An Ideal Math Curriculum for Physics Students

03 April 2026

I occasionally fantasize about an “ideal math curriculum” for Physics graduate students, based on my experience, in school and out. Which topics make sense, which don’t, what should count as reasonably expected knowledge, what is actually useful?

There are also some textbooks, several of which having been published after I left school, that I would like to use (actually: like to have used) in the appropriate classes.

It is assumed that the following math curriculum is given in addition to the usual theoretical Quadrivium, consisting of Classical Mechanics, Electricity and Magnetism, Quantum Mechanics, and Statistical Mechanics, which itself should definitely be completed through a good course on Quantum Field Theory.

I did not attempt to break down the topics into “weekly class hours”. I imagine a semester system, with two classes per (calendar) year. The specific choice of topics will vary with the length of the academic year; the main point here is the general outline, not the individual topics.

Track A: Analysis

The school I attended initially (University of Göttingen) made all its incoming physics students take the first year math classes. That seemed like an irritating detour at first, but I feel that it has served me well in years since: it has been useful to develop an appreciation for rigorous mathematical arguments from the start. It also created an intuition for the kinds of edge cases that can trip up the mathematically naive (conditional vs absolute, pointwise vs uniform convergence): at least one blinks before interchanging summation and integration, or is aware that a sequence of perfectly analytic functions may end up converging to a singular one.

Many of the rigorous arguments from real and complex analysis are never (consciously) needed again; but having been exposed to that mode of thinking is foundational.

A course on ordinary differential equations is intentionally missing from this track; we will come back to that issue below (in Track 3).

Class 1: Real Analysis

Point set topology, open/closed sets, epsilon/delta arguments, modes of convergence (conditional, absolute, pointwise, uniform).

Class 2: Complex Analysis: Theory of Functions

Complex numbers, analytic functions, complex integration, Taylor and Laurent series, residue calculus and contour integrals: corresponding to the first half (Part I) of Ablowitz and Fokas.

Class 3: Partial Differential Equations, Boundary Value Problems, Fourier Theory, and Distributions

All the topics listed in the title are closely related and are best taught together, rather than in separate classes that specialize on only one of them. In particular, the setting within Fourier Theory makes it possible to treat distributions (“Dirac’s Delta Function”) in a somewhat more rigorous, but much more useful way than the usual, extremely hand-waving treatment in the typical “Mathematics for Physicists” class.

There is a truly outstanding book to serve as foundation for such a class: G. B. Folland’s Fourier Analysis and Its Applications. The choice of topics and the level of treatment are perfect for Physics students.

Track B: Algebra and Probability

This is the “non-Analysis” math track; otherwise, the topics don’t necessarily have much in common.

I believe that it makes sense to spend extra time on linear algebra: the topic is surprisingly rich, much of it is not “intuitively obvious” in the way that most of Analysis is, and hence worth presenting explicitly and in some detail. Many of the more advanced topics are also surprisingly useful in applications.

In my experience, a solid grounding in mathematical Probability Theory has traditionally been absent from a Physics education: a few casual remarks, at the beginning of the Statistical Mechanics class, had to be enough. I think this is a mistake. A good course on Probability (but without measure theory) pays of handsomely: it lends depth and perspective to theoretical questions, and it is amazingly useful outside of academia, in applied endeavors. It is also very good fun.

Class 1: Linear Algebra 1

Vector spaces, basis, linear independence, linear transformations, eigentheory, maybe Jordan Normal Form. (Linear systems of equations are of little relevance to the Physicist, except that they are needed to define eigenvalues, and hence are required.)

There is an excellent text for this material: Matrix Analysis and Applied Linear Algebra by C.D. Meyer (also see my review). The link is to the second edition, which I consider less successful than the first — but it is still an excellent text, and the best one there is.

Class 2: Linear Algebra 2

No fixed set of topics, but further explorations of the material. Example topics may include: condition numbers and matrix norms, Singular Value Decomposition, pseudo-inverses, pre-conditioning, other matrix decompositions, matrix functions, Frobenius theory, Gershgorin circles.

The second edition of Meyer provides plenty of material for a course like that.

Class 3: Probability Theory

Probability, Events and Sample Spaces, Combinatorics and Conditional Probability, Random Variables and Expectation Values, Distributions, Generating Functions.

The canonical reference for this material is Feller, Vol I; a more modern text is Grimmett and Stirzaker. Both of these books include more material than I would expect in an introductory class; in particular stochastic processes can and should be left for another time.

Track C: Applied Math and Perturbation Theory

The topics treated in this track are important for Physicists, but do not fit into a “pure” math syllabus. Hence they have to be addressed somewhere — usually in the form of (often horrible) “Math for Physicists” classes. I think one can do better: by shifting more material to proper math classes (in particular complex analysis, Fourier analysis, and distributions), but extending the reach and quality of the topics covered here (in particular through the inclusion of formal perturbation methods).

The treatment of ordinary differential equations in this curriculum is probably rather unconventional, and for that reason deserves some special discussions.

First of, there is widespread consensus that nobody knows how to teach ordinary differential equations: Klaus Jänich has described the topic as “amorphous” and denies the existence or even possibility of general teaching conventions, Gian-Carlo Rota stated that writing a textbook on the topic was the greatest mistake of his career, David Sánchez has written an entire book on the differential equations — and then written another book on how to teach them!

My suggestion, in particular in addressing the “amorphous” nature of the topic, is to strip the treatment of ordinary differential equations to its very core, and in particular, to leave out everything that can naturally be treated elsewhere. This means that boundary value problems are gone (they are much more naturally presented in the context of separation methods for partial differential equations), there is no mention of numerical methods (they belong in a class on, you guessed it, numerics — which requires a completely different mindset than a math class), and Laplace transformations are best discussed in a class on digital signal processing or control theory (for those who actually do this kind of work). Non-linear systems are a specialized, advanced topic by themselves; there is very little that can be said about them at an introductory level beyond basic stability theory near fixed points.

I agree with an emerging consensus to rigorously trim traditional topics that are of limited practical usefulness (homogeneous equations, exact equations, integrating factors; but also series solutions and special functions). But, counter to the consensus, I believe that it makes sense to address existence and uniqueness, and to discuss in some detail the concepts of general and singular (or particular) solutions: in particular the latter are not necessarily obvious “from the Physics”, and have a definite potential to trip up the naive or unwary.

Overall, it seems to me that the essential minimum in differential equations itself is actually quite small; it’s the applications of ordinary differential equations to all and sundry that lead to the overwhelming, confusing, and “amorphous” impression of the subject. But if we view the curriculum in its entirety, rather than as isolated courses, then we can leave the applications to the appropriate places, and keep the central core small and transparent.

The other somewhat unusual aspect of the current proposal is the inclusion of formal perturbation methods. Surprisingly, this is a topic that is rarely mentioned in the typical Physics curriculum at all. Perturbation theory comes up all the time (in classical mechanics, in quantum mechanics, in quantum field theory), but is never treated as a mathematical subject, with its own results and techniques, in its own right. As a consequence, most physicists, I believe, are unaware of things like singular perturbation theory or boundary layers — they are occasionally re-discovered, for sure, but wouldn’t it be useful if they were just generally known, from the get-go? (It is a valid question how useful, for example, boundary layer methods are to the typical working physicist, but the fact that they, together with WKB methods, keep being re-discovered suggests that there is something useful there, there.)

Class 1: Applied Analysis

Ordinary Differential Equations

First-order separable equations; existence and uniqueness.
Second-order linear equations, including inhomogeneous problems.
Linear systems and “state-space methods”, matrix exponential.
How to set up differential equations from real world problems, “modeling”.
Mention power-series as an approach briefly, demonstrate using sin/cos, and them move on.
Maybe discuss behavior and linearization of dynamical systems near fixed points.

David Sánchez has produced a short text that outlines a similar idea, in more detail — but it is essentially an essay on curriculum development, not a textbook.

The chapter on Ordinary Differential Equations in Jon Mathews “Mathematical Methods of Physics” (which, I believe, is based on Feynman’s post-Los Alamos lectures at Cornell) has a similarly focused and practical choice of topics; unfortunately, it is completely unsystematic and therefore very confusing.

Most texts on ordinary differential equations take a very different, “comprehensive” approach, attempting to include too many side topics, which obscures the core of the topic. My favorite is the classic by Tenenbaum and Pollard: read selectively, picking appropriate chapters to match the syllabus given above, I find it very helpful.

Vector Calculus Grad, Div, Curl, and all that; line and surface integrals.

I believe it makes sense to discuss these topics, briefly, outside the class on electricity and magnetism, where they naturally arise. By tradition, the standard text seems to be Schey’s Grad, Div, Curl, and all that. A slightly more advanced, but still compact, treatment would be desirable.

Class 2: Perturbation and Variational Methods

Formal perturbation series, asymptotic expansions of integrals, algebraic and differential equations, boundary layers, WKB. Calculus of variations and variational approximation methods.

A very good introductory treatment is the book by Holmes. This book presents the methods of perturbation theory in a systematic manner, and that makes it much more practically useful than the fascinating, but bewildering classic by Bender and Orszag, which mostly consists of a surfeit of disparate examples.

For variational methods, the little book by Gelfand and Fomin is useful, and possibly more accessible than the classic by Courant and Hilbert.

Track D: Computers

This track might be combined with Track C to give a three-class sequence; but conceptually, it’s really different. (Applied math is not numerics, and, even more so, numerics is not applied math.)

The topic “Computers in Physics” is even more amorphous and fragmented than ordinary differential equations. Before any discussion can take place, one must first clarify the scope and the natural divisions of the topic, so that one can then state which of these areas one is addressing.

The main distinctions are:

Nature of the work: Numeric, Symbolic, or Simulation (the latter split into Monte Carlo and Molecular Dynamics)
Scale and effort: Small and casual; “serious”; career-defining

Depending on which of those $3 \times 3$ (or even $4 \times 3$) choices has been made, a totally different class results.

Another complication is that any discussion of computer methods involves a choice of “tools”: from integrated work environments (like Matlab, Mathematica, Scilab), over libraries (such as NumPy/SciPy), to programming languages for glue code or bespoke work (Python, C/C++, Fortran?). In particular in Physics there is no clear winner (as is the case in engineering with Matlab), and hence an explicit choice has to be made — which is mostly a choice about what will not be covered! Finally, some of the tools are not free (Matlab, Maple, Mathematica), and therefore not infinitely portable, even if the department has a license. (Student editions tend to be very reasonably priced, professional ones less so.)

Class: Basic Numeric and Symbolic Tool Usage

In this somewhat crowded terrain, I have a modest suggestion to make: leaving aside simulations (which are a really a separate field, all to themselves), should basic numeric and symbolic computer skills, for casual, ad-hoc computations, become as standard as manual differentiation and integration are today? In other words, if we expect a graduate student to be able to integrate (say) $f^\prime - \alpha f = 0$ and $\int 1/(1+x) dx$, should we also expect the same graduate student to produce a numeric solution of $f^\prime - f^2 = \sin(x)$ or employ a computer-algebra system to find the partial fraction decomposition of $(1 - 2x + 2x^2)/(1 - 4x + 5x^2 - 2x^3)$ quickly? This precisely means not just familiarity, but fluency with the appropriate tools, and therefore implies a choice of tools in the first place. (The skills don’t transfer that well: the concepts do, the syntax doesn’t, and that’s what matters, for casual usefulness.)

In other words, both for simple numerics and for straightforward algebra, the use of an appropriate package should be as familiar as using a pocket calculator to find $\sqrt[3]{17.81}$. (Again, pocket calculators all work the same, whereas numerical and algebraic packages don’t, but still…)

I believe these skills could be taught within a one-semester class. Thus, the current track could be combined with Track C to form another three-class track on “applied” methods.

I am not sure what is standard in contemporary curricula, but I believe we are “not there yet”. There is a handful of books which could be used for such a class, most of them using various Python libraries. That’s a reasonable choice of tools, but certainly not the only one. Again, as far as the tools are concerned (in particular the open-source ones), we are “not there yet”, either.

Finally, “serious” and large-scale calculation and simulation are a separate topic altogether, and not part of the general curriculum, but steps towards specialization and research. In that context, the question arises what level of software “meta-skills” should be included (software engineering best practices, software architecture, build and development tools). But that’s a different topic.