Key Points in Differential Equations
- Mammoth structures blur the key points.
Example. Compatibility conditions [Spi, vol. 1, p.117, l.1-l.4]. Spivak would
help readers gain insight if he had added the following statement: The determinant is smooth
across the fibers.
- [Mun00, p.495, l.13-l.14] provides the key point of constructing a
covering space that satisfies the condition given in [Mun00, p.495, Theorem 82.1].
- In order to highlight the key points of a long proof, we should use the following
method rather than repeat the key points many times.
- Find an appropriate example so that the key points may emerge vividly and the reasoning may flow smoothly
[Cod, p.139, (1.3)].
- Before we start a proof, we must initiate a plan whose strategy should correspond to that of the above example
[Cod, p.152, l.-15-l.-6].
- Pinpoint the exact locations where the key points appear in the proof.
- The key point is often buried in the sea of means that are necessary to make the
formulation precise and accurate.
The key point of [Har, p. 12, Theorem 3.1] is [Har, p.11, Corollary 2.1]. However, in order to use rigorous language to articulate the key idea, we must use two pages
[Har, pp.13-14] as means that are necessary to make the formulation precise and
- We focus on the key point by neglecting trivial matters.
In [Zyg, vol.1, p.61, l.-19], Zygmund says, "The
curves y = Sn(x) …
condense to the interval 0£y£G(p)
of the y-axis." What does "condense" mean? Does it mean that these
curves stay inside the interval or does it mean that they stay inside a neighborhood of the
interval? If we read the general definition of Gibbs' phenomenon [Zyg, vol.1,
p.61, l.-12], all we are concerned with is that Sn(p/n)
® G(p). Thus, we may
neglect the meaning of the word "condense".
In order to prove [Zyg, vol.1, p.78, Theorem 1.26], we first prove that if n/k
is bounded, then sn, k
® s as sn
® s [Zyg, vol.1, p.80, l.4]. Next, we let k = [ne]+1
[Zyg, vol.1, p.80, l.15]. We may ignore the rest of material contained in [Zyg, vol.1, p.79,
- It is difficult to study Tauber's theorem for the first time using the
material in [Ru3, pp.208-209,
§9.1] or that in [Zyg, vol.1, p.81, Theorem
1.36] because the authors add many complications. In essence, Tauber's theorem is simply the converse of
Abel's theorem for certain classes of series [Sak, p.132, l.-12].
- In order to stress the key point, we must hit the nail on the head.
In order to clarify the point, [Wat1, p.365, l.-14-l.-10] hits the nail on the head, while [Guo, p.353, l.-5-l.-4] only touches the point lightly.
- The key point to facilitating a theorem's application
For applications, [Perr, p.239, Satz 10] is more convenient than [Perr, p.237, Satz 9]. The key point to this transformation is the inequality given in [Perr, p.238, l.-11].