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Key Points in Differential Equations

1. 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.
   
   
2. [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].
   
   
3. 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.
   1. Find an appropriate example so that the key points may emerge vividly and the reasoning may flow smoothly [Cod, p.139, (1.3)].
   2. 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].
   3. Pinpoint the exact locations where the key points appear in the proof.
   Example. [1].
   
   
4. 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 accurate.
   
   
5. We focus on the key point by neglecting trivial matters.
       In [Zyg, vol.1, p.61, l.-19], Zygmund says, "The curves y = S_n(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 S_n(p/n) ® G(p). Thus, we may neglect the meaning of the word "condense".
   
   
6. In order to prove [Zyg, vol.1, p.78, Theorem 1.26], we first prove that if n/k is bounded, then s_{n, k} ® s as s_n ® 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, l.1-p.80, l.11].
   
   
7. 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].
   
   
8. 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.



9. 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].