  On transformation to the Fourier space $({\bf k}, \omega)$, the partial
differential Maxwell equations simplify to algebraic equations, and the
Helmholtz theorem of vector calculus reduces to vector algebraic projections.
Maxwell equations and their solutions can then be separated readily into
longitudinal and transverse components relative to the direction of the wave
vector {\bf k}. The concepts of wave motion, causality, scalar and vector
potentials and their gauge transformations in vacuum and in materials can also
be discussed from an elementary perspective. In particular, the excessive
freedom of choice associated with the gauge dependence of the scalar and the
longitudinal vector potentials stands out with clarity in Fourier spaces. Since
these potentials are introduced to represent the instantaneous longitudinal
electric field, the actual cancellation in the latter of causal contributions
arising from these potentials separately in most velocity gauges becomes an
important issue. This cancellation is explicitly demonstrated both in the
Fourier space, and for pedagogical reasons again in space-time. The physical
origin of the gauge degree of freedom in the masslessness of the photon, the
quantum of electromagnetic wave, is elucidated with the help of special
relativity and quantum mechanics.
