The completeness of a discussion of Fourier analysis is determined by how extensively it interfaces with the theory of light.
Express (the energy associated with the frequency range dκ) in terms of the Fourier transform g1
(of the wave profile f ) [Dit, p.95, 4(75)].
A long wave train is associated with a narrow range of
frequencies [Dit, p.71, l.12-l.13; pp.96-97, §4A10,
Distribution of Energy for a damped harmonic wave [Dit, p.98, 4(94)].
A wave group spreads slowly as it advances in a dispersive medium [Dit, p.100, l.8].
When the wave profile is given, the total energy can be calculated simply using Parseval's theorem
[Dit, p.100, l.-4].
The origin of diffraction: The spatial Fourier expansion of a non-plane wave
contains components whose wave vectors have different directions [Lan2, p.154, l.14-l.16].
Advanced-level interfaces between electromagnetism and the calculus of
variations viewed from multiple perspectives (Dirichlet's problem [Jack,
Jackson relates [Jack, p.44, (1.63)] to the minimal energy [Jack, p.41, (1.54)].
[Jack, p.44, (1.65)] can also be proved by a formula similar to [Cou,
vol. 1, 192, (25)] or to the formula in [Cou, vol. 1, p.241, l.-5].
[Jack, p.46, l.12-p.47, l.6] shows the advantage and drawback of the Rayleigh-Ritz method
[Cou, pp.175-176, §IV. 2.2], which is a simulation of the calculus of variations
that reformulates the problem of finding the minimum of a functional into
a problem of finding an ordinary
minimum [Cou, vol. 1, p.176, l.9].
[Jack, p.47, l.-15-p.48, l.2] establishes the
variational nature of the relaxation method.
[Jack, pp.64-65, §2.6] uses the concept
charge to specify the value of ¶GD/¶n'
in [Jack, p.39, (1.44)] for the case where the boundary S is a sphere, and
thereby gives the physical origin of the Poisson kernel [Ru4, p.41, l.1
or Ru2, p.255, Theorem 11.10]. Thus, by [Jack, p.59, (2.5)], the Poisson kernel essentially is the
surface charge density induced on the sphere due to a unit charge.
Remark. Hilbert and Jackson have the vision to spot the advanced-level interfaces
between electromagnetism and the calculus of variations considered from multiple
when studying Dirichlet's problem. [Fom]
and [Wangs] fail to discuss Dirichlet's problem. [Ru2, p.255, Theorem 11.10]
discusses Dirichlet's problem, but fails to relate it to its physical origin and
the calculus of variations. The shortcomings of these three books show that some physicists and mathematicians simply do
not have enough background to capture the theme and fully digest the material on this subject.
The discussion of thermal properties of solids requires the background of thermal physics. When reviewing this background, we should link
our discussion to the fundamental concepts of thermal physics: heat bath [Iba, p.86, l.-12],
the density of states[Iba, p.86, l.7], and the canonical distribution [Iba, p.86, l.-11-l.-8].
Thus, whenever we encounter a macroscopic physical quantity, we automatically associate it
with its average value in the probabilistic sense [Iba, p.87, (5.11)]. In
contrast, [Hoo] links the discussion only to the main results such as the
formula in [Hoo, p.49, l.-7]. Hook's link cannot
explain why a harmonic oscillator in a heat bath cannot have an eigenstate [Iba.
p.86. l.11]. Elementary considerations (Compare [Kit2, p.132, (42)] with [Iba,
p.96, (5.43)]) usually serve to give a quick and intuitive prerequisite that uses only rough estimates for evaluation. Therefore, the argument of elementary considerations should
rigorous because it fails to analyze the interfaces between two
topics with sufficient care.
Suppose we interface an abstract theory and a concrete application. On the one
hand, we can locate the essential point of the abstract theory; on the other
hand, the application may reveal the direction of the theory's further development.
For example, in order to prove [Ashc, p.270, l.10], we study [Coh, Complement EVI]
and find that [Coh, p.752. (67)] is the most important result in the complement.
If we study a theory alone, it would be difficult for us to recognize its generalizability and limitations. By interfacing Kirchhoff's laws with Maxwell's equations, we find the natural generation [Wangs, p.450, (27-5)] and the
limitation [Wangs, p.451, l.13-l.14] of Kirchhoff's laws.