References

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. 

• Equations of Mathematical Physics, V. S. Vladimirov
  Another classic from 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.

• There is an experimental observation that "Duration of urination does not change with body size" for mammals. (Here's a nice infographic from National Geographic.) The observation won an igNobel prize. I recommend arxiv:1403.6369 for an example for how this observation can be used to test hypotheses simply based on dimensional analysis. 

Linear Algebra

• Linear Algebra Done Right, Axler

• Linear Algebra Done Wrong, Sergei Treil

Linear Algebra: tensors

• "Introduction to Tensor Calculus," Kees Dullemond & Kasper Peeters (booklet) 2010. This is a reasonable introduction to the "indexology" that we use in physics. The booklet is much more rigorous than we will be. It may help motivate the types of rules that we will build into our index notation. If you enjoy this reference, you can find more in the first few chapters of most general relativity textbooks, for example Sean Carrol's Spacetime and Geometry.

• "Tensors: A guide for undergraduate students," Franco Battaglia, American Journal of Physics 81, 498 (2013); (accessible through UCR VPN)This is an excellent and readable introduction to to tensors. In the last half of the article (beyond our scope) one starts to get a taste of how linear algebra is a local approximation to calculus on curved spaces.

• There's a fantastic history of vectors in mathematics and physics, Vector: a surprising story of space, time, and mathematical transformation by Robyn Arianrhod. It's the kind of history that a physicist would love, connecting some familiar names with a story of the development of an idea that we tend to ignore in our education. 

• One of my favorite examples of a vector space is color space. The three types of cone cells in our eyes  are sensitive to different pieces of the visual spectrum. They define a basis for a three-dimensional vector space corresponding to neurological responses to the electromagnetic spectrum. Other animals can have more types of cone cells, in which case their color spaces are higher dimensional. Here are a couple of nice videos discussing the concept. There are references to more academic papers in our course notes.

  1. • "RGB to XYZ: The Science and History of Color" by John Austin" at the Strange Loop Conference, 2019

     • "Color Spaces: Explained from the Ground Up" via- Video Tech Explained

Moment of inertia tensor and geometric algebra

We posed the question of why the moment of inertia tensor is a tensor and not a matrix. Many textbooks get this wrong. One way to see this is that the moment of inertia tensor properly has two lower indices. It may help to write the cross prodcut with respect to the Levi-Civita tensor in 3D. Some references:

• 1. • This answer on physics.stackexchange. The presentation is a little technical. It is similar to what one finds in V.I. Arnold's Mathematical Methods of Classical Mechanics in the section on rigid body motion.

     • A really great reference is this article on bivectors: "Teaching Rotational Physics with Bivectors," Steuard Jensen, Jack Poling. (arXiv: 2207.03560) The bivector construction may seem a little unfamiliar, but they do a good job of introducing it. In 3D, the bivector is a nicer way to represent the cross product. However, bivectors (and their cousins, k-forms) generalize to higher dimensions. The appendix describes the moment of inertia tensor in higher dimensions.

     • Along the lines of the above, here's a more systematic introduction to geometric algebra, including a nice discussion of the moment of inertia tensor: "Unifying the inertia and Riemann curvature tensors through geometric algebra" by M.. Berrondo, J. Greenwald, and C. Verhaaren (Am. J. Phys. 80, 905–912).

     • If you want to really dive deeper into this field, check out "Spacetime algebra as a powerful tool for electromagnetism" (arXiv:1411.5002) by Dressel, Bliokh, and Nori. 

     • A nice introduction to the ideas of geometric algebra on YouTube: "A Swift Introduction to Geometric Algebra."

     • A slightly more sophisticated set of videos: "QED Prerequisites Geometric Algebra: Introduction and Motivation"

     • A YouTube explainer on rigid body dynamics using geometric algebra.

The continuum limit

We talked about how the N to infinity limit of histogram space is not well defined. We can see this in the challenges in defining the inner product. The formal reason is that the normalizable states do not contain the limit points of all possible sequences of normalizable states.

• 1. •  I refer to Appel's Mathematics for Physics & Physicists, Chapter 9 for a thorough discussion. In quantum mechanics, we assume a Hilbert space, which is a space where the N to infinity limit makes sense.

     • See also Karen Smith's notes on infinite dimensional vector spaces from her Math 513 course in U. Mich. These are very well written for a physics audience.

Complex Analysis

• Your favorite undergraduate math methods course is a good starting point. I am partial to the textbook by Boas. 

• Visual Complex Analysis, Needham. A little unusual in the exposition, but the visual approach is well thought-out. That being said, the approach is a little idiosyncratic and may not align with more familiar treatments.

• "Guide to Cultivating Complex Analysis" by Jiří Lebl seems to be a nice formal discussion of complex analysis.

• What are Riemann sheets good for?

  • "Resonances and poles in the second Riemann sheet," by Thomas Wolkanowski. A MSc thesis on the significance of the second Riemann sheet in quantum field theory.

  • "Scattering amplitudes and contour deformations," by Gernot Eichmann, Pedro Duarte, M. T. Peña, Alfred Stadler

Probability and Statistics

• Probability Theory: The Logic of Science, Jaynes

• Statistics: A Guide to the Use of Statistical Methods in the Physical Sciences, Barlow

• "Why isn't every physicist a Bayesian?" Robert Cousins, American Journal of Physics (1994)

Cosmic Microwave Background (Fall 2026)

The Fall 2026 version of this class will use the cosmic microwave background (CMB) as a case study for the mathematical physics ideas in this course. No prior cosmology or general relativity is required. Here are some background materials for those who are interested:

• Wayne Hu's CMB tutorials; a good starting point is his Scientific American article from 2004.

• "Cosmic Microwave Background Anisotropies," Wayne Hu, Scott Dodelson [astro-ph/0110414]

• Neelima Sehgal's TASI 2016 lectures on the CMB [Lecture 1, Lecture 2, Lecture 3, Lecture 4]

• Daniel Baumann's TASI lectures; see also his textbook 

• V. Mukhanov's Physical Foundations of Cosmology (the CMB is toward the end, but the first few chapters give a great introduction to cosmology as the overlap of relativistic, quantum, and statistical physics. 

• Chapter 28.6 of Modern Classical Physics by Thorne and Blandford is a succinct introduction to the physics of the CMB packaged in a deeply thoughtful textbook for first year graduate students. (Though the fact that this is towards the end of a gigantic five volume tome perhaps suggests that it may be a subject more appropriate for the end of a first year course.)

• Ned Wright's CMB tutorial. (Also a nice 'cosmological relic' of what websites looked like in the late 90s.)