Notations, Formulas, Terminologies and Language in Differential Equations

  1. Abuse.

    1. If an arc has a parametric representation which is smooth, then we say the arc is smooth. If the given parametric representation of an arc is not smooth, it would be very difficult to determine whether the arc is smooth. This is because smoothness is actually the attribute of one of the arc's parametric representation rather than the arc itself [Cou2, vol.2, p.88, l.- 3-l.- 1].

    2. The language used in the paragraph, [Cou2, vol.2, p.648, l.-19-p.649, l.2], is awkward and confusing. It requires a more technical language such as topology (more precisely, the topic of topological subspace) to clarify its meaning.

    3. People want to know the true meaning of mathematics rather than who did what in mathematics.
          In [Hob, p.265. l.-1; p.371, l.7-l.13], Hobson wants to impress his readers  and notes that he has improved other people's work. If I grade someone's work, my evaluation will be based heavily on originality rather than improvement of others' ideas. The above passages of Hobson remind me of a politician's trick. In politics, if a government has made a mistake, the politician who is supposed to be responsible for the mistake creates a crisis to divert people's attention. However, in mathematics, we judge the skill of a mathematician only by his performance. Let us examine a section of Hobson's work. In [Hob, p.282, l.4], Hobson says that m has the values n, n-1, n-2, …. Actually, it is -m that has the values n, n-1, n-2, …. He has made the same mistake in [Hob, p.283, l.10]. The mistake continues until he says that the values of m are n+1, n+2, … in [Hob, p.283, l.-7]. In my opinion, the first requirement for impressing readers is to be free of mistakes. The second requirement is to be free of misprints. [Hob] has quite a few misprints. Some of them are obvious. For example, Qnm(m)p=Qm-n-1(m) in [Hob, 204, l.2]should have been Qnm(m)=Qm-n-1(m). Others are difficult to detect. For example, The right-hand side of the equality in [Hob, p.265, l.12] should have a minus sign. As far as I can remember, the textbook containing the most misprints is [Bir], an undergraduate textbook on differential equations written by a Harvard professor teamed with a MIT professor. Almost every page of this book has misprints. This exposes a serious problem with academics in America. Most scholars are more enthusiastic about boosting their personal fame through hasty publication and less about the quality of their work. The third requirement is clarity. It is wiser to spend more space to clarify a confusing point (e.g., in terms of clarity, [Hob, 220] is poorly written) than to applaud oneself.

    4. Strictly speaking, the statement given in [Gon, p.706, l.9] is incorrect because uniform convergence requires at least two variables. However, we can easily make a correction. Write z = reiq. The statement can be interpreted as follows: (reiq)a+1 f(reiq) 0 uniformly in q as r0. It is true that the interpretation of the formula given in [Gon, p.714, l.5] is more informative if we interpret the formula using the concept of uniform convergence rather than that of convergence.

    5. What does f mean?
          In [Zyg, vol.1, p.40, l.-10], "f is an integral" means "f L1 is absolutely continuous". In [Zyg, vol.1, p.6, l.-6], "any function f (x)" means "any f L1"; otherwise, [Zyg, vol.1, p.7, (4.3)] cannot be defined. However, in [Zyg, vol.1, chap. II, 3], we should not assume that f L1.

    6. The angle q given in [Zyg, vol.1, p.99, l.1] is confusing. In order to avoid ambiguity, the two variables of an angle function, its initial side and its terminal side, should be specified. If a real value of the function is given, the angle will be determined if we follow convention and consider counterclockwise positive.
      Remark. "f(r,x) oscillates finitely" given in [Zyg, vol.1, p.99, l.4] may not be a descriptive statement. Zygmund should have said that q may have several sequential limits.
    7. A summation should not just list a few first terms; it should indicate the last term. Notations should not be awkward; they should be able to reveal simple relations and should be easy to handle. Compare the equalities given in [Hob18, p.107, (7); p.108, (10)] with those given in [Wat, p.33, l.17-l.18].

  2. Useful rules and notations.

    1. Due to the good use of the vector notation, the proof of [Ru1, p.190, Theorem 9.12] can be established word by word from its 1-dim counterpart. Consequently, the proof of [Cou2, vol.2, p.55, l.10-p.57, l.10] seems awkward in comparison to that of [Ru1, p.190, Theorem 9.12].

    2. The five-step criteria given in [Haw, pp.165-166] and [Haw, pp.167-168] give a short summary of how to obtain derivatives. The underlined logic follows the criteria in a stepwise fashion. The complete details of the proof of the criterion would be messy, but its key idea is very simple. For example, [Haw, p.167, (11-18)] is obtained by applying [Haw, p.166, (11-14)] twice; [Haw, p.169, l.-9] is obtained by applying [Haw, p.162, (11-3) & p.166, (11-15)].

    3. Notation assignments should be organized according to the proof's pattern.
          Weatherburn assigns his notations in the beginning of the proof [Wea1, p.90, (1)], so they fail to reflect any information about the proof's pattern. In fact, the massive number of his notations impedes recognition of the proof's pattern. In contrast, the way that Forsyth organizes his notation assignments is based on updated knowledge about the proof's pattern. He assigns similar notations for similar arguments. Therefore, his notations facilitate the completion of the proof. In fact, the proof of [For, p.45, l.-13-l.-11] can be almost immediately obtained from [For, p.45, l.9-l.17] by replacing the unprimed notations with primed notations.