Typical examples reveal the center of a discussion.
Generalization emphasizes the formal deduction of general properties. If we say that a general theory includes special cases, it refers to their general properties rather than their constructions.
[Pon, p.264, Theorem 25] is decorated with new terminology, formalism and a sophisticated argument. A trivial example of [Pon, p.261, Definition] is also provided [Pon, p.262, l.10]. Indeed, this theorem has no defects
in logic. However, unless we have the picture of [Arn1, p.56, Fig.45] in mind, the abstract theorem alone [Pon, p.264, Theorem 25] is like a satellite without a planet and like a wedding without a bride.
Typical examples provide the big picture and our long range goal.
Suppose our long range goal L is to study the stability in [Arn1, p.56, Fig.45]. Then [Bir, p.325, Theorem 6] is our middle range goal M and [Pon, p.146, Theorem 12] is our short range goal S. If we aim at L, we will probably end up with M. If we aim at M, we probably end up with S. If we aim at S, we will probably end up nowhere.
Typical examples bridge gaps and solidify a theory.
The goal of math is to produce and illustrate significant examples. Theorems and theories are nothing but footnotes and by-products of these examples. Only after finding some nontrivial examples [Arn1, p.56, Fig.45] that satisfy the hypothesis of a definition [Pon, p.261, Definition], may we say the definition is
meaningful or definable. Similarly, only after nontrivial examples show that the hypothesis of a theorem can be realized, may we say the theorem is not vacuous. [Inc, p.148, Example, 49.2] gives a typical example that the assumption in [Inc, p.148, l.5-l.7] can be realized.
Typical examples indicate that the old method is inadequate and a new method has to be found.
J−n(x)= (−1)nJn(x) [Inc, p.167, l.7-l.8] shows that [Bir, p.242, Corollary] is inadequate. [Bir, p.242, Theorem 8] provides a second independent solution in the exceptional case.
Typical examples point out the correct direction of the theory's development.
Example. The Clifford numbers [Che, p.61, §XI] show how the potential of a division ring can be fully developed.
Vivid examples provide an arena for geniuses to tap their great ideas.
In [Sne, p.21, Theorem 5], we give a synthetic
method to find the primitive. With concrete examples, we can improve the method's effectiveness according to various situations [Sne, p.26, chap.1, §6].
A typical example [Col, p.164, Fig. III.5 or p.186, Fig.III.18] injects life into a theory so that we have a concrete image to work
with. Particularly, it makes the artificially assigned values [Boundary conditions] more natural. Furthermore, the above two examples naturally motivate us to search for the general case [Col, p.173, III.12].
(Generalization and specification)
An example [Boro, p.281, Example 10] has two functions:
We may generalize the argument to create a theorem [Boro, p.273, Theorem 13] from the example.
It indicates how we obtain effective methods to satisfy the theorem's assumptions [Boro, p.276, l.−6-p.277, l.20].
If we were to compare a theory to a space, we would see that theorems are like points. There are
important links between points that are not covered by theorems. A theory
requires examples to fill these gaps. In addition, a theory sometimes requires examples to
complete the final touch.
Example. For parallel transport on a sphere, [Kre, p.232, Theorem 79.3] considers great circles only, while [Kre, p.234, Example 80.1] considers
any type of circles on the sphere. As a second example, consider [Kre, p.240, l.-7-l.-6]
in light of [Kre, p.239, Theorem 82.1; p.240, Theorem 82.2].
A crucial example of a definition can reveal that the definition is designed
to make the best of a situation despite the poor conditions we encounter.
Examples. The definition of an asymptotic expansion is designed to estimate the
error [Wat1, p.150, l.-2-l.-1] even when the series diverges
[Wat1, p.150, l.12-l.13]. Watson leaves out the only example that can reveal what
the idealized case is when he discusses asymptotic expansions in
[Wat1, §8.1, §8.2,
§8.21]. This idealized example is given in [Jef,
p.499, l.-12]. To prove that the example satisfies
the definition is nontrivial: we have to use [Ru2, p.229, Theorem 10.26].
The meaning of a vague idea can be honed to sharp precision only through examples.
Example. Compare the statement given in [Wat1, p.114, l.13-l.14] with [Gon,
p.683, Corollary 9.5].
If an author proves a theorem but fails to provide an example, then we may not be able to follow his proof and proceed any further if the author leave out something
in his proof due to negligence. However, if he provides an example, we may
recreate the situation, figure out his intention, and supply the information
required to fill the gap. For example, in order to establish the first
equality of [Tit, p.23, (1.12.4)], we have to let T = t0
- 2d. In order
to establish the last equality of [Tit, p.23, l.-1],
we use the fact that
that d/t0
is small. The example given in [Tit, p.24, l.10] may easily supply the necessary
information.
The method used for simple cases provides the key to solving the complicated cases.
The ratio test can be used to prove the divergence of the asymptotic series given in [Inc1, p.172, l.8-l.9]. The same test can be used to prove the divergence of the asymptotic series given in [Wat1, p.368, l.-7-l.-5].
Examples are required for proving a certain claim.
Without examples, it is difficult to convince people that the passage given in [Perr, p.205, l.12-l.16] is true. The
required examples are provided by [Perr, p.209, (16)®(17);
p.239, Example 1].
Examples satisfying the assumptions within an assumption
The existence of V given in [Har, p.38, l.-15] is an assumption within an assumption because it must satisfy the condition [Har, p.38, l.23-l.24, (iii)] which assumes that the solutions of the given ODE are known.
Studying a theorem containing assumptions within an assumption can be difficult if we do not have a example
satisfying these assumptions. [Har, pp.37-40, §8] fails to provide such an example. We do not know whether the development of the theory will lead to a contradiction.
Thus, an abstract theory can easily become an empty theory. However, the existence of an example ensures that a contradiction will never occur. In this case, V can be considered generalized energy function and -dV/dt generalized dissipation function
[www.stanford.edu/class/ee363/lectures/lyap.pdf, p.4 & p.21]. [www.stanford.edu/class/ee363/lectures/lyap.pdf, p.6, l.2-l.4] says that Lyapunov theory is used to make conclusions about trajectories of a system dx/dt
= f(x) without solving the differential equation. This sounds as if the Lyapunov
method was more efficient and the method of solving the ODE was a detour. In fact, the Lyapunov theory is merely a qualitative theory
and what it can do is quite limited: Given partial but essential information
(e.g., the property of eigenvalues given in [Pon, p.208, Theorem 19]) of solutions, we may derive some related results which can be used to explain
some physical phenomenon. These results are more general than those derived
by solving the ODE, so the former results are less effective.
For a counterexample [Arn1, p.60, l.−7], we would like to point out the key factor which prevents us from reaching the conclusion. Even though sin t is an odd function, its power series at t=0 contains higher terms. In view of this fact, a simpler case would be gt x=x+t3.
Responses to a counterexample (Sub + Quotient ≠ Quotient + Sub [Mas, p.250, l.14-l.22]).
Both [Dug, p.122, Theorem 2.1] and [Mas, p.251, Proposition 4.2] list some sufficient conditions for Sub and Quotient to
be commutative. However, for the case in [Mas, p.166, l.7], we have to use the method in [Mas,
p.251, Proposition 4.2] rather than its conclusion, because this case does not satisfy any of the above conditions.
If a slight change of a theorem's hypothesis can lead to a entirely
different conclusion, then the theorem is called a sensitive theorem. To
thoroughly understand such a theorem, it is not enough to understand its proof alone.
Traditional education only guides students to the correct path. Explaining why
an incorrect method is not part of the typical curriculum. If you get lost in a maze or fall into a trap, traditional teaching will not help you
return to the right track. Example. If x and t are independent variables, then
¶2P/¶x¶t=¶2P/¶t¶x.
However, If x depends on t, then (d/dx)(dP/dt)¹(d/dt)(dP/dx).
Suppose we try to prove [Guo, p.179, l.-5]. If we
improperly apply the above theorem, we will not obtain the correct
answer. There are three steps to
learn from our mistakes next time. First, discard the method that leads to a wrong answer.
If one stops here, one might still commit the same mistake. Second, compare the
correct method with the incorrect method and find their slight differences. The
comparison will make you more careful next time. Third, find a counterexample
and convince yourself that a slight change does lead to a completely different conclusion.
To give an counterexample for the theorem about interchanging the order of differentiation, let x = 4t(1-t).
A counterexample serves to show how the omission affects the conclusion if we omit an assumption of a theorem.
We should not be content with a counterexample that barely meets the
requirement. A trivial counterexample may only lead to a very superficial
understanding of the theorem. Sometimes, even a few nontrivial examples [Edw,
pp.21-22, Example 4; p.23, Remark 2] are inadequate for covering all the
possible consequences. In order to deeply and completely understand the
consequences of the omission, we must construct a significant example which
shows the worst possible case [Har, p.18,
l.6-l.12].
Counterexample. [Har, chap. II, §5].
Remark 1. In order to satisfy the condition [Har, p.21, (5.22)], 2n+1
in [Har, p.21, (5.21)] should have been replaced by 2n. Hartman made a mistake in calculation here.
Remark 2. We may use [Dug, p.302, Theorem 5.2] to prove the fact that U(t, u)
has a continuous extension on the (t, u)-plane [Har, p.22, l.-9-l.-8].
In order to systematically produce divergent continued fractions, we first characterize the convergence of continued fractions by finding its necessary and sufficient conditions [Perr, p.276, Satz 38], and then weaken the conditions [Perr, p.277, Satz 39].