... rank1.1
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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... respectively3.1
Note that we define the magnitude of the vector $ \vA$ (written either $ A$ or $ \vert\vA\vert$ ) in terms of the inner product:

$\displaystyle A = \vert\vA\vert = +\sqrt{\vA \cdot \vA} = (A_x^2 + A_y^2 + A_z^2)^{\frac{1}{2}} $

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...contraction3.2
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 $ i$ th index is summed ``out'' of the expression, so that the result has fewer indices on the left than it has on the right.
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... equations11.1
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.
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... separates11.2
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) $ \Vec{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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... that11.3
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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...real11.4
Whoops! You mean $ \hat{\bf n}$ doesn't have to be real? See below. Note also that we are implicitly assuming $ \epsilon $ and $ \mu$ are real as well, and they don't have to be either!
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... real11.5
Heh, heh.
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... starters11.6
Note Well! The $ \Hat{n}$ we are using here is not the direction of $ \Vec{k}$ , it is the direction of the normal to the surface, that is to say $ \Hat{z}$ .
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... medium11.7
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.
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... ``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.
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... form11.9
Remember the algebra where we got the square root in the first place? Well, do it backwards.
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... fields11.10
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 $ x^2$ where $ x$ is the displacement from equilibrium, corresponding to a linear restoring force to lowest order.
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... motion11.11
You do remember Newton's law, don't you? Sure hope so...
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... electron11.12
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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... function13.1
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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... function13.2
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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... time13.3
Heh, heh, heh...:-)
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... wavelength13.4
We will learn to treat certain exceptions, believe me.
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... series13.5
Taylor? Power? Laurent? Who can remember...
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... that13.6
This really isn't an assumption. We could equally well write $ \lapl$ 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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... reasons13.7
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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...exactly13.8
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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... solution13.9
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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... functions14.1
From now on, this term is generic unless clearly otherwise in context.
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... sphere''16.1
Hyuk, hyuk, hyuk...
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... anything16.2
Even if it's true ...
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... reflection16.3
Sorry...
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... frames17.1
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''17.2
``Hyperbolic'' because of the relative minus sign between $ x^2$ and $ ct^2$ . More on this later.
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... past17.3
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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...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 $ \epsilon _{ijk}$ is a third rank fully antisymmetric tensor.
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... verify18.2
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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... is19.1
Note that I've rearranged this slightly to avoid having to do lots of stuff with $ g$ sandwiches below.
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... tanstaafl20.1
There Ain't No Such Thing As A Free Lunch. No kidding.
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... us21.1
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''21.2
Trouble such as particles capable of lifting themselves up by their own metaphorical bootstraps...
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