Books

This course has no official course textbook. The instructor will post course notes as we go through the course.

Feel free to look for some that you can access electronically from the UCR Library.

Undergraduate Level

I expect that for most of this course you can use your favorite undergraduate math methods textbook. I list a few here.

Mathematical Methods in the Physical Sciences, Mary Boas.
This is the book I used as a student. It is an excellent introduction to many topics. I still refer to it I need a reminder of how to get started with some calculations. Howeer, it does not directly cover Green's functions.
Mathematical Methods in Engineering and Physics, Felder and Felder
A wonderful book for self study, though studnets may find it a bit annoying because of its narrative style. The authors expect you to engage with a a motivating example at the beginning of each chapter before working through each section methodically. This can be effective for learning a subject the first time, but may be less efficient when used as a quick reference.

Graduate Level

Mathematical Methods for Physicists, Arfken, Weber, Harris
This giant tome may be a good reference for our course. It covers many of the topics that we will review. I never got around to using it because it was too heavy to lug around in my backpack.
Mathematical Methods for Physics and Engineering, Riley, Hobson, Bence
Another fairly comprehensive tome.
Mathematics of Classical and Quantum Physics, Byron and Fuller.
Inexpensive gem as a Dover edition.
Mathematical Methods of Physics, Mathews and Walker.
Another gem from a prior generation. The text is an excellent combendium of topics leading up to a chapter on eigenfunctions, eigenvalues, and Green's functions.
Mathematical Physics, Eugene Butkov
Similar to Matthews & Walker, an excellent text on the kind of functional manipulations that were the bread and butter of physicists in the 1960s.

Specific Topics

Dimensional Analysis

"Dimensional analysis, falling bodies, and the fine art of not solving differential equations," Craig Bohren. American Journal of Physics 72, 534 (2004); (access through UCR VPN)This article really captures the spirit of this course and is a non-trivial demonstration of the power of dimensional analysis. It gives us a reason to pause and think about what it means for our idealized models of nature to be reasonable approximations to the complex reality around us.
"Natural Units and the Scales of Fundamental Physics," Robert Jaffe, Supplementary Notes for MIT’s Quantum Theory Sequence, Feb 2017. Jaffe's notes have plenty of examples of dimensional analysis as well as a thorough introduction to natural units.
"Dimensional Analysis, Chaos and Self-Organised Criticality" (Ch. 10) in Longair, Theoretical Concepts in Physics An Alternative View of Theoretical Reasoning in Physics
"Dimensional analysis as the other language of physics" R. W. Robinett, American Journal of Physics 83, 353 (2015). Very nice introduction to dimensional analysis.
"Section 11: The Method of Similarity," in Mathematical Methods of Classical Mechanics by V.I. Arnold. This is a delightful book about differential geometry that is disguised about a book about mechanics. In section 11 Arnold describes how to use scaling relations to relate relate different orbits of a central potential (we mention this example in our class notes). Much more fun are the two problems at the end of the section.
"Can One Take the Logarithm or the Sine of a Dimensioned Quantity or a Unit?," Matta, Massa, Gubskaya, and Knoll, Journal of Chemical Education 88 (2011). Via Fermat's Library. (See also John Baez / Don Koks on Physics FAQ)
"Cursed Units," Joseph Newton (YouTube). A collection of cursed scientific units.
The book Scale by Geoffrey West is about dimensional analysis, though I've never gotten around to reading it. Let me know if you've read it and recommend it.

Linear Algebra

Linear Algebra Done Right, Axler
Linear Algebra Done Wrong, Sergei Treil

Complex Analysis

Visual Complex Analysis, Needham
A little unusual in the exposition, but the visual approach is well thought out.

Probability and Statistics

Probability Theory: The Logic of Science, Jaynes
Statistics: A Guide to the Use of Statistical Methods in the Physical Sciences, Barlow

Advanced Topics

Here are some mathematical physics references for topics that are a bit beyond the scope of this course but may be up your alley if you enjoy the topics in this course.

Advanced Mathematical Methods for Scientists & Engineers I, Bender and Orzag
This is the textbook that Steven Strogatz used in his "Asymptotics and perturbation methods" course over the pandemic.
Physical Mathematics, Kevin Cahill
Great for digging more deeply into some of the topics in this course.
Mathematics for Physics, Stone & Goldbart

Geometry

First Steps in Differential Geometry, McInerny
Geometry, Topology, and Physics, Nakahara
"Topology of Fibre bundles and Global Aspects of Gauge Theories,"
Andres Collinucci, Alexander Wijns
https://arxiv.org/abs/hep-th/0611201
Differential Geometry and Lie Groups for Physicists, Fecko
Geometrical Methods of Mathematical Physics, Schutz
Electricity and Magnetism for Mathematicians, Garity

Popular Level

Road to Reality, Penrose.
Not a textbook, per se, but a wonderful tome to that complements the course well. The first 400 pages or so rapidly touch on many mathematical ideas that form a deep connection to theoretical physics. It may ostensibly be written for a lay audience, but I think it is ideally geared to a PhD candidate in physics: someone who has met and used some of the ideas before, and who is thus equipped to appreciate the hints to deeper connections.
Explorations in Mathematical Physics, Don Koks
I think this would be a wonderfully fun book to lecture from some day.

Meme via Steve McCormick, Oct 17, 2023
Original source unknown, TinEye points to this 2019 template.

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