  The goal of this article is to derive the reciprocity theorem, mutual energy
theorem from Poynting theorem instead of from Maxwell equation. The Poynting
theorem is generalized to the modified Poynting theorem. In the modified
Poynting theorem the electromagnetic field is superimposition of different
electromagnetic fields including the retarded potential and advanced potential,
time-offset field. The media epsilon (permittivity) and mu (permeability) can
also be different in the different fields. The concept of mutual energy is
introduced which is the difference between the total energy and self-energy.
Mixed mutual energy theorem is derived. We derive the mutual energy from
Fourier domain. We obtain the time-reversed mutual energy theorem and the
mutual energy theorem. Then we derive the mutual energy theorem in time-domain.
The instantaneous modified mutual energy theorem is derived. Applying
time-offset transform and time integral to the instantaneous modified mutual
energy theorem, the time-correlation modified mutual energy theorem is
obtained. Assume there are two electromagnetic fields one is retarded potential
and one is advanced potential, the convolution reciprocity theorem can be
derived. Corresponding to the modified time-correlation mutual energy theorem
and the time-convolution reciprocity theorem in Fourier domain, there is the
modified mutual energy theorem and the Lorentz reciprocity theorem. Hence all
mutual energy theorem and the reciprocity theorems are put in one frame of the
concept of the mutual energy. 3 new Complementary theorems are derived. The
inner product is introduced for two different electromagnetic fields in both
time domain and Fourier domain for the application of the wave expansion.
