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Motivating Questions in Differential Equations

  1. 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 picture.

    1. 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.
      1. 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].
    2. Are Rn (n>2) and R2 homeomorphic?
      Answer: 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].
    3. Are any pair of surfaces among S2, P2, T, and T2 homeomorphic [Mun, p.356, Corollary 7.7]?
      Answer: no. The fundamental group of S2  is trivial. The fundamental group of P2  is Z2. The fundamental group of T  is ZZ. 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].
    4. 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.
    5. If mn, are Rm and Rn homeomorphic? Answer: no [Mun, p.350, Corollary 6.4; Dug, p.350, Theorem 6.3; Dug, p.359, Theorem 3.3].

  3. [Inc1, p.229, l.16-l.25] motivates us to formulate and prove [Bir, p.270, Lemma 3].