Theory and Its Application in Differential Equations
[Cou, vol.1, p.6, l.- 11-l.-
10] is an application of [Jaco, vol. II, p.59, Theorem 11, part 2]. Through this
translation, we discover that
Without the theorem's guidance, it would be difficult to find a solution
for the application.
Without any application, it would be difficult to understand the real
meaning of this theorem.
A textbook on advanced calculus should not be confined to theoretical
development. It should also build an arsenal of weapons for students to use when attacking
physics problems. In other words, in the course of its theory's development, a textbook of advanced calculus should provide a link to physics by proving the essential formulas
of physics. [Cou2, vol.2] proves all the formulas in [Cor, the front cover] (e.g., [Cou2, vol.2, p.62, (29)]), while [Ru1] proves very few of them.
A delicate (advanced) device designed for a complicated task is usually an
awkward and ineffective method when the device was applied to a simple task.
Example. Let the device be Barnes' integral representation [Guo, p.166, (7)].
If g-a-b ¹ integer, the case is simple. If
g-a-b is 0 or a negative integer, the case is complicated. Compare the method used in [Guo, p.167, l.1] with that used in [Guo, p.160, (4)].
After we recognize that certain patterns exist in the theory of space curves, we use these patterns as
a guide to study the theory of surfaces.
Examples. [Lau, p.21, l.3-l.6; p.28, l.-12-l.-7;
p.6, (1.15) ® p.63, (5.25); p.11, Exercise 1.7.4 ® p.62, Theorem 5.3.1].
A good theory extracts as much information as possible from studied material
and has a vision for organizing its results by expressing them in certain forms so that
its intrinsic properties can be easily recognized. If we can
express geometric or physical quantities in terms of tensors, we are convinced that relations among these quantities retain the same form under the coordinate
transformations (Compare [Lau, p.65, Theorem 5.5.1] with [Kre, p.91, Theorem
27.1]). Thus, expressing formulas of differential geometry in tensor form not
only enriches their meanings, but also is a necessity for physical applications.
[Chou, p.130, l.8-l.10] tells us when and how to apply the method of
separation of variables.
A theory should be mingled with examples in proper timing and proportion.
Otherwise, the theory will become a list of statements that all appear equal in
importance and lack a purpose.
Birkhoff uses good timing for his discussion of the singularities of the hypergeometric DE,
because he puts his discussion
in the section titled "Singular points at infinity". In order to study the
hypergeometric DE's singularity at infinity, we need the indicial equation at infinity [Bir,
p.250, (39)]. In contrast, the statement given in [Guo, p.135, l.-5-l.-4]
is too many pages away from its theoretical base [Guo, §2.6].
Furthermore, Guo fails to provide the important tool of the indicial equation at infinity
to facilitate the study of the regularity singularity at infinity for
the hypergeometric DE.
As a theory's development reaches a mature stage, discussion about it will fall into
patterns. The patterns dictate what we shall include in our discussion.
The formulation of the commutability of the operation of inducing a subspace topology with
the operation of attachment
[Dug, p.128, Theorem 6.4] is similar to that of the operation of inducing a subspace topology with
the operation of constructing a
Cartesian product of topologies [Dug, p.99, Theorem 1.2 (3)]. The discussion on
completeness [Dug, p.224, Theorem 1.4] follows the patterns of discussion on compactness [Dug,
p.295, Theorem 2.5]. Except when a pattern is encountered for the first time,
everything is routine, i.e., there is nothing creative.
The disadvantages of this pattern approach:
(1). The argument of a proof will proceed too slowly.
(2). In order to make the discussion complete, each step of the proof will be burdened with generalizations and clumsy structures.
(3). The key points will be buried in the sea of distractions and digressions.
It is more inspiring if we introduce the definition of a strong deformation retraction through the path
given in [Mun, p.343, l.5-p.345, l.9] rather than the one given in [Dug,
p.315, l.1-p.324, l.-1]. This is because the former
path is introduced when the need arises and because the former path highlights the trail of the concept's origin and applications.
Suppose system A and system B that are nearly the same. It is better to read
two books: one using system A and the other using system B. One should not read
just theorems in a single system and try to deduce the corresponding theorem in the other
system. That would be too troublesome
for frequent references.
The proof of
[Mun00, p.382, Lemma 62.2] uses some properties of compactness. [Dug, p.222,
Definition 1.1] and [Per, p.56, l.-2]
define compactness in slightly different ways. Munkres adopts the latter
definition. In linear algebra, [Her] defines a vector space over a field, while
[Jaco, vol. 2] defines a (left or right) vector space over a division ring.
Jacobson's definition has the advantage of helping us see that "column rank" = "row rank"
can be derived directly from the structure of a division ring rather than via the rank of a determinant [Jaco, vol. 2, l.-13-l.-5].
In electromagnetism, [Jack] uses SI units, while [Lan2] uses Gaussian units.
In order to gain a comprehensive understanding of a definition [a Lyapunov
function], we must go through several stages: the literal meaning
[Har, p.38, l.-20-l.-17]
® clarification [Pon, pp.204-205, (B)] ®
origins ® construction of a significant example [Pon,
p.206, l.-8-p.208, l.3]
® applications [Har, p.38, Theorem 8.1].
[Har, p.38] jumps directly from the literal meaning of a Lyapunov function to
its application and fails to substantiate the definition's meaning by
constructing a
significant example. Consequently, this approach
leaves a serious gap in the theory. Remark. Although we may use [Pon,
p.206, (E)] and [Har, p.39, Theorem 8.2] to prove [Pon, p.208, Theorem 19], the
proof given in [Pon, pp.209-211] gives an effective error estimate. Similarly,
the proof of [Pon, p.202, (A)] is more effective than that of [Har, p.38,
Theorem 8.1].
We use [Zyg] as our model to illustrate how to build a theory
(Managing resources) An average mathematical textbook will
say that a theory is founded on a few axioms. This view can only be derived from hindsight. A good mathematical textbook [Zyg]
shows that a theory (note the similarity between [Ru2, p.86, l.-13]
and [Ru2, p.96, (1)]) is synthesized from specific and concrete examples. In
order to establish or fully understand a theory, we must study the theory case
by case (Fourier series [Zyg, vol.1, p.7, (4.4)] and Fourier-Stieltjes series [Zyg,
vol.1, p.11, l.7]; note that the Riemann interal and the Riemann-Stieltjes
integral [Ru1, §6.1 and
§6.2] can be unified by means of complex
measures [Ru2, p.128, l.8]). For a theorem in the theory, we must find the most precise
error estimate and the most effective proof for each case (Example 1: Compare [Zyg, vol.1,
p.48, Theorem 4.12] with [Zyg, vol.1, p.45, Theorem 4.4]; Example 2 [Zyg, vol.1,
p.66, l.-15-l.-14]).
We would like to eliminate superfluous assumptions and keep the conclusion valid.
This allows us to keep a theory lean and helps us identify the exact cause that
leads to the conclusion. Example. [Wat1, p.85, l.-5-l.-2].
If a counterexample exists [Zyg, vol.1, p.303, l.-3-l.-1] after we weaken
a theorem's [Zyg, vol.1, p.63, Theorem 10.5] hypothesis [Zyg, vol.1, p.63,
l.14], we should try to strengthen our hypothesis somewhat to see if we can keep
the conclusion valid [Zyg, vol.1, p.63, Theorem 10.7].
(Precision) [Zyg, vol.1, p.63, Theorem 10.7] discusses a function's
condition which ensures the convergence of its Fourier series. We would like to
strengthen the theorem's hypothesis so that we may estimate the error term of
the convergence [Zyg, vol.1, p.64, Theorem 10.8]. If a counterexample shows that we
cannot improve the precision of the error term [Zyg,
vol.1, p.315, Example 10], we would like to further strengthen the hypothesis so
that the error term will meet the precision requirement
[Zyg, vol.1, p.64, Theorem 10.9].
By
comparing and analyzing the strength of two methods, we can expand a theory to a new frontier.
Theory A is stronger than theory B if theory A contains more theorems than
theory B [Bou, p.27, C4]. Similarly, method A is stronger than method B if
method A contains more conditions than method B; statement A is stronger than
statement B if statement A implies statement B. Examples. The summability of
a series using a specific summuation method refers to the convergence of its
corresponding partial sums. If a partial sum of
method A can be expressed as a linear combination of partial sums of method
B by a regular matrix [Zyg, vol.1, p.74, l.-17],
then we say summation method A is stronger than summation method B. For a
regular linear combination see the formula
given in [Zyg,
vol.1, p.74, (1.1)], where a_{ni}
can be treated as a probability distribution [Boro, p.5, l.15] when n is large [Zyg,
vol.1, p.74, (iii)]. For a trigonometric series, the method of summation given in [Zyg, vol.1, p.113, (12.11)] is stronger than the (C,1)-method [Zyg, vol.1, p.113, l.-2-l.-1].
Proof. Use [Zyg, vol.1, p.112, (12.5)].
[Zyg, vol.1, p.74, Theorem 1.2] shows that the summability of a series by a weak method implies the summability
of the series by a strong method.
Let statement B be (B_{H} Þ
B_{C}), statement B_{1}
be (B_{H} Þ B'_{C}),
and
statement B' be (B'_{H} Þ
B'_{C}). Suppose B is true but B_{1}
is false. If we desire
to establish a conclusion B'_{C} stronger than B_{C},
we must change B_{H} to a
stronger hypothesis B'_{H} so that statement B'
can be
true.
Example 1. Statement B: At every point x where f has a jump d, S'[f] is summable by
the logarithmic mean to d/p [Zyg, vol.1, p.107,
l.7-l.10]. If the (C,1) mean s_{n}
of a sequence {s_{n}} converges s, then the
logarithmic mean t_{n}
of the sequence {s_{n}}
converges to s [Zyg, vol.1, p.107, l.4]. The example given in [Zyg,
p.107, l.4-l.5] shows that the converse is false. Thus, the method of
the logarithmic mean is strictly stronger than the (C,1)-method.
We wish to establish a stronger conclusion: S'[f] is summable (C,1) to d/p.
[Zyg, vol.1, p.134, Example 3] shows that in general this is false. However, we
can prove its validity [Zyg, vol.1, p.107, Theorem 9.3] by adding an extra
assumption: F is of bounded variation.
Example 2: In view of the formula given in [Zyg, vol.1, p.80, l.-10], the procedure given in [Zyg, vol.1, p.109, l.-1]
strengthens the inequalities given in [Zyg, vol.1, p.109, l.-4].
It is difficult to go from [Zyg, vol.1, p.109, l.-4]
to [Zyg, vol.1, p.110, Theorem 10.2] because the goal requires a more refined method.
In contrast, it is easy to go from [Zyg, vol.1, p.110, Theorem 10.2] to [Zyg, vol.1, p.109, l.-4].
All we need is to use the formula given in [Zyg, vol.1, p.80, l.-10].¬
Assumptions
The origins and design of assumptions
Given assumptions in a theory, we must understand not only their literal
meanings but also their origins. For example, in [Zyg, vol.1, chap. III,
§2], [Zyg, vol.1, p.85, (2.6)] originates
from [Ru1, p.174, (76) (ii)]; [Zyg, vol.1, p.85, l.-2,
(A)] originates from [Ru1, p.174, (79)]; [Zyg, vol.1, p.86, l.5, (B)] originates
from [Ru1, p.174, (78)]; [Zyg, vol.1, p.86, l.-15,
(C)] originates from [Ru1, p.174, (80)]. Thus, the assumptions given in
[Zyg, vol.1, chap. III, §2] can be
considered a composition formed by mingling [Zyg, vol.1, p.74, (i), (ii), and (iii)]
with [Ru1, p.174, Theorem 8.13].
Why these the assumptions are necessary [Ru1, p.177, l.11-l.14; Zyg, vol.1, p.86, l.-8-l.-5].
If a new theory (Fourier integrals) is essentially the same as an old theory
(Fourier series), we should not allow them to go separate ways [Wat1,
§9.7; Ru3, chapter 7]. Instead, we should constantly
link the new theory back to the old one during the new theory's development by
proving that the corresponding concepts and theorems are essentially the same. If two theories are essentially the same,
we should establish a theorem [Zyg, vol.2, p.242, Theorem 1.3] which may enable us to transform most
of the theorems in one theory directly to
corresponding theorems in the other theory and vice versa. [1]
(Rigor and completeness)
Sometimes 0 and ¥ should be treated differently from
other numbers because there are many orders of magnitude of smallness and
largeness. However, for
complex variables we often treat 0, ¥, and
other complex numbers the same. Textbooksin complex variables usually treat ¥
as an exceptional point and do not consider it until they introduce a
function continuous or analytic at z = ¥. In
contrast, [Sak] puts ¥ and complex numbers on equal
footing from the beginning of the book. Saks' approach ensures that any property
valid for z = a, where a is a complex number,is also valid for z = ¥.
To understand the definition of functions continuous at z = ¥,
one must know the fact that the one-point compactification [Dug, p.246,
Theorem 8.4] of R^{2} is homeomorphic to S^{2}. To understand
the definition of functions analytic at z = ¥, one
must have a background in linear fractional transformations [Sak, chap. I, 14; Ru2, p.298,
l.1-p.300, l.8].