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Hello everybody I am kind of struggling with an assignment question related to general relativity. I am just going to type what the question is and explain what I am having trouble with:

Consider an electromagnetic plane wave in Minkowski space. We assume that in stationary coordinates $(t,x,y,z)$ the wave propagates in the positive $x$-direction, and that the components of the electric field $E$ and the magnetic field B are only a function of $u = t-x\ ($here $c = 1)$:

$E = E_{y}(t-x)\frac{\partial}{\partial{y}}+E_{z}(t-x)\frac{\partial}{\partial{z}}$

$B = B_{y}(t-x)\frac{\partial}{\partial{y}}+B_{z}(t-x)\frac{\partial}{\partial{z}}.$

Moreover, we assume E and B are compactly supported in $u$.

(1) Use Maxwell's equations to show that:

$B_{y}=-E_{z},$ $\space$ $B_{z}=E_{y}$.

First of all sorry if the formatting isn't that great. Now I am having trouble with this because I am not exactly sure what $E$ and $B$ are... by that I mean I know $E$ and $B$ are supposed to be vector fields but in that format they look like scalar fields. Is there another way to represent those two equations, I feel that if I understand the notation it would make doing this question pretty easy. Any help would be greatly appreciated. I just want to understand the notation properly and how to work with it so that I can actually apply Maxwell's equations on them and figure out those identities. Thank you and have a good day!

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Well the reason I wrote it here versus a more physics oriented forum is my question is more regarding mathematics than physics. Do you think I ought to ask there anyways just in case? –  InsigMath Mar 10 '13 at 20:28
    
Well I asked them too now... I guess it does not hurt to have multiple people looking at the question. –  InsigMath Mar 10 '13 at 20:31
    
@Insig: It doesn't if you link both questions to each other to avoid duplication of efforts. –  joriki Mar 10 '13 at 20:32
    
@Insig: I removed the general-relativity tag; this has nothing to do with general relativity; in fact it's not even specifically related to special relativity; it's just electrodynamics and vector calculus. –  joriki Mar 10 '13 at 20:36
    
@joriki: fair enough, this is for a class on GR so that is why I used that tag but vector-analysis is probably more appropriate. –  InsigMath Mar 10 '13 at 20:38
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1 Answer

up vote 3 down vote accepted

This is a notation from differential geometry that's unusual in physics (as far as I'm aware). The operators $\frac{\partial}{\partial y}$ and $\frac{\partial}{\partial z}$ are basis vectors for vector fields; see e.g. this PlanetMath article for an introduction (in particular where it says "in some sense" :-)

If you don't want to worry about differential geometry, you could write the field in good old-fashioned vector notation as

$$ E=\pmatrix{0\\E_y(t-x)\\E_z(t-x)}\;,\quad B=\pmatrix{0\\B_y(t-x)\\B_z(t-x)}\;. $$

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Thank you so much, I think this makes it a lot more clear! –  InsigMath Mar 10 '13 at 20:48
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