Show that, when the monochromatic, complex electric field is of the form E0ei(kr−ωt)E, where E^ is the field polarization, that the following identities are valid:
∇⋅E=Re(ik⋅E0ei(k⋅r−ωt)),
∇×E=Re(ik×E0ei(k⋅r−ωt)),
∇2E=Re(−(k⋅k)E0ei(k⋅r−ωt)), That is, for monochromatic plane waves, ∇→ik and ∂/∂t→−iω.
∂E/∂t =Re(−iωE0ei(k⋅r−ωt)).