An advanced textbook should discuss points in greater depth. In [Jack,
pp.95-96, §3.1], we assume the potential has
the form of [Jack, p.95, (3.2)]. It seems that the discussion of the solutions
which do not have the form is left out. However, [Lev2, p.124, l.9] shows
that in most cases of physical interest this product form includes all the
solutions [Lev2, pp.124-125, (6.12)-(6.16)].
Using symmetry allows us to reduce mechanical calculations [Sad, p.104,
l.3-l.5; p.126, l.7-l.12].
The circular symmetry in [Sad, p.114, Fig. 4.7; p.270, Fig. 7.8(a)] allows us to avoid evaluating the integral of the first
term in [Sad, p.115, (4.24); p.270, l.10] (see [Sad, p.115, l.6-l.10; p.270,
l.13]).
Choosing a cylindrical surface symmetric with respect to both z-axis and xy-plane makes the calculations in [Sad, p.127,
§4.6.B] more effective than those in [Sad,
pp.112-114, §4.3.A].
Using Gauss's (Ampere's) law to find the electric (magnetic) field.
Given symmetric charge (current) distributions, it is possible to find an
integral surface (path) over which the electric (magnetic) field is constant in
magnitude. Examples. [Sad, pp.126-132, §4.6;
pp.274-281, §7.4].
Suppose we begin with an effective algorithm [Hei, p.25, (5.5);
p.36, l.9-p.37, l.-20; p.38, l.1-l.21]. Then we
give some definitions along the way so that we have some terminology to refer back
to as we proceed.
Without the effective algorithm, the introduced definitions would be meaningless and the
theory would be baseless.
Heine introduces the concepts "group representation
[Hei, p.26, l.13-16], a basis [Hei, p.28, l.-14] and
reducibility [Hei, p.33, (5.16)]" using the same example given in [Hei, p.25, l.-11;
p.26, l.-16] as a guide. Thus, Heine's introduction is natural
and effective. In contrast, Tinkham introduces the concepts "group
representation [Tin, p.18, l.-5] and reducibility
[Tin, p.20, l.4]" in abstract settings. The method that Tinkham uses to derive
the
irreducible representations from his example [Tin, p.19, l.9-l.21] fails to use
basic functions and thus is not effective as Heinr's. Without an effective algorithm as
the foundation, his introduction is artificial and leaves unsolved the problem
of how to find effective methods to check the definitions using a basis.