Why Fourier Transform moves us to the time domain?
Circles, lots and lots of circles...
When we derive \(F_{time}\), we are bending to form a circle around which time travels by applying constrains and conditions. Fourier transform represents everything from \(t=\cfrac{x}{c}\) in circles, \(e^{-i2\pi f t}\). Both paths leads to Rome, but Fourier transform does not conserve energy. Fourier coefficients are not constrained by,
\(a^2_{t1}+a^2_{t2}+...a^2_{tn}+...=A.E_s(b_{s1},b_{s2},...b_{sn}...)\)
where \(a_{tn}\), \(n=1,2,3..\) are coefficients of the transform in time and \(b_{sn}\) are the coefficients/parameters in space. \(E_s(...b_{sn}...)\) is a function that provides the total energy of the system in space. \(A\) is a constant, for the system, ie. for the set of equations that provides \(b_{sn}\).
Fortunately, the energy change in moving between two valid points (energy states) in a conservative system is irrespective of the path taken from one point to the other. Since energy is conserved between space and time, the discrepancy will always be \(\cfrac{1}{4}\) the total energy in space, as shown in one instance in the post "Energy Accounting With Fourier" dated 08 Jul 2016.
We naturally derive physical parameters in per unit time as the "force in a field" was mistaken as the Newtonian force and not power, the change in energy per second.
\(\cfrac{1}{4\pi r^2}\cfrac{q|_{a_\psi}}{\varepsilon_o}=F.c=\cfrac{\partial\,KE}{\partial\,t}\ne F_{newton}\) --- (*)
The notion of flux was abandoned a long time ago.
The unit meter in \(c\) from the (*), unlike the "per unit time" mentioned above, remains unaccounted for. This could be the reason for the confusion between intensity and power.
You know the bull is going round and round when what it unloads forms into a circle.
