Any matrix can be standardized to a triangular form [Halm, p.107, Theorem
2]. To prove det eX = e Tr
(X) [Kna, p.8, proposition 1.1(e)], it is enough to consider the
triangular case. Thus standardization can be a very effective tool.
Solving DE=s effectively.
(Pattern identification) There are some basic patterns to solve DE.=s.
We would like to identify [Inc, p.26,
l.8] with [Bir, p.16, (16)].
Recognizing the existence and uniqueness intuitively.
The rectification theorem [Arn, p.48, Theorem 7.1] is intuitively plausible
([Olv, p.30, Proposition 1.29] actually uses a pictorial argument to convince
the readers of this theorem), although the actual proof of existence of the
local diffeomorphism is very complicated. Therefore, we may reduce the proof of
existence and uniqueness [Arn, p.50, Corollary 1 & 2] to the trivial case [Arn,
Destandardization: Finding a proper setting for application.
To simplify a theory, we would like to standardize our model. However, if we
follow the argument of the standardized theory (xn=0 [Joh, p.57,
(2.9)]), we may deviate from the track for solving a practical problem (φn
=0 [Joh, p.59, (2.24)]). For immediate application, we
have to reverse a part of the procedure of standardization. In this process of destandardization, we should measure the scope of influence precisely [Joh,
p.58, (2.13)], and design a setting (Principal part [Joh, p.57, l.!3])
which has enough room for the general case and at the same time is lean enough
to be efficiently used.
For Cauchy Problems, we use approximate solutions [Joh, p.137, (1.46)] to
standardize the spacelike [Joh, p.138, (1.54)] initial surface t=φ
(x) to t=0. Uniqueness: Formula [Joh, p.129, (1.14)] (see [Joh, p.139,
The strategy of standardizing the Cauchy problem:
Reducing the higher-order system to the first-order [Pet, p.16,
Reducing the initial condition to 0 [Pet, p.18, l.!17-l.!6].
We would like to standardize initial conditions so that we may
interpret a solution term by term. Please compare [Joh, p.139,
(1.14)] with [Pet, p.80, (4,12), (5,12)].
(Structures) Sometimes the representative nature of a standard example
can only be recognized through isomorphisms [Mas, p.256, l.!11].
Standardization serves to facilitate calculations. In solving y'
all we need is to use diagonalization to standardize A (see [Rut, p.234, l.!2]).
It is not necessary to require that the basis be standardized. However, in [Arn1,
p.173, l.!1-p.174, l.!3],
Arnol=d seems to like to standardize
everything in sight. For the standard basis, he uses coordinate x [Arn1, p.174,
l.!20]; for the eigenvector basis, he uses
coordinate C [Arn1, p.174, l.!8]. His
practice of transforming
coordinates back and forth is really much ado for nothing.
Standardization eliminates insignificant generalization. In a sense, a
standard version is the most generalized version. For any
theorem (For example, [Col, p.42, l.!15]) involving
the generalized norm defined in [Col, p.38, (I.47)], we may use the standard
norm [Col, p.37, (I.46)] to replace the generalized norm
without invalidating the proof. We will need trivial modifications to interpret
In the calculus of variations, the linear case eh
can represent the general case y(x,e)
without loss of generality [Cou, vol. 1, p.186,
Sometimes referring to a standard object can be a random choice. In this
case, it is more appropriate to define the concept by using the method of
abstraction [Cou2, vol.2, p.196, footnote 1; p.200, footnote 1].
We can metrize the topological vector space (C¥[0,1],
t) by the metric d1
given in [Ru3, p.27 (1)] or the metric d2 given
in [Dug, p.301, l.-15]. The problem would become
quite complicated if we
were to try to directly prove the equivalence of the above two metrics. However, if we
treat t as the standard and prove that d1
is compatible with t [Ru3, p.28, l.1-l.12], then we
use the same scheme to prove that d2 is
compatible with t. This strategy applies whether the
summation index n is related to the size of a domain or the order of a derivative