Notations, Formulas, Terminologies and Language in Differential
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.-
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.
People want to know the true meaning of mathematics rather than who did what
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.
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
uniformly in q as r®0. 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.
What does f mean?
In [Zyg, vol.1, p.40, l.-10], "f is an integral"
means "f Î L1 is
absolutely continuous". In [Zyg, vol.1, p.6, l.-6],
"any function f (x)" means "any f Î 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
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
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].
Useful rules and notations.
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].
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)].
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.