References

Students may use any auxiliary reference books they wish.

The following textbooks are recommended. These books are the closest to our goals in this course.

A Couse in Modern Mathematical Physics by Peter Szekeres (Cambridge). [UCR Library] This is the closest textbook I have found to our couse in material and notation. The book does include a lot of other material. We focus on Chapters 3-6, 7, 9, and 12.
A Student's Guide to Vectors and Tensors, Fleisch, Daniel. Cambridge University Press; 2012 [online though UCR library; additional resources on author's site]. Presents the picture of tensors as linear maps that transform in specific ways. What's missing: eigenvectors, determinants, complex vector spaces.
Linear Algebra: Step by Step, Kuldeep Singh. Oxford University Press; 2013. Covers the standard material in a first linear algebra course in a conversational and inquisitive tone. What's missing: complex vector spaces. Unfortunately there is no digital copy available, but you are welcome to substitute in any comparable textbook.
An Introduction to Tensors and Group Theory for Physicists by Jeevanjee. Springer Press. Digital version available through UCR library. The first part of this text is written at a slightly more formal level, but the topics and approach are similar to our course.

Prof. Tanedo's past course notes
- 2024 Course Notes
- 2023 Course Notes
- 2022 Course notes (hand written)
- P231 Course Notes (graduate-level)

Complementary material: the following material contains chapters that help fill some of the gaps of the required textbooks relative to our course. You are not expected to purchase all of these references!

Recommended: "The Mathematics of Quantum Mechanics," Laforest; 2015. Lecture notes from the Quantum Cryptography School for Young Students. This is a fairly good summary of the material that we'll cover.
Recommended: Basic Training in Mathematics: A Fitness Program for Science Students, R. Shankar.
Recommended: Chapter 1 ("Mathematical Introduction") of Principles of Quantum Mechanics, R. Shankar. The first chapter of this book covers all of the main ideas in our course.
Recommended: Mathematical Methods for Engineers and Scientists 1: Complex Analysis and Linear Algebra, Tang, Kwong-Tin. Springer; 2022 [online through UCR library] The second half of the book is an excellent reference for complex linear algebra. The first half of the book on complex analysis will serve you well in future graduate coursework.
Appendix ("Linear Algebra") of Introduction to Quantum Mechanics, D. Griffiths. An excellent reference on how linear algebra appears in quantum mechanics.
Optional: Linear Algebra Done Right, Axler: available free via UCR library. (2023: 4th edition now available free online.) Excellent reference for the more mathematically inclined who prefer a concise, abstract approach.
Sergei Treil's "Linear Algebra Done Wrong" is a delightful 'response' to Axler's book that is more in line with our course. It is slighlty more technically dense than our course, but otherwise covers very similar topics.
Optional: Linear Algebra, Serge Lang. This was the book from which I first learned linear algebra.
Additional chapters from: Mathematics for Quantum Mechanics, Jackson (low priced Dover edition available)
Quantum Computing for Everyone, Bernhardt (paperback now available): applications to quantum computing and an overall fun read.
A little more advanced: "Tensors: A guide for undergraduate students," American Journal of Physics 81, 498 (2013); https://doi.org/10.1119/1.4802811 (use the UCR VPN to access it)
Similarly more advanced: "Introduction to Tensor Calculus," Kees Dullemond & Kasper Peeters
An Introduction to Tensors and Group Theory for Physicists, Nadir Jeevanjee. Excellent examples in chapter 3.
For quantum computation, see Chapter 15 of the excellent textbook by Moore and Mertens, The Nature of Computation. This is the primary reference for the last 2 weeks of the course.
David Tong's introductory lectures on Quantum Mechanics, section 3 on formalism
Quantum Mechanics: A Graduate Course by Horatiu Nastase. Chapters 1-2 on the mathematical foundations.

Web references

3Blue1Brown: Linear Algebra mini course
Most relevant: chapters 1-4, 9, 13-16
Immersive Linear Algebra website
Most relevant: chapteers 2-3, 6-7, 9-10
"Inside Einstein's head," by Juho Ojala. Visualizations of thought experiments in special relativity.

Introductory / background

Vector: A surprising story of space, time, and mathematical transformation, Robyn Arianrhod. Not a physics book, but any student of physics will appreciate this history of linear algebra.
"The Geometric Tool That Solved Einstein’s Relativity Problem," J. Howlett for Quanta. An introduction to tensors.

Neat discussions

"Do tensors outside of physics transform like a tensor?" Reddit thread from 2023. There is a lot of good discussion here (I do not agree with all of it), but after this class you should be equipped to identify which comments are accurate and which are less so.
"Are there basis vectors for the set of continuous functions?" Reddit thread from 2024. Addresses the idea of a "histogram basis" for the space of functions. This is one of those slippery ideas that we use as a crutch to motivate Fourier analysis, things get tricky in the continuum limit.
Rigged Hilbert space references. The space of integrable functions in quantum mechanics is called a Hilbert space. Apparently there a mathematical "but actually" here where physicists are actually using a rigged Hilbert space. Here are a few discussions on the idea.
- "Rigged Hilbert space and QM" reference request from physics.stackexchange in 2012.
I've been looking for good examples of transformations of spaces as a starting point for talking about transformations of the vector spaces "on top" of those spaces. This is a more advanced topic that we may or may not cover in the class. However, one possible example is MC Escher's "Print Gallery." There's a really neat website and article about it.
In class we mention the Levi-Civita/totally antisymmetric tensor. This is related to alternating forms and wedge products. There's a great discussion on Reddit.

More advanced stuff

This class is a bridge to some of the more advanced mathematical methods in physics. Here are some references for those who want to get a head start.

Mathematical Linear Algebra

Here's a nice article on the mathematical view of linear algebra. Unlike our physics approach, which embraces indices, the mathematical approach is "coordinate free" and has no indices.

Green's functions

My Physics 231 class is a stepping stone from linear algebra to Green's functions in first year physics.
There are several older textbooks that are fantastic but harder to find. Among these are the mathematical physics textbooks by Matthew & Walker, Butkov, and Byron & Fuller. There's one by Feldman & Feldman which is great and readily available, but very expensive.

Analytical Mechanics & Differential Geometry

A good starting place is the classical mechanics textbook by Arnold, Mathematical Methods in Classical Mechanics.
There's a neat discussion at this math.stackexchange post about why momentum is a one-form/dual vector. This is a weird idea: we learned that dual vectors eat vectors and return nubmers. In what way does a momentum do this?
Related to the idea of the momentum as a dual vector: here's a physics.stackexchange post about connecting the Legendre transform to the underlying covector picture.

Relativity & Differential Geometry

A good place to start is your favorite general relativity textbook. There are lots of really great ones. Sean Carroll's book is one of my favorites (the appendices are a short course in more formal differential geometry), but you will also find great books by Schutz, Hartle, and Zee.
More advanced general relativity: the big book known as MSW (Misner Thorne Wheeler's Gravitation) is a fantastic read if you have a chunk of time and want to dig really deeply. There was an online celebration for its 50th anniversary. The similarly-giant book by Thorne and Blandford (Modern Classical Physics) also uses many of the same mathematical foundations.
"Topology of Fibre bundles and Global Aspects of Gauge Theories," Andres Collinucci, Alexander Wijns (hep-th/0611201). As a graduate student, this was one of my favorite sets of lecture notes on differential geometry in physics. It's a good lead in for more sophisticated refernces like the textbooks by Nakahara or Frankel.
Visual Differential Geometry and Forms by Needham. Most books suffer from a lack of pictures. Needham's "visual mathematics" books are one of the fantastic exceptions.

Group Theory

An Introduction to Tensors and Group Theory for Physicists by Jeevanjee, reviews linear algebra and introduces group theory.
Jan Gutowski's Symmetries and Particle Physics course is how I learned group theory. The approach is rooted in geometry.

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