A definition requires a thorough justification. Using a
incompletely justified formula in a definition may result in application an
unproven theorem.
Many people think that it is legitimate to leave out the essential steps of
an idea's development simply because
they are under the impression that a definition does not require a proof. [Ru3,
p.143, Definition 6.12] says that formula (1) is motivated by [Ru3, p.136, (2)]. Rudin's
justification is obviously not enough because he only checks the case n=1 for
infinitely differentiable functions. He should have also checked the
n-dimensional case [Joh, p.89,
(6.6)]. All existing old concepts or formulas must be consistent with the new ones.
Distribution theory is designed to solve partial differential equations.
Rudin's failure to use the divergence
theorem [Joh, p.79, l.11] to prove the Lagrange's
identity is exactly how he misses the point [Sne, p.145, l.-2].
A definition is like the door to a theory. Its setting
should be able to represent the general case.
Example: Domain of influence & domain of dependence [Sne, p.223, (12); p.222,
Fig. 38; p.224, Fig. 40 & 41]. The situation of [Joh, p.41, (4.13); p.42, Fig.
2.2] is too narrow to be typical. To see the complete relationship
between a solution and its domain of dependence, we must directly construct the
solution by characteristics [Pet, p.68, Fig. 5].
Generalization versus effectiveness.
When a concept is generalized, it allows more cases to be included. A
generalized concept may have various equivalent definitions [Orientable:
War,
p.139, Proposition 4.2]. However, only in some restricted cases [War,
p.140, Example (b)] will we have an effective method to check the definition.
Furthermore, if the manifold is a surface in ú^{3},
then the method has an intuitive geometric interpretation [O'N,
p.178, Proof of Theorem 7.5].
When several definitions are applicable, we should use the one
that applies to the smallest category.
There are two definitions for elliptic equations: Definition 1 (discriminant)
is for second-order PDE's [Joh, p.35,
l.15] and Definition 2 (definite forms) is for mth-order PDE's
(see [Joh, p.60, l.16] or [Ru3, p.198, l.18]). Definition 2 must be compatible
with Definition 1. When checking if a second-order PDE is elliptic, we
should use Definition 1 because its testing rule is simpler and more
efficient.
Countability.
To define a set, we have to consider how it can be effectively constructed. For the definition of a differentiable manifold, there are three
points of view on adjoining convenient charts [Spi, vol.1, p.37, l.!5]:
(Warner) Define the maximal set ö
[War, p.6, l.4].
Defects: (i). The resource to supply a candidate element may be unknown.
(ii). Even if the resource is known and we have a method to adjoin a new element
to the old set, we can only increase the legitimate elements one by
one. Such a slow pace may not give us any hope that we can exhaust
the resource and complete the construction.
(Olver) Exclude the uncountable case [Olv, p.3, l.7].
Defect: Can we choose the required coordinate system in [War, p.18, l.2] from Olver's collection [Olv, p.3,
Definition 1.1]?
(Arnold) Use the concept of equivalent atlases [Arn, p.235, Definition
5].
Defect: Arnold attempts to avoid using Zorn's
lemma which is based on the axiom of choice [Dug, p.31, Theorem 2.1]. Actually,
the problem still remains. From a logician's
point of view, "x is equivalent to y"
is either true or false. However, in reality, there may be is no effective
method to make such a quick decision. In addition, there may not even be enough
resource to provide promising candidates.
Remark 1. Arnold's attitude.
The goal of mathematics is not to determine whether a statement is true or
false. It is to propose a correct attitude. Before application, a theory
is hypothetical. The theory has a real meaning only after application.
How many atlases exist in the class is not the issue. Our main concern is
whether the required atlas is available
when a situation arises. In other
words, we would like to adjoin a map and thus create a new atlas whenever
necessary. We still have the same differential structure. In practice, checking
whether the new map is compatible with others can be easy.
Flexible representative. The list of its representatives grows only if
our knowledge does. We choose the representative we need and ignore the rest.
Remark 2. We isolate problematic parts to preserve the quality of a
theory. The theory of topological manifold has a good quality. It is
differential structure that is problematic. Separating differential structure
from topological manifold [Boo, p.53, Definition 1.2] is an effective method of
preserving the quality in what we have accomplished. In other words, the whole
theory of differentiable manifold will not be completely ruined by this tiny
problematic part. When we apply the theory to practical examples, all we need to
do is to watch and check this small area of concern.
In order to reveal more relationships, some definitions should have
been derived from basic concepts through identifications.
Examples. (1). The tensor product in [Spi, vol. 1, p.159, l.7]: The
identification in [War, p.58, l.!4]. (2). The wedge
product in [Spi, vol. 1, p.275, l.!1] or [War,
p.60, (4)]: The identifications in [War, p.60, l.2].
Consistency
(Style) The way Warner defines the Lie derivative [War, p.69, (1) and
the figure] is consistent with the traditional way: f N(x)=
lim (f (x+h)! f(x))/h, while Spivak's
[Spi, vol. 1, p.28, l.4] is not.
(Idea) Wedge product, cross product, and dot product.
(Not finished)
(Coordinates) The exterior derivative of 1-form [O'N,
p.154, l.11-l.15].
(Imitation) The new definition of direct product [Po2, p.17, Definition
10'] serves to create a large group by imitating [Po2, p.17, G)] the old model
[Po2, p.16, Definition 10]. In this large group the new direct product agrees
with the old one [Po2, p.17, H)].
General method of construction.
We may give the general method of constructing a covering space
either before or after its definition:
(Massey). Definition [Mas, p.145, Definition]
® Examples [Mas, pp.146-149]
® Using the concept of homotopy to
characterize a covering space and propose a possible construction [Mas,
p.165, l.12-l.18] ® [Mas, p.165,
Proposition 8.2].
(Boothby). Massey's
construction [Boo, p.97, Theorem 8.3]
®
Definition [Boo, p.101, Definition 9.1]
®
No other methods [Boo, p.102, Theorem 9.3].
Adopting a selective viewpoint according to application.
There are many aspects involved in the definition of a global Lie group. How
do these aspects affect its application to differential equations? Which aspect
should we use as the foundation? Local or global? See [Po3, p.283, l.!9-l.!6].
With application as a goal, one necessarily presents a complicated definition in a
form edited to emphasize usefulness. Like going
through a maze, only few selective approaches turn out to be successful.
Without loss of generality we would like to make the setting as
specific as possible.
We use Lie groups to act on manifolds. It seems that we may generalize the
concept of action by using topological transformation groups instead. [Po3,
p.343, Theorem 7.5] shows that there is no loss of generality in using Lie
groups.
Sometimes a tremendous amount of computation and comparison is required to
verify that a given example satisfies the conditions of a definition. We would
like to find a better version of the definition which might reduce the
amount of work [Po3, p.517, l.!7-l.!4].
The criterion in [Cou, vol.1, p.62, l.5] gives a geometric picture about
linear dependency and, thereby, provides a more informative resource than [Cou,
vol.1, p.50, l.13] to work on.
The criterion in [Cou, vol.1, p.62, l.15]
provides a more effective means than [Cou, vol.1, p.50, l.13] to determine
whether a finite set of continuous functions is linearly dependent.
We would like to adopt a version of a definition that is as simple as possible as long
as it serves our current purpose. A simple and intuitive definition always leads
to a simple and intuitive proof.
Example. The proof of [Cou2, vol.2, p.49, (15b)] is simpler than that of [O'N, p.24,
Corollary 5.5] because the definition in [Cou2,
vol.2, p.49, (15a)] is simpler than [O'N,
p.23, Definition 5.2].
On should develop a complicated definition in stages rather than give the final form
alone without any explanation. This is because we want to preserve the motives of the way that the definition is constructed.
Example 1 (defining P_{n}^{m} on the complex plane).
Stage 1. The method [Guo, p.247] used to solve [Guo, p.247, (1)] leads us to consider the solution form in [Hob, p.185, (8)].
Remark. For motivation, we should provide a general method rather than discuss a special case. [Wat1, p.326, l.11] gives the definition of P_{n}^{m} without any motivation.
We have no clue about where we should start to study the definition. The definition of v given
in [Hob, p.183, l.3] illustrates the usefulness of integral
transform, but fails to explain why we choose the Euler transform and how we
form the integrand. In contrast, [Guo,
p.85, l.11-16; p.247, l.-8-l.-4] provide
a general method to answer these these questions.
Stage 2. We establish the relationship between the Pochhammer integral and the beta function [Hob, p.186, (c)].
Stage 3. We express the solution form in terms of the Pochhammer integral and a hypergeometric series [Hob, p.186, l.-5].
By comparing Case m=0 with the definition of P_{n},
we determine the value of C_{n}^{0}
[Hob, p.187, (9)].
Stage 4. Using [Hob, p.187, (9) & l.-1], we determine
the the value of C_{n}^{m} [Hob, p.188,
l.4].
Example 2 (defining Q_{n}^{m} on the complex plane).
Stage 1. The method [Guo, p.247] used to solve [Guo, p.247, (1)] leads us to consider the solution form in [Hob, p.193,
l.6].
Stage 2. We express the solution form in terms of a hypergeometric series [Hob, p.194, l.10-l.11].
By comparing Case m=0 with the definition of Q_{n},
we determine the value of f_{n}^{0}
[Hob, p.194, l.-8].
Stage 3. Using [Hob, p.194, l.-6], we determine
the the value of f_{n}^{m} [Hob, p.194,
l.-1].
Remark. It is awkward that Guo uses [Guo, p.251, (1)] to prove [Guo, p.254, (6)]
because [Guo, p.254, (6)] can be proved more simply by the method of analytic
continuation. In my opinion, using [Hob, p.194, l.-6]
to determine the value of f_{n}^{m} is a
more economical approach to justifying the definition of Q_{n}^{m}
on the complex plane.
We can effectively determine whether or not a curve on a surface is principal [O'N,
p.223, Definition 5.1] if we use the definition given in [Wea1, p.66, l.-2-l.-1].
However, we cannot make this determination if we use the definition given in [O'N, p.199, Definition
2.3]. This is because the existence of the maximum and the minimum in [O'N,
p.199, Definition 2.3] is assumed by an axiom in the general case. In contrast,
[Wea1, p.66, (20)] gives the explicit equations of the principal directions, and
[kre, p91, l.3] provides an effective to determine the extremum.
Therefore, it can be said that O'Neill's approach is shallow and ineffective.
(A differentiable mapping from one manifold into another) It seems that
dividing a procedure into two small steps will facilitate our understanding of
the procedure. However, this is not necessarily true for formulating a
definition. In order to facilitate our understanding of a definition, we must formulate the algorithm to check the definition in a
concrete fashion. For example, the domain and the range of y◦f◦x^{-1}
given in [Spi, vol.1, p.41, l.1] are concrete. In contrast, both the domain of h
and the domain of hT given in [Kre, p.277, l.16-l.19] are abstract. This is
because Krezszig factors T_{j}^{-1}TT_{i}
into two parts: T_{j}^{-1}TT_{i}
= (hT_{j})^{-1}◦ hTT_{i}.
The definition given in [Kre, p.277, l.16-l.19] is derived from the abstraction of
the above factorization.
The definition of an evolute given in [Mur, p.272] is natural. In contrast, the definition given in [Wea1,
§10] or [Kre, §15] reverses the natural thinking process.
[Ches] defines characteristic curves twice [Che, p.4, l.14; p.165, l.2]. The
second definition is constructed by by the method which leads to [Che, p.164,
(8-3)] in xyu-space. The first definition can be viewed either as the projection
of the second definition onto the xy-plane or as the result of applying the same
method directly to the xy-plane. The method and the projection are commutative.
Redundant definitions.
[Spi1, p.98, l.-10] defines ¶I^{n}.
[Spi1, p.98, l.5] defines ¶c. The former definition
can be regarded as a special case of the latter one. For logic, it is
unnecessary to repeat the definition for this special case [Spi, vol. 1, p.339,
l.-6]. However, for motivation and visualization, we
would like to define the boundary on the standard n-cube first.
A theorem should not be regarded as a definition.
The definition given in [Spi1, p.98, l.-3] can be derived from
[Spi1, p.98, l.-7] and [Spi1, p.98, l.-5].
How we choose a definition that is tailored to our needs when there are several options available.
When we choose a definition, our choice is based on the following order of priority: convenience for application,
the key idea, and generality.
[Sag, p.25, l.10-p.26, l.16] gives three definitions of the differential of a
functional. Each one is stronger than its successor. That is, each one is more
general than its predecessor. [Fomi, p.12, l.2] chooses the first one, while
[Sag] chooses the third one. The proof of [Formi, p.13, Theorem 2] is incorrect.
We can use [Sag, p.26, Theorem 1.5, p.28, Lemma 1.5.2, and p.34, Theorem 1.7] to correct Gelfand's mistake.
In other words, the key idea of [Formi, p.13, Theorem 2] is [Sag, p.34, Theorem
1.7]. On the one hand, based on generality or the key idea, we are tempted to
adopt the weakest differential as our definition. On the other hand, [Fomi,
p.13, Theorem 2] gives a stronger result than [Sag, p.34, Theorem 1.7] because
the Fréchet differential [Sag, p.25, l.13]
is stronger than the Gâteaux variation [Sag,
p.26, Definition 1.5]. However, for the case given in [Sag, p.34, l.-12],
the most frequently used case in physics, both the Fréchet
differential and the Gâteaux variation lead
to the same result [Fomi, p.15, l.5; Sag, p.28, (1.5.4)]. Thus, if we consider
the necessary condition for a relative minimum of a functional, it makes no
difference no matter which differential we choose to use. Now suppose we
consider the sufficient condition for a relative minimum of a functional. If we
want our condition to be practical, we must use the second Gâteaux
variation instead of the second Fréchet
differential [Sag, p.39, l.10-l.14].
The fundamental periods of Tan z
For l > -1, 2k_{1}>0 is defined as the area bounded by
G and the axis on the upper half-plane [Gon1, p.390,
l.14]. By [Gon1, p.389, Definition 5.9], 2k_{1 }is
the smallest positive period of Tan z. 2k_{2} is
defined in [Gon1, p.402, (5.9-9)] and can be represented as in [Gon1, p.404,
5.11-1)]. [Gon1, p.406, (5.11-5)] shows 2k_{2}>0.
[Gon1, p.403, (5.9-15)] shows that 2k_{2} is a
positive period of Tan z. [Gon1, p.408, l.-5-p.409,
l.8] shows that 2k_{2} is the smallest positive
period of Tan z. Therefore, 2k_{1}=2k_{2}.
González should not have concluded that 2k_{1}=2k_{2}
as stated in [Gon1, p.406, l.6-l.8] before presenting [Gon1, p.408, l.-5-p.409,
l.8].
A definition is designed to exclude abnormal cases.
Example: The absolute convergence of an infinite product [Con, p.162, l.1-l.13].
Remark 1. In contrast, [Gon1, p.186, Definition 3.2] is given without
justification. The lack of justification makes it difficult to understand why absolute convergence implies
convergence [Gon1, p.190, l.2].
Remark 2. [Gon1, p.190, Theorem 3.10] is a converse of [Gon1, p.189, Theorem 3.7]. In order to prove the divergence of the second series [Gon1, p.191, l.13], González should have considered
d_{n}<
-1/4 rather than d_{n
}> -1 [Gon1, p.191, l.7].
(Avoiding inherent drawbacks)
In [Guo, p.536, l.-12-l.-8],
Guo has a hard time determining the signs because he uses [Guo, p.531, (7)]
to define cn and dn [Guo, p.531, (5)]. In contrast, González
relates cn and dn directly to Tan [Gon1, p.421, Definition 5.11], so he will not
suffer the consequences caused by problematic definitions [Gon1, p.433,
(5.18-16)].
If we want to emphasize what s(z) is, then we use
[Gon1, p.461, (5.33-1)] as our definition. If we want to emphasize the origin of s(z), we use [Wat1,
p.447, l.7-l.9] as our definition. The definition of s(z)
given in [Guo, p.473, (1) & (2)] is the compromise of the above two
definitions.
The way we define a term should be similar to the way an commander
select a vantage point for his command post during a war. He must collect
intelligence about the enemy, understand the area's people and topography, and take
into consideration the
location of allied troops and his own troops, as well as the effective range of weaponry. Similarly, when we define a mathematical term, we should consider
how the definition applies to the theory. For example, [Wat1, p.492, (A)] defines sn as the series solution of the differential equation given in [Wat1, p.478, l.-12;
p.464, l.10-l.13], while [Gon1, p.421, (5.17-1)] defines sn as a function of
Tan. González's definition is better because
we may more easily visualize the geometric meaning of sn through Tan than
we can through a series. In addition, González's
definition has the advantage in the following applications:
The similarity between sn and sin: [Gon1, p.421, (5.17-1); Wat1, p.479, l.7].
The similarity between their periods 4K and 2p:
[Gon1, p.404, l.14; Wat1, p.479, l.10; p. 504, l.-13].
The similarity between cn and cos: [Gon1, p.421, (5.17-2); Wat1, p.500, l.14].
Remark. Of course, we should not start every proof with the definitions given in [Gon1, p.421, (5.17-1)-(5.17-3)].
Being flexible is the key to success. In order to calculate the residues of the
Jacobian elliptic functions at z = iK', we should switch to the method of
[Wat1, §22.341] after we obtain [Gon1,
p.429, (5.17-20)-(5.17-22)]. The calculation given in [Wat1,
§22.341] is simpler than that given in
[Gon1, p.434, §5.20] because it is much
easier to divide one series by another than it is to use the method given in
[Gon1, p.434, §5.20].
A definition can be defined in terms of logic alone [Roy, p.56, l.-6-l.-5].
This definition provides no effective algorithm to directly determine whether a
given set is measurable. Only after we study further may we acquire significant
examples of measurable sets in our arsenal. For example, after proving [Roy,
p.59, Lemma 1], we may use the properties of a s-algebra
to prove that all Borel sets are measurable.
Remark. In my opinion, the definition given in [Roy, p.56, l.-6-l.-5]
is more accessible than that given in [Ru2, p.53, l.6; p.43, l.-8-l.-7].
(Inverse relations) Given a relation xRy, we define its inverse relation by
interchanging x and y in the definition of xRy. Examples. [Lan8, p.9, Definition 2]
® [Lan8, p.9, Definition 3]; [Lan8, p.21, Definition 9]
® [Lan8, p.21, Definition 10]; [Lan8, p.35,
Definition 18] ® [Lan8, p.35, Definition 19]; [Lan8,
p.45, Definition 30] ® [Lan8, p.45, Definition 31].
Expanding the domain of definition
Legendre polynomials can be effectively defined by Rodrigues' formula [Wat1, p.303, l.-10], while the Legendre function
of degree n of the first kind is ineffectively defined by the integral given in
[Wat1, p.307, l.7].
The wording of a definition should avoid possible ambiguity, otherwise the definition may easily cause misunderstandings.
Example.
(Stability) [Cod, p.314,l.5-l.18]
The word "a" given in [Har, p.38, l.-12] should be replaced with "any". The drawback of the definition given in [Pon, p.202, l.20-l.28] is that it cannot distinguish y(0) given in [Cod, p.314, l.12] from y(t) given in [Cod, p.314, l.14].
If the theory of an old item is well established and we want to study a new item which is closely related to the old one, it is unnecessary to repeat the same type of theory for the new item. All we have to do is define the new item in terms of the old one.
Example: [Wat, p.77, l.19-l.20]. [Wat, p.77, (1)] is obtained by replacing z in [Wat, p.38, (1)] with iz. The theory of J_{n} is well eatablished, so we define I_{n}(z) in terms of J_{n}(z). Why is the definition of I_{n}(z) divided into the two cases as shown in [Wat, p.77, l.19--l.20]? By [Guo, p.374, (6); p.375, (11)(i)], K_{0}(z) is expressed in terms of ln (z). Consequently, the domain of I_{n}(z) should be -p< arg z £ p. By [Guo, p.366, (3); p.367, (6)(i)], Y_{0}(z) is expressed in terms of ln (z). Consequently, the domain of J_{n}(z) should be -p< arg z £ p. The above definition transforms from the natural domain of I_{n}(z) to the natural domain of J_{n}(z). The factors before J_{n} are designed to make the new definition compatible with the old definition given in [Wat, p.77, (2)].
The domain of J_{n} is a Riemann surface; ze^{-3pi/2} must go around in a circle on a sheet to reach ze^{pi/2}. If arg z = p/2, J_{n}(ze^{pi/2}) and J_{n}(ze^{-3pi/2}) may be different, but e^{-npi/2 }J_{n}(ze^{pi/2}) = e^{3np/2 }J_{n}(ze^{-3pi/2}).
Since the concept of principal plane plays a crucial role in classification of conicoids [Fin, §364], the definitions for general form given in [Fin, §362] are better than those for special form given in [Bel63, p.101, l.11-l.-2].