To summarize what we've covered so far: Our study of electrodynamics is going to be founded on real and complex numbers that represent physical quantities with units, so we learned a bit about these kinds of (scalar) numbers. Since it is a kind of a map of what happens in space and time, we need to understand coordinates, vectors in a coordinate system, and various ways to multiply vectors. That led us to consider both tensor forms and coordinate transformation, as both of these will prove to be very useful if not essential. Coordinate transformations (at least) often form groups so we learned what a group was (and realized that we've been using e.g. the multiplication group all of our lives without realizing it.

It should come as no surprise that the remaining chunk of math we will
need is calculus. After all, Newton invented calculus so he
could invent physics, and electrodynamics is very much a part of
physics. I'm not going to cover every single thing you learned in
calculus classes in the past here (the chapter would be as long or
longer than the entire book if I did) but rather will focus on showing
you the path between the plain old calculus you already know (I
profoundly hope) and the vector calculus you probably don't know
anywhere near well enough unless you had a really extraordinary course
in multivariate calculus *and remember it all*.

Let's begin pretty close to the beginning, with ordinary
differentiation. Even here our treatment won't *quite* be
ordinary, because we will not be reviewing this purely in the abstract.
In all cases, where I refer to various (scalar and vector and possibly
even tensor) *functions*, you should be thinking of those functions
as numerically representing definite physical quantities, with units.
The calculus we need is not *abstract*, it is *descriptive*,
and it is this (possibly subtle) differentiation that separates the
mathematician from the physicist.

Both a mathematician and a physicist may talk about doing things to or
with a function
, but the physicist is always thinking about
functions
that actually ``stand for something'' and
will usually
be replaced by traditional symbols in application. To many
mathematicians,
is just
- some function, *any* function -
and it may or may not mean anything at all besides its own shape or form
if even that is specified.

- Scalar Differentiation
- Vector Differentiation

- The Gradient
- Vector Derivatives

- Second Derivatives
- Scalar Integration

- Vector Integration
- The Fundamental Theorem(s) of Vector Calculus

- Integration by Parts

- Integration By Parts in Electrodynamics