k-forms on a manifold are defined in a very artificial way. [War, p.17,
l.3] may give us a trace of inspiration.
Why it is so hard to recognize the origin.
General formulation makes the original meaning fall apart.
For a local
1-parameter group, compare [War, p.39, l.16] with [Olv, p.18, Definition
1.20 & p.20, Definition 1.25].
The purpose has been diluted or eliminated.
(Exterior derivative) The formula (*) in [Spi, vol.1, p.289, Theorem 13] is
designed for efficient computation (see [O'N, p.154, Definition 4.4]). The
important tools (coordinate patch [O'N, p.150, Exercise 4(b)] and basis [O'N,
p.153, Lemma 4.2]) for this purpose disappear after Spivak's generalization.
We should develop a concept step by step. If we miss one step, we
will leave a gap there.
[Boo, chap.1, §5]
indicates that the concept of real projective space naturally leads to that
of tangent bundle, and vice versa.
A byproduct has a real meaning only when it is attached to the main
idea. Local group [Olv, p.18, Definition 1.20] is a byproduct of local
isomorphism.
(Global setting, local operator) [Po3, p.133, Definition 30] → (Local
setting) [Po3, p.137, D)] → (Local setting, local operator) [Po3, p.138,
H)].
Sometimes a generalized theorem has two original resources. Ascoli's
theorem [Ru3, p.369, l.−9-l.−1] is a partial converse [Po3, p.189,
l.17-l.18] of the original proposition. It is also a generalization of [Ru1,
p.35, Theorem 2.41 (a) Þ (c)] (see [Ru3, p.369, Theorem A4 (c)
Þ (b)]). To
facilitate understanding a generalized theorem, its formulation
should be oriented towards preserving the original ideas.
Please compare [Ru3, p.369, A4 & A5] with [Dug, p.267, Theorem 6.4]. See
also [Ru3, p.32, l.−11-l.−7; p.34, l.1-l.8].
The key step has been omitted, merged, generalized, or obscured. Compare the proof of [Perr, p.258, Satz 26] with those of [Wal, p.50, l.13] and [Scott and Wall: A convergence theorem for continued fractions, p.158, Theorem B].¬
Why we trace origins.
Origin is a central guide for development. Féjer's theorem [Ru1, p.176,
Theorem 8.15] can be generalized in many areas:
Hilbert spaces [Ru2, p.90, Theorem 4.18 (b)].
Topological groups (Peter-Weyl) [Po3, p.225, Theorem 32]. Once we match
our topic with the original model, a generalization can be inspired by a
major point [Po3, p.225, Theorem 32] or a minor point [Po3, p.232, Example
58].
To completely understand the development of a concept, we should consider
the contributing factors from all angles and develop the concept step-by-step
from the origins.
C(Ω) [Ru3, p.31, §1.44] → (vector space, seminorms) [Ru3, p.26, Theorem 1.37]
→ σ-compact metric space [Dug, p.272, 8.5] → c-topology [Dug, p.257,
Definition 1.1].
If we trace to the origin [Col, p.179, l.10] of Green's function for a
boundary-value problem, we will immediately have a clear picture of δ-function
[Col, p.160, Fig. III.2] and the general solution [Col, p.179, l.13]. If we
twist the meaning of Green's function somewhat [Arn1, p.55, l.2], the
intuitive physical meaning of the generalization process is completely lost.
Thus employing a twisted version or a generalization halfway through the process [Col,
p.182, (III.37)] only makes it unnecessarily difficult to recognize the general
solution. By the way, it is simple and natural to demonstrate the concept of
δ-function with [Col, p.160, Fig.III.2]. In contrast, the introduction to
δ-function [Arn1, p.53, l.−3] is artificial and awkward.
Origin versus definition.
The theory of differential forms originates from consideration of what to do with integrands when
we transform a curved surface to
a plane and try to make the meaning of orientation precise in proving
Stokes' theorem [Cou2, vol.2, pp.611-615, §5.10.a].
The design of the definitions in [Cou2, vol.2, p.318, l.12; p.319, (61b)] was
oriented toward this purpose.
The concept of partitions of unity originates from the attempt to
define an integral over an oriented simple surface [Cou2, vol.2, p.634, l.-10-l.-2].
The origin of the fundamental solution of a partial differential
operator [Ru3, p.192, l.-7]: [Lan2, p.158, l.-8-p.159,
l.3].
The Poisson distribution serves to establish the similarity between molecular collisions and a game of chance [Reif,
p.465, l.-12-p.467, l.18]. The distribution
involves two sides
(molecules and dice) of the story. Theoretical statisticians use an abstract
language to describe molecular collisions and omit the original beautiful
story of molecules. This omission makes the restrictions imposed in [Reic,
p.192, l.-1] look artificial.
The osculating plane, the normal plane, and the rectifying plane [Kli, p.20, l.1-l.10].
The binormal [For, p.4, l.-3-l.-2].
Multiplication of matrices [Wu, chap. 2, §1].
[Cou2, vol. 1, p.378, §4.2.b]
discusses many properties of a catenary, but leaves out its origin, the
differential equation of a catenary. For the catenary's origin, see [Col,
chap. III, pp.164-167, §2.4].
For a complicated formula, if we can trace back to its original simple form, we will still be able to
grasp its essence despite its complicated appearance.
Examples.
Closed forms.
A closed curve (i,e., a closed singular 1-cube) [Spi, vol. 1, p.342, l.1-l.2]
®
A closed k-chain [Spi, vol. 1, p.342, l.2]
®
A closed k-form [Spi, vol. 1, p.342, l.5] because of the formal similarity between d and
¶ [Spi, vol. 1, p.342, l.9-l.10].
If we fail to trace back to a theory's origins, a definition will look
artificial and the discussion will look disorganized and uninsightful.
The Frobenius method [Guo, §2.4 &
§2.5] and the theory of asymptotic
expansions originates from the same ideas: power series and analytic
continuation. If we fail to study [Was, §4
& §5], the discussion in [Guo,
§2.4 & §2.5]
will look disorganized and uninsightful.
Remark. If we fail to study [Was, p.28, l.10-l.-1], the
definition given in [Bir, p.233, Definition; Guo, p.56, l.-8-l.-7]
will look artificial.
The confluent hypergeometric equation [Cod, p.132, l.15].
Genesis of an ordinary differential equation [Inc1,
pp.4-11, §1.2-§1.24].
Consideration of the origins of ideas may help us review the course of mathematics' historical developments.
Example. Singular points [Inc1, p.69, l.-5]
® (critical points, equilibrium states, stability) [Bir, chap. 5,
§7-§8].
The concept of the adjoint equation
originates from the concept of an integrating factor [Inc1, p.123, l.17-l.21;
p.124, l.4; l.11-l.17].¬
Why we need to study the firsthand math literature
A secondhand math book may leave out an important message.
The similarity between the series given in [Wat1, p.433, l.-4]
and that given in [Wat1, p.434, l.3] inspires us to study the function
Ã(z) [Wat1, p.434, l.1]. However, [Gon1, p.288, l.-8-p.292,
l.-5] fails to point out this important similarity.
In a secondhand book, a headline may be deemphasized and buried in a long passage.
Example. Compare [Wat1, p.513, l.-11] with [Guo,
p.546, (4); p.547, l.-4-l.-3].
A second-hand book may leave a gap in a proof. [Wat1, p.449, l.-6-l.-5] points out that
the zero (or pole) of the product and the zero (or pole) of
f(z) at the
origin are of the same order of multiplicity. The proof of [Guo, p.479,
(8)] neglects this consideration and thereby Guo leaves a gap in the proof.
The top priority of mathematics is to solve a problem with the most
effective method rather than to generalize theorems. Suppose a firsthand book solves each problem using the most effective
method. A second-hand book may use the most complicated method to solve a
problem regardless of whether or not the problem is complicated. Example. [Wat1,
p.115, l.8-l.11] says that the results of [Wat1,
§6.221] remain true if we replace the condition of [Wat1,
§6.221] with a less stringent condition.
The latter case is a generalization of the former case. The proof of the
former case uses a simple method, while the proof of the latter case requires a more
delicate method: Jordan's lemma. A secondhand book may preserve only the
latter case simply because the former case is a special case of the latter
case. The omission of the simpler method may lead readers to believe that
they must use the more complicated method to
prove this type of theorems in all cases. This
indiscriminate approach would contradict the goal of mathematics: Solve
a problem with the most effective method. Remark. In [Gon, p.693, l.-4],
González uses Jordan's lemma to prove [Gon, p.695, (9.11-5)]. In [Gon, p.701,
l.1], González uses Jordan's lemma to prove [Gon, p.701, (9.11-23)]. In these
two cases, González uses Jordan's lemma as a pass to prove the above two
formulas, but he fails to point out the unique feature of Jordan's lemma: only
the case a =1 [Gon, p.693, l.11; p.700, l.-3]
requires the use of the delicate Jordan's lemma; for the case
a >1, the use of [Gon, p.682, Lemma 9.3] would be
sufficient [Wat1, p.115, l.8-l.11]. ¬
Suppose the firsthand book organizes formulas into groups and in each
group the book gives an effective method to build an exhaustive list of
formulas from a single formula. The secondhand book may only give some sample
formulas from different groups and mingle them into a single pile. They may not give any method of building a
complete list of formulas for each group.
Example. Compare [Wat1, p.487, Example 1; p.488, Example 2] with [Guo, pp.523-524, Exercise 2].
A secondhand book may avoid the important and dwell on the trivial, and
may thus become superficial. If a passage of the original book is difficult
to read, the authors of secondhand books should provide more clues to make it
more accessible to readers. On the contrary, these authors may either not say
a single word about it or just lightly touch on the difficult topic.
Sometimes, they may even misinterpret the passage.
Examples. The value of cn (K+iK') is determined by a path of integration
[Wat1, p.502, l.2 & l.20]: The minus sign given in [Wat1, p.502, l.20] comes from the
minus sign given in [Wat1, p.502, l.5]. In contrast, [Guo,
p.536, l.-11-l.-8] fails
to explain this difficult point.
The purpose of the statement given in [Wat1,
p.502, l.7] is to generalize the statement dn K = +k' from the case 0 < k <1
[Wat1, p.499, l.5-l.6] to the case k Î C\((-¥,0)È(1,+¥
)). In [Gon1, p.399, l.2-l.3], González claims that the two cases given in
[Gon1, §5.23] are special. In fact, from the
viewpoint of analytic continuation, there is nothing special about these two
cases. [Gon1, §5.23] only discusses the
cases (l = 1-2c [Gon1,
p.421, l.-4]) in which the domain of the function 2iK'
[Wat1, p.501, l.3], one of Tan's periods, falls on
the branch cuts given in [Wat1, p.501, l.17-l.18].
Remark. Of course,
a secondhand book may make some improvements to
the original book. For example, the derivation of [Guo, p.504, (1)] is clearer
than that of the left-most equality given in [Wat1, p.466, l.-6].
As a second example, [Gon, p.680, Lemma 9.2] is a generalization of [Wat1,
p.115, l.12-l.15]. For a third example, read [1].
In the above, I emphasize the importance of first-hand books because it is
possible that a second-hand book may
fail to fully convey the right messages.
The origin of asymptotic expansions [Wat1, .150, l.4-p.151, l.3].
Bessel functions of the second kind [Guo, p.365, l.-2-p.366, l.-7].
[Har, p.57, (5.7)] is originated from [Har, p.57, (5.5)].
The formula given in [Har, p.54, l.-5] is originated from [Har, p.48, (2.5)].
The origins that require a proof
(The minimal surface area with a fixed boundary) The definition of a minimal surface with a fixed boundary is given in [O'N, p.207, Definition 3.6]. However, it fails to give an explanation. A justification is given in [Fomi, p.24, Example].
The spectrum of an S-L system [Bir, p.261, l.10-l.23].