Regarding Electromagnetic Plane and Maxwell equations

Question

I asked this on the math.stackechange but I was told that it might be a good idea to ask here too since my problem is physics/math! Here is the question:

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!

Answer 1

Your problem is that you're not used to $\partial/\partial y$ being treated as a basis vector. This is a common concept in differential geometry (and thus applicable to relativity), where "vectors" are really partial derivatives (that is, the directional derivative that that vector is associated with).

It is somewhat uncommon to actually have the $E$ and $B$ fields written out in this way, however. Usually by this point, you're firmly entrenched in index notation, and you just deal with the components of the Faraday bivector, usually denoted as $F_{\mu \nu}$.

Overall, though, you should be able to convert $\partial/\partial y$ into your preferred notation for a unit basis vector and then do Maxwell's equations as you're most familiar.