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Theorems in Mechanics

  1. A theorem's formulation (Compare [Rei, pp.211-212, 6.4] with [Pat, pp.53-61, 3.2]).
    1. In formulating a theorem, we should not conceal the original problem that we want to solve (see [Rei, p.211, l.7]).
    2. In formulating a theorem, we are not allowed to ignore the effects of the value of a variable on the conclusion. For instance, whether the a in [Rei, p.211, l.-16] is 1010 or 1020 does not affect the validity of [Pat, p.59, (33)]. However, a must be large enough [Rei, p.211, l.-16] to allow the statistics to make sense.
    3. In a theorem, we must distinguish a constant from a variable [Rei, p.211, l.7].
    4. One should not coin new terminologies in a theorem (see [Pat, p.136, 6.1]).

  2. When applying a theorem, we use the theorem's result instead of its proof.
        [Lev2, p.116, (5-121)][Lev2, p.117, (5.122)] is a theorem. After applying the theorem twice, we obtain [Lev2, p.117, l.10]. After applying the theorem k times, we obtain [Lev2, p.117, (5.123)]. Levine's repeated use of the theorem's proof [Lev2, p.117, l.8] is not the right way to apply a theorem.

  3.  
    1. [Jack, pp.2-5, I.1] discusses the origin of the electric and magnetic fields, the situations to which the classical Maxwell's equations apply, and the situations that require a quantum mechanical treatment of the electromagnetic fields.
    2. [Jack, p.9, l.18-21] gives the lower limit and the upper limit of the orders of magnitude in length scale where the photon mass can be taken to be zero and the inverse square law holds.
    3. The precision of the inverse square law is given in [Jack, p.7, l.16 & l.25].

  4. From the macroscopic point of view, the charge distribution and the electric field of a conductor are not continuous across its surface [Jack, p.21, l.5]. However, from the microscopic point of view, these two functions are continuous across its surface [Jack, p.21, Fig. I.5].

  5. The discrete and continuous versions of a theorem to which linear superposition can apply are equivalent.
        For example, [Jack, p.26, (1.4)] [Jack, p.26, (1.5)]. From discrete values of q to a continuous function r, we use the limit of sums. From a continuous function r to discrete values of q, we use the properties of the d-function [Jack, p.27, (1.6)].

  6. The validity of Maxwell's equations.
        At its core, physics is an experimental science. Therefore, we must establish the validity of the laws in physics through experiments.
    1. The validity of the inverse square law [Jack, p.9, l.3-l.21].
    2. The validity of Maxwell's equations in vacuum or the microscopic fields inside atoms and nuclei.
          The first question is whether Maxwell's equations are a system of linear differential equations. The second question is what is the origin of small nonlinear effects. The answers are given in [Jack, p.13, l.6-l.11].
    3. Solving Maxwell's equations in macroscopic media.
      1. The General relation [Jack, p.13, (I.9)].
      2. Weak fields (other than ferroelectrics or ferromagnets): [Jack, p.14, (I.10)].
      3. Strong fields (other than ferroelectrics or ferromagnets): [Jack, p.16, (I.12)].

  7. It is unnecessary to prove the corollary of a theorem by repeating the same argument of the theorem. We should derive the corollary directly from the theorem.
    Example. Huang derives the corollary in [Hua, p.13] directly from Carnot's theorem [Hua, p.12], while Reichl derives the corollary by repeating the same argument of the theorem [Reic, p.26, l.-10]. In other words, Reichl fails to take advantage of the logical shortcut to the corollary.

  8. A formula must be compact, organized and good for application. Each term must have a physical meaning.
    Examples.
    Conservation theorem: [Rei, p.529, (14.5.1)] (good); [Hua, p.96, (5.14)] (bad).
    Time dilation: [Rin, p.27, l.17] (good); [Lan3, p.8, (3.2)] (bad).

  9. The partition theorem simultaneously specifies coordinates and momenta. According to the Heisenberg uncertainty principle, it requires [Rei, p.246, (7.4.3)] to make the simultaneous specification valid. Examples:[Rei, p.247, l.-16-l.-3; p.241, (7.2.10)]. By contrast, [Rei, p.247, l.-2-p.248, l.3] provides an example in which the simultaneous specification is incorrect. Consequently, if a theorem is valid in statistical mechanics, we must ask further whether it is valid in the classical regime or in the quantum regime.


  10. For harmonic oscillators, [Rei, p.253, (7.6.12) gives the general formula in quantum mechanics. In the classical regime [Rei, p.253, (7.6.13)], E is given by [Rei, p.253, (7.6.14)]. In the quantum regime [Rei, p.253, (7.6.15)], E is given by [Rei, p.253, (7.6.16)].