Motivating Questions in Differential Equations
- Motivating questions drive a theory's development. Each solution brings us
to a new world. After we solve one question, another question arises. For example,
after we solve the Cauchy problem for quasi-linear PDEs, we want to solve the
non-linear case. First, we must increase the number of characteristic equations
from two to four (Compare [Ches, p.176, (8-36)] with [Ches, p.164, (8-3)]). In
order to formulate a sensible question, the Cauchy problem should be posed for
an initial strip rather than an initial curve [Ches, p.176, l.-5-p.178,
l.1]. A textbook may choose to expose the origin that motivates us to define a
strip [Ches, p.177, l.3] or to bury it [Sne, p.63, l.11]. If a textbook chooses
to expose motivating questions, readers will have an easier times seeing the big
- Are S2 and S1 homeomorphic?
Answer: no. The fundamental group of S2 is trivial [Mun, p.328, Lemma 2.3], while the fundamental group of
S1 is infinitely cyclic
[Mun, p.340, Theorem 4.4].
Remark 1. [Mun, p.339, Theorem 4.3] provides the crucial link between covering spaces and the fundamental group.
Remark 2. If we compare the proof of [Mun00, p.368, Theorem 59.1] with that of
[Mun, p.348, Theorem 6.1] from the viewpoint of motivating questions, we find
that the former proof is more structured and less bloated (see [Mun, 348, l.-8-p.349,
l.-6]). However, the more convincing argument given
in [Mun, p.348, l.13-l.9] is unfortunately omitted in the second edition.
- To what extent does the fundamental group depend on the base point?
Answer. [Mun, p.328, Corollary 2.2] or [Mun, p.330, Corollary 2.5].
- Are Rn (n>2) and R2 homeomorphic?
no. Deleting a point from Rn leaves a simply connected space
[Mun, p.350, Corollary 6.3], while deleting a point from R2 does not
[Mun, p.340, Theorem 4.4; p.343, Theorem 5.1].
- Are any pair of surfaces among S2, P2, T,
and T2 homeomorphic [Mun, p.356, Corollary
Answer: no. The fundamental group of S2
is trivial. The fundamental group of P2
is Z2. The fundamental group of T is Z´Z.
The fundamental group of T2 is not
abelian [Mun00, p.374, Theorem 60.6]. In fact, the fundamental group of T2
is a free group on two generators [Mun00, p.432, Example 1].
- Does a compact 2-manifold have topological dimension precisely 2? Answer: yes [Mun00, p.308, l.10-l.15].
Remark. This question shows that the question and its answer constitute a
natural unit: there is no clear dividing line between topology and algebraic
topology. Thus, it is obvious that any attempt to classify mathematics into
various subjects is artificial and superficial. Subject classification is
only good for cataloguing materials in a library. Sometimes, we even doubt if it can serve that purpose. For example,
shall we classify [Cou, vol. 2] under the subject of theoretical physics or that
of partial differential equations? Theoretic physics is partial differential
equations, so the book belongs to both subjects. Due to the
confusing title of the book, one may not be able to find the book in the section
of partial differential equations at a mathematical library. Some may say that the
classification is useful for someone to declare his or her expertise in order to
apply for a position or grant. If that is the case, the subject classification
originates from a motive similar to
that of a male leopard that urinates around trees to mark its territory, to claim
the area's females, and to warn other males that they will be banished if they
intrude into its domain.
- If m¹n, are Rm and Rn homeomorphic?
Answer: no [Mun, p.350, Corollary 6.4; Dug, p.350, Theorem 6.3; Dug, p.359,
- [Inc1, p.229, l.16-l.25] motivates us to formulate and prove [Bir, p.270,