- ... rank
^{1.1} - Some parts are
simpler still if expressed in terms of the geometric extension of the
graded division algebra associated with complex numbers: ``geometric
algebra''. This is the algebra of a class of objects that includes the
reals, the complex numbers, and the quaternions - as well as
generalized objects of what used to be called ``Clifford algebra''. I
urge interested students to check out Lasenby's lovely book on Geometric
Algebra, especially the parts that describe the quaternionic formulation
of Maxwell's equations.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... respectively
^{3.1} - Note that we define the magnitude of the vector
(written
either
or
) in terms of the inner product:
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...contraction
^{3.2} - See the chapter
coming up on tensors to learn what a contraction (and a tensor) is, but
in the meantime, this just means that the
th index is summed ``out''
of the expression, so that the result has fewer indices on the left than
it has on the right.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
equations
^{11.1} - Or is it four? These are
*vector*partial differential equations, so one can break them up into*eight*distinct equations relating particular components, although it isn't clear that all eight will be independent. Alternatively, as we will see later, we can reduce them to just*two*tensor partial differential equations in a relativistic formulation, and will be able to see how one might be able to write them as a*single*tensor equation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... separates
^{11.2} - In case you've forgotten: Try a
solution such as
, or (with a bit of
inspiration)
in the differential
equation. Divide by
. You end up with a bunch of terms that can
each be identified as being constant as they depend on
separately. For a suitable choice of constants one obtains the
following PDE for spatial part of harmonic waves.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... that
^{11.3} - Yes, you should work this out termwise if
you've never done so before. Don't just take my word for
*anything*.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
**real**^{11.4} - Whoops! You mean
doesn't have to be real? See
below. Note also that we are implicitly assuming
and
are real as well, and they don't have to be either!
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... real
^{11.5} - Heh, heh.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
starters
^{11.6} *Note Well!*The we are using here is*not*the direction of , it is the direction of the normal to the surface, that is to say .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
medium
^{11.7} - Indeed, you should have learned something about this in
elementary physics studying the reflections of wave pulses on a string,
and again when studying thin film interference (a phenomenon where
accounting for this inversion is critical to getting the right answers).
If you've never see this and it doesn't make sense to you please ask for
help.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... ``easy''
^{11.8} - Easy enough to write down in the book in an
intelligible form. Of course it is straightforward to
*compute*it with e.g. a computer for arbitrary incident angles - this is why God*invented*computers, because human brains were not really up to the task. Unless, of course, they belong to complete masochists.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
form
^{11.9} - Remember the algebra where we got the square root in the
first place? Well, do it
*backwards*.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... fields
^{11.10} - Why? If
you don't understand this, you need to go back to basics and think about
expanding a potential well in a Taylor series about a particle's
equilibrium position. The linear term vanishes because it is
equilibrium, so the first surviving term is likely to be quadratic.
Which is to say, proportional to
where
is the displacement
from equilibrium, corresponding to a linear restoring force to lowest
order.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... motion
^{11.11} - You do remember Newton's law, don't you?
Sure hope so...
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... electron
^{11.12} - I certainly hope you can derive this result,
at least if your life depends on it. In qualifiers, while teaching
kiddy physics, whenever.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
function
^{13.1} - Note that this expression stands for: ``The
generalized point source potential/field developed by Green.'' A number
of people criticize the various ways of referring to it - Green
function (what color was that again? what shade of Green?), Greens
function (a function made of lettuce and spinach and kale?), ``a''
Green's function (a singular representative of a plural class referenced
as a singular object). All have problems. I tend to go with the latter
of these as it seems least odd to me.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... function
^{13.2} -
Note well that both the Green's ``function'' and the associated Dirac
delta ``function'' are not
*functions*- they are defined in terms of*limits of a distribution*in such a way that the interchange of limits and values of the integrals above make sense. This is necessary as both of the objects are*singular*in the limit and hence are meaningless without the limiting process. However, we'll get into*real*trouble if we have to write ``The limit of the distribution defined by Green that is the solution of an inhomogeneous PDE with a source distribution that in the same limit approaches a unit source supported at a single point'' instead of just ``Green's function''. So we won't.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... time
^{13.3} - Heh, heh, heh...:-)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... wavelength
^{13.4} - We will learn to treat certain
exceptions, believe me.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... series
^{13.5} - Taylor? Power? Laurent? Who can
remember...
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... that
^{13.6} - This really isn't an assumption. We could
equally well write
in spherical polar coordinates, separate
variables, note that the angular ODEs have spherical harmonics as
eigenstates (``quantized'' by the requirement of single-valuedness on
e.g. rotations of
in
) and reconstruct the separated
solution. But that's too much work and we already did it at least once
in our lives, right? So we'll ``assume''.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... reasons
^{13.7} - A cop-out phrase if there ever
was one. It translates as: because that's the way it turns out at the
end.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
*exactly*^{13.8} - Well, in a uniformly convergent expansion, which is
kind of exact, in the limit of an infinite sum. In the mean time, it
is a damn good approximation. Usually.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... solution
^{13.9} - This suggests that there are some interesting
connections between the conjugation symmetry and time reversal
symmetry. Too bad we won't have time to explore them. You may on
your own, though.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... functions
^{14.1} - From now on, this term is
generic unless clearly otherwise in context.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... sphere''
^{16.1} - Hyuk, hyuk,
hyuk...
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... anything
^{16.2} - Even if it's true
...
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... reflection
^{16.3} - Sorry...
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... frames
^{17.1} - If we relax this requirement and allow
for uniform expansions and/or contractions of the coordinate system, a more
general group structure, the
**conformal group**, results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... rotation''
^{17.2} -
``Hyperbolic'' because of the relative minus sign between
and
. More on this later.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... past
^{17.3} - Don't think too hard about this sentence or you'll
start to go slightly nuts because it is self-referential and hence
Gödelian.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{18.1} - The
**rank**of a tensor is determined by the number of indices it has. Scalars are 0th rank, vectors are 1st rank, 2D matrices are 2nd rank, and our old friend is a third rank fully antisymmetric tensor.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... verify
^{18.2} - And should! That's right, you students, you know
who I'm talking to. So here's a question for you: Are
a real isomorphism to complex numbers? What
would the various results of the introduction to complex numbers look
like expressed in terms of these two matrices? What
*in particular*does multiplying by a unimodular ``complex number'' such as look like? Hmmm... veeeery interesting.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... is
^{19.1} - Note that I've rearranged
this slightly to avoid having to do lots of stuff with
sandwiches
below.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
tanstaafl
^{20.1} - There Ain't No Such Thing As A Free Lunch. No
kidding.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... us
^{21.1} - It is
interesting to meditate upon the fact that
*your*event horizon and*my*event horizon are not coincident, which leads in turn to an interesting problem with logical positivism.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... ``trouble''
^{21.2} - Trouble such as particles capable of
lifting themselves up by their own metaphorical bootstraps...
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .