- 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 g
_{1}(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, §4A11, §4A12].
- 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].

- Express (the energy associated with the frequency range dκ) in terms of the Fourier transform g
- Advanced-level interfaces between electromagnetism and the calculus of
variations viewed from multiple perspectives (Dirichlet's problem [Jack,
p.39]).
- 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
of image
charge to specify the value of ¶G
_{D}/¶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.

- Jackson relates [Jack, p.44, (1.63)] to the
- 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 not be considered 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 E_{VI}] 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.