Maxwell Equations and Fourier Expansions

[JTPF]

Homework Statement

The field E(r,t) can be written as a Fourier expansion of plane waves $$E(r,t)=∫E(k,w)e^{i(kr-wt)}d^{3}kdw$$ with similar expansions for other fields.

Need to show the derivation of $$kXE(k,w)=wB(k,w)$$ from Faraday's law $$∇XE(r,t)=-∂B(r,t)/∂t$$ and also the derivation of $$kXH(k,w)=-wD(k,w)$$ from Ampere's law $$∇XH(r,t)=∂D(r,t)/∂t$$

Homework Equations

$$∫e^{ax}=(1/a)e^{ax}$$

The Attempt at a Solution

I thought the Fourier expansion expression for E meant to integrate with once with respect to w and three times with respect to k, so get:

$$∫E(k,w)e^{i(kr-wt)}d^{3}kdw = (1/r)x(1/r)x(1/r)x(-1/t)e^{i(kr-wt)}=(-1/r^{3}t)e^{i(kr-wt)}$$

But that clearly doesn't give the result, no k or w there at all... what am I doing wrong? I get this isn't tricky but can't figure it out.

 

[cosmic dust]

Just try to evaluate curlE : (curlE)_i = ε_ijk∂_jE_k and compare the result with the definition of cross product : (A×B)_i = ε_ijkΑ_jB_k. You don't have to try to evaluate the Fourier integral; actually you can't, since E(k,w) is unknown (if known, then the field E(r,t) would also be known). Make the partial derivations of the Fourier integral by derivating directly the integrand.

 

[JTPF]

I still don't get how to do this

So I should take $$(∇E)i = ε_{ijk}∂_{j}E_{k}$$ and compare the result with the definition of cross product $$(A×B)i = ε_{ijk}Α_{j}B_{k}$$?

So $$(∇E) = ∂_{j}E_{k} = A_{j}B_{k}=AXB=kXE$$?