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- 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.

- 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.

- Generation is a special case of abstraction.

- 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.

- 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].

- 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].

- 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.

- 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 R^{3}[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 R^{n}because the concept of being C^{¥}-related becomes trivial in a concrete setting.

(Vector fields) In R^{n}, 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 R^{n}[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].

- 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.

- [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.
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