...wikinote¹

Wikipedia: http://www.wikipedia.org/wiki/wikipedia. A wikinote is basically a footnote that directs a student to a useful article in the Wikipedia. There is some (frankly silly) controversy on just how accurate and useful the Wikipedia is for scholarly work, but for teaching or learning science and mathematics on your own it is rapidly becoming indispensible as some excellent articles are constantly being added and improved that cover, basically, all of electrodynamics and the requisite supporting mathematics. Personally, I think the objections to it are largely economic - in a few more years this superb free resource will essentially destroy the lucrative textbook market altogether, which honestly is probably a good thing. At the very least, a textbook will have to add significant value to survive, and maybe will be a bit less expensive than the $100-a-book current standard.

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... class^1.1

This is a transparent ploy to make you hand it in on time. But I mean it! Really!

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... rank^2.1

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.

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...field^4.1

Wikipedia: http://www.wikipedia.org/wiki/Field_mathematics. ;

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... algebra^4.2

Wikipedia: http://www.wikipedia.org/wiki/Division_algebra. .

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...$\mu$^9.1

In SI units, now that Jackson 3rd finally dropped the curséd evil of Gaussian units. Mostly, anyway. Except for relativity.

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... separates^9.2

In case you've forgotten: Try a solution such as $u({\bf x},t) = X(x)Y(y)Z(z)T(t)$, or (with a bit of inspiration) $\mbox{\boldmath $E$}({\bf x})e^{-i\omega t}$ in the differential equation. Divide by $u$. You end up with a bunch of terms that can each be identified as being constant as they depend on $x,y,z,t$ separately. For a suitable choice of constants one obtains the following PDE for spatial part of harmonic waves.

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... that^9.3

Yes, you should work this out termwise if you've never done so before. Don't just take my word for anything.

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...real^9.4

Whoops! You mean $\mbox{\boldmath $n$}$ doesn't have to be real? See below. Note also that we are assuming $\epsilon$ and $\mu$ are real as well, and they don't have to be either.

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... real^9.5

Heh, heh.

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... fields^9.6

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 $x^2$ where $x$ is the displacement from equilibrium, corresponding to a linear restoring force to lowest order.

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... motion^9.7

You do remember Newton's law, don't you? Sure hope so...

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... electron^9.8

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

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... function^11.1

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.

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... function^11.2

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.

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... time^11.3

Heh, heh, heh...:-)

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... wavelength^11.4

We will learn to treat certain exceptions, believe me.

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... series^11.5

Taylor? Power? Laurent? Who can remember...

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... that^11.6

This really isn't an assumption. We could equally well write $\nabla^2$ 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 $2\pi$ in $\phi$) 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''.

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... reasons^11.7

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

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...exactly^11.8

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.

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... solution^11.9

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.

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... functions^12.1

From now on, this term is generic unless clearly otherwise in context.

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... sphere''^14.1

Hyuk, hyuk, hyuk...

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... anything^14.2

Even if it's true ...

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... reflection^14.3

Sorry...

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... frames^15.1

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

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... rotation''^15.2

``Hyperbolic'' because of the relative minus sign between $x^2$ and $ct^2$. More on this later.

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... past^15.3

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

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...Abelian^16.1

Wikipedia: http://www.wikipedia.org/wiki/Abelian group. ;

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... group^16.2

Wikipedia: http://www.wikipedia.org/wiki/Lie group. ,

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... Lorentz^16.3

Wikipedia: http://www.wikipedia.org/wiki/Lorentz group. ,

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...$k$^16.4

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 $\epsilon_{ijk}$ is a third rank fully antisymmetric tensor.

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...manifold^16.5

Wikipedia: http://www.wikipedia.org/wiki/Manifold. i

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... verify^16.6

And should! That's right, you students, you know who I'm talking to. So here's a question for you: Are ${I,\sigma_3\sigma_1}$ 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 $\cos(\theta) I + \sin(\theta) \sigma_3\sigma_1$ look like? Hmmm... veeeery interesting.

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... tensor^16.7

Wikipedia: http://www.wikipedia.org/wiki/Electromagnetic tensor. Note that I'm not completely comfortable with the signs for the covariant form of the potential in the Wikipedia article, although its main conclusions are sound enough.

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... is^17.1

Note that I've rearranged this slightly to avoid having to do lots of stuff with $g$ sandwiches below.

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... tanstaafl^18.1

There Ain't No Such Thing As A Free Lunch. No kidding.

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... us^19.1

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.

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... ``trouble''^19.2

Trouble such as particles capable of lifting themselves up by their own metaphorical bootstraps...

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