linear algebra, then this book should be your first choice. As prerequisites, I recommend a rigorous proof-based linear algebra course on the level of Axler or Lang. Further, an abstract algebra course is absolutely required.

 

[micromass]

• Author: Serge Lang
• Title: Basic Mathematics
• Amazon Link: https://www.amazon.com/dp/0387967877/?tag=pfamazon01-20
• Prerequisities: Mathematics before high school
• Level: Motivated high-school students, college students
• Contents: Real numbers, solving equations, logic, geometry, trigonometry, functions, complex numbers, induction, determinants

User comments:

• micromass:
  This book is truly one of a kind. This is a book on basic mathematics, written from the point-of-view of an advanced mathematician. Do not expect many plug-and-chug exercises that just drill the method into you. Do not expect silly notions such as "it is important to rationalize the denominator", it is not. What you should expect is a book that teaches what mathematics really is about. It teaches proofs and logic as a foundation of mathematics.
  However, Lang's style of writing is a bit weird, it takes some time to get used to it. The book does contain some (typographical) errors, but if you are aware of this then this shouldn't bother much.

 

[micromass]

• Author: John M. Lee
• Title: Introduction to Topological Manifolds
• Amazon Link: https://www.amazon.com/dp/1441979395/?tag=pfamazon01-20
• Prerequisities: Real Analysis course involving epsilon-delta and preferebly metric spaces, group theory
• Level: Grad students
• Contents: Basic topology: open, closed, connected, compact; classification of surfaces, homotopy theory, covering spaces, homology

User comments:

• micromass:
  Absolutely one of the best topology books out there. Lee is a real master at writing books. He makes everything very clear and his explanations are superb. This book is the perfect book for those who want to go into differential geometry. The results are really focused towards geometry, which means that some material that is important for analysis is left out. For example, Tychonoff's theorem is not covered and neither are nets and filters.
  Interested reader absolutely must be comfortable with epsilon-delta proofs and continuity. Some knowledge of metric spaces would be nice as well, although Lee provides an appendix that has everything you need to know. For the later chapters, you will need group theory.

 

[BloodyFrozen]

• Author: Apostol
• Title: Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra
• Amazon Link: https://www.amazon.com/dp/0471000051/?tag=pfamazon01-20
• Prerequisities: PreCalculus/Algebra & Trig ; Ideal with a basic knowledge of calculus however, it's fine without it.
• Contents: Set theory, limits, continuity, integration, differentiation, applications, series, differential equations, complex numbers, vector algebra, linear spaces

User comments:

• BloodyFrozen:
  A very good book for a motivated reader. It requires the reader to be mature, and the intellectual level is challenging. It may require a lot from the reader (especially a beginner), but definitely worth the study.
• micromass:
  This book is a very good treatise on calculus. It is as rigorous as Spivak on many accounts. The main difference between Apostol and Spivak seems to be that Apostol's exercises are more computational (but still quite hard), while Spivak is more theoretical in many aspects.
  A good grasp of high-school mathematics is a must and I also advise having seen a bit of calculus already.

 

[micromass]

• Author: Michael Artin
• Title: Algebra
• Amazon Link: https://www.amazon.com/dp/0132413779/?tag=pfamazon01-20
• Prerequisities: High-school mathematics, proofs
• Level: Undergrad
• Contents: Matrices, Groups, Vector spaces, Symmetry, Representations, Rings, Modules, Fields, Galois theory

User comments:

• micromass:
  Artin is a top notch mathematician and this is very apparent from this book. The book treats the basics of abstract algebra in a really nice way. Furthermore, there are some nice additions such as symmetry of plane figures. If you want to start studying abstract algebra and you're looking for a nice first book, then this is the ideal book for you. Don't expect the book to be easy though. A course on proofs and logic seems necessary before doing this book.