When I first started teaching classical electrodynamics, it rapidly
became apparent to me that I was spending as much time teaching what
amounted to remedial mathematics as I was teaching physics. After all,
to even write Maxwell's equations down in either integral or
differential form requires multivariate calculus - path integrals,
surface integrals, gradients, divergences, curls. These equations are
rapidly converted into inhomogeneous partial differential equations and
their static and dynamic solutions are expanded in (multipolar)
representations, requiring a knowledge of spherical harmonics and
various hypergeometric solutions. The solutions are in many cases
naturally expressed in terms of complex exponentials, and one requires a
certain facility in doing e.g. contour integrals to be able to (for
example) understand dispersion or establish representations between
various forms of the Green's function. Green's functions themselves and
Green's theorem emerge, which in turn requires a student to learn to
integrate by parts in vector calculus. This culminates with the
development of *vector* spherical harmonics, Hansen functions, and
dyadic tensors in the integral equations that allow one to evaluate
multipolar fields directly.

Then one hits theory of special relativity and does it all *again*,
but now expressing everything in terms of *tensors* and the *theory of continuous groups*. It turns out that all the electrodynamics
we worked so hard on is much, much easier to understand if it is
expressed in terms of tensors of various rank^{2.1}.

We discover that it is *essential* to understand tensors and tensor
operations and notation in order to follow the formulation of relativity
theory and relativistic electrodynamics in a compact, workable form.
This is in part because some of the difficulties we have encountered in
describing the electric and magnetic fields separately result from the
fact that they are *not, in fact, vector fields!* They are
components of a second rank field strength tensor and hence *mix*
when one changes relativistic frames. Tensors are indeed the natural
language of field theories (and much else) in physics, one that is
unfortunately not effectively taught where they are taught at all.

The same is true of group theory. Relativity is best and most generally
derived by looking for the group of all (coordinate) transformations
that preserve a *scalar* form for certain physical quantities, that
leave e.g. equations of motion such as the wave equation form invariant.
There are strong connections between groups of transformations that
conserve a property, the underlying symmetry of the system that requires
that property to be conserved, and the labels and coordinatization of
the physical description of the system. By effectively exploiting this
symmetry, we can often tremendously simplify our mathematical
description of a physical system even as we deduce physical laws
associated with the symmetry.

Unfortunately, it is the rare graduate student that already knows
complex variables and is skilled at doing contour integrals, is very
comfortable with multivariate/vector calculus, is familiar with the
relevant partial differential equations and their basic solutions, has
any idea what you're talking about when you introduce the notion of
tensors and manifolds, has worked through the general theory of the
generators of groups of continuous transformations that preserve scalar
forms, or have even *heard* of either geometric algebra or Hansen
multipoles. So rare as to be practically non-existent.

I don't blame the students, of course. I didn't know it, either, when I
was a student (if it can honestly be said that I *know* all of this
now, for all that I try to teach it). Nevertheless filling in all of
the missing pieces, one student at a time, very definitely detracts from
the flow of teaching *electrodynamics*, while if one *doesn't*
bother to fill them in, one might as well not bother trying to teach the
course at all.

Over the years in between I've tried many approaches to dealing with the
missing math. The most successful one has been to insert little
minilectures that focus on the math at appropriate points during the
semester, which serve to both prepare the student and to give them a
smattering of the basic facts that a good book on mathematical physics
would give them, and to *also* require that the students *purchase* a decent book on mathematical physics even though the ones
available tend to be encyclopediac and say far too much or omit whole
crucial topics and thereby say far too little (or even both).

I'm now trying out a new, semi-integrated approach. This part of the book is devoted to a lightning fast, lecture note-level review of mathematical physics. Fast or not, it will endeavor to be quite complete, at least in terms of what is directly required for this course. However, this is very much a work in progress and I welcome feedback on the idea itself as well as mistakes of omission and commission as always. At the end of I list several readily available sources and references that I'm using myself as I write it and that you might use independently both to learn this material more completely and to check that what I've written is in fact correct and comprehensible.