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Abstraction in Differential Equations

  1. Abstraction complements the insufficiency of concreteness. Any attempt to visualize a 3-dim wave through concrete geometry contains intrinsic defects [Hec, p.27, r.c., l.8], so using the abstract wave equation [Hec, p.14, (2.11)] to generalize the concept of 1-dim wave [Hec, pp.10-14, §2.1] is the only possible and natural alternative.

  2. Given a concrete solving procedure for the Boltzmann integral equation [Rei, p.536, (14.7.12); p.539, (14.8.4)]. If we use a functional analysis approach to make the procedure abstract, the effectiveness of constructing solutions will be reduced. In functional analysis, we assume that the test functions are given [Ru3, p.136, l.−1]. In contrast, in statistical mechanics, to solve the Boltzmann integral equation, there are clues to find the pertinent and useful test functions [Rei, p.540, (14.8.5)]. Furthermore, in statistical mechanics, the relationships between solutions and test functions [Rei, p.537, l.21 & l.−16] are clear.

  3. Generation is a special case of abstraction.

  4. Abstraction clarifies logical structures, but eliminates its concrete meanings by chunks.
        For the "smoothing" method [Boro, p.131, l.−14] in the theory of Fourier transforms, Rudin's choice [Ru2, p.197, l.−1] serves its purpose in a general setting, but Borovkov's choice [Boro, p.131, l.10] is more meaningful in the probability model.

  5. The same theorems in different settings can have different meanings, functions, and goals.
        In probability theory, Fourier transforms are interpreted as characteristic functions and Parseval's equality assumes the form of [Boro, p.135, Assertion 3].

  6. Compare [Ru2, p.150, Theorem 7.8] with [Cou1, vol. 2, chap. IV, 3.1].
        For abstraction, one can select a trivial aspect (e.g., s-algebra) and expand it. On may easily blow it out of proportion. However, understanding a theorem's abstract structure is not beneficial for application [see Cou1, vol. 2, chap. IV, 3.3].

  7. Executing operations on an abstract surface.
    Example. [Kre, p.233, l.-3-p.234, l.5] provides an effective implementation for parallel transport on an abstract surface.

  8. Some concepts are more easily visualized in an abstract setting, and others in a concrete setting. If the latter is the case, the concept defined in the concrete setting can always be used as a guide to understand that defined in the abstract setting.
    Examples. The tangent space is independent of coordinate systems [Spi, vol. 1, p.103, l.-5]. This statement is more easily visualized on a surface on R3 [Spi1, p.115, l.19-l.23] than on a manifold. Coordinate systems are C-related [Spi, vol. 1, p.36, l.-3]. This statement is more easily visualized on an abstract manifold than on a k-dim manifold in Rn because the concept of being C-related becomes trivial in a concrete setting.
    (Vector fields) In Rn, both the vector field G on W and the vector field H on V refer to the vector field F on the manifold M [Spi1, pp.115-116]. The basic scheme of defining a vector field on an abstract manifold should be the same, but it takes quite large amount of work to accomplish this task [Spi, vol. 1, p.97, l.9; p.111, l.-9 & l.-6].
    (Orientations) The definition of a k-dim oriented manifold in Rn [Spi1, p.117, l.11-p.119, l.5] is somewhat easier to undeition givenrstand than the definition of an oriented abstract manifold [Spi, vol. 1, p.117, l.1-l.4; p.284, l.-7].

  9. The readers may easily get lost in a long abstract theory. An abstract theory requires concrete examples to support its development. It is inadequate if the author provides examples only at the beginning stage of a theory's development: the later stages of a theory's development should also be followed up with significant examples. For instance, [Jaco, vol. II, chap. III, 6-10] in abstract algebra should be backed up with [Inc1, 6.4-6.53] in the theory of differential equations.

  10. [Pon, p.20, Theorem 2] is a special case of the theorem given in [Inc1, p.71, l.21-l.23]. The continuity of [Pon, p.19, (3)] embodies the Lipschitz condition given in [Inc1, p.71, l.-11-l.-10]. It is easy to obtain the Lipschitz condition from the continuity of the partial derivatives, but not vice versa.

  11. Links {1, 2}.