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References in Differential Equations

  1. A textbook should not just be written for oneself as a reminder; it should be made easy for other readers to understand. For example, there is a small gap in the last equality in [Guo, p.366, l.- 1] and a large gap in [Guo, p.367, (5)]. Guo should have referred the readers to [Wat, p.58, l.4-l.5] to fill the small gap and [Wat, p.61, l.12-p.62, l.6] to fill the large gap. Omitting the references and leaving the readers to fill the tricky gaps on their own is like asking someone to climb to the roof and then taking his ladder away. There are many new things for one to learn. It is not worthwhile to spend time and energy in rediscovering other people's tricks.


  2. In [Hob, p.206, l.5], Hobson fails to explain why he differentiates [(m+1)/(m-1)]m/2 with respect to m. This is because he wants to apply the l'Hpital's rule. For this small gap, Erdlyi's approach is to follow suit: Hobson leaves a gap, so Erdlyi does the same [Erd, vol. 1, p.149, l.9]. Guo's approach : Hobson leaves a small gap in his proof; Guo does not even bother to refer to  any available proof [Guo, p.261, l.7]. Both approaches are confusing to the readers, a better approach is to quote [Hob, p.205, l.-14-p.206, l.-4] and make a comment to fill the gap.

  3. [Wat] is a classic because there are few mathematicians have such profound knowledge now. However, most of the textbooks that it refers to are from the early 1900's. Only the old schools have these old books. Some old books are very fragile. When I borrowed [Hob] (1931) from the library, I found that many pages of the book contained large cracks. These cracks are most likely due to old wounds. Perhaps 70 years ago, someone touched the pages and slightly damaged the paper. At that time they did not crack, but now they have because the paper of the book has become like withered leaves. You cannot dog-ear any page of the book because the dog's-ear will fall off. If Cambridge University Press had thought that their publications could not last more than 80 years, it would have been appropriate to use this kind of cheap paper. Otherwise, they should have used better paper so that readers from later generations would not have to deal with a pile of ashes.

  4. A reference should ask readers to fetch a simple, modern and powerful weapon to attack a special type of problem. It should not ask readers to go to a graveyard to exhume skeletons. [Wat] was published in 1944. When dealing with the justification of term-by-term integration, it was appropriate at that time for Watson to refer his readers to Bromwich, Theory of Infinite Series (1925), 176 [Wat, p.385, l.-7]. In contrast, [Guo] was published in 1989. Guo imitates Watson and still refers the readers to the same book. When I went to the library and found the book, the book's binding was severely damaged. It required a box to put its entire contents together just like an old man requires assistive devices to live. To prove that the integral of a sum equals the sum of integrals, Bromwich's method is too complicated (Bromwich unnecessarily divides his method in 175 & 176 into too many cases) and out of date. In contrast, the theory of the Lebesgue integration is a modern and powerful tool to deal with this type of problem. Thus, a more proper and up-to-date reference would be [Ru2, Theorem 1.27 & Theorem 1.34]. Perhaps this is one of the reasons why Watson says that [Wat, Chap. XIII] needs to be rewritten [Wat, vi, l.-14].

  5. A reference should be as specific as possible.
        If the application and the quoted source are expressed with different wording, the reference should be as specific as possible. Otherwise, the readers will have a hard time finding the needed reference. For example, it would be better if [Wea1, p.83, l.-7] referred the readers to [Wea1, p.80, l.15] instead of Art. 35.

  6. Practicing line-by-line references.
        Line-by-line references are a good practice adopted in N. Bourbaki's books. They explain a theory by quoting the needed sources exactly and explicitly. No matter how deep the theory is, the explanations he provides remain clear. This practice shows the prudence of a scholar and the rigor of his reasoning.
        [Wea1, p.83] is written in a style as though it were an article presented in a modern journal. There are no references. Weatherburn assumes that his readers read his book from the first page and memorize all the material in between. Anything said before will not be mentioned again. A poor memory is not an excuse.
        Sometimes a textbook fails to include a needed reference that is not in the book. The readers must expand their readings beyond the textbook. For example, [Wea1. p.46, l.14-l.15] should have referred to readers to [Kre, p.51, l.-10-l.-7] and [Wea1, p.99, l.18-l.16] should have referred the readers to [Kli, p.16, Proposition 1.4.1].

  7. A reference should be as original as possible.
        The reference in [For, p.48, l.-2-l.-1] shows the years that Mainardi and Codazzi published their works, while the reference in [Wea1, p.95, l.1-l.4] does not.

  8. [For, p.50, l.-1] refers to 173 of a book of its 3rd edition. Hale Library at Kansas State University has only the book's 2nd edition and the 5th edition. I wonder if 173 of the 5th edition is the same as 173 of the 3rd edition. Book companies issue a new edition every few years in order to make more money. Some popular books even have more than ten editions. Librarians do not want to keep ten copies of the same book in the library, so they usually only retain the newest edition. All the references to the previous editions become nearly useless. Scientists will be blocked and puzzled at every turn. However, if the book is a textbook, the situation is entirely different. A university science library will purchase only the first edition. This is because the university fears that if the library purchases the updated editions, the student will not buy the expensive textbook and the university bookstore will lose business. As a result, the library prevents its patrons from finding the updated information. Making money is applauded by capitalism, but is devastating to science.

  9. References should not be used for rhetoric. Clarity and precision rather than big fancy words are essential to quoting a reference.
        [Kre, p.185, l.18] says, "The solution of this composition problem can be obtained by means of function theory." It should have said, "The solution of this composition problem can be obtained by means of the Riemann mapping theorem [Ru2, p.302, Theorem 14.8]." [Kre, p.185, l.20-l.21] says, "Difficulties may be overcome by the theory of uniformization." I wonder what the theory of uniformization is. Maybe it is common in German textbooks but not in English textbooks. In fact, we do not need a theory. All we need is a few lines of explanation such as follows: Step 1. Based on [Spi, vol. 1, p.624, Theorem 1(a)], we choose countable local mappings whose domains cover the entire surface. Step 2. We use the Riemann mapping theorem to compose two local mappings [Kre, p.185, Fig. 58.1].  Step 3. Using Step 2 and  mathematical induction we extend the domain of the mapping to the entire surface.

  10. References should focus on the point being discussed and should not divert attention from the text to promote someone else's work.
        In [Spi, vol.1, p.200, l.-4-l.-3], Spivak says "If f: U Rn is Ck, then the flow a: (-b,b)Ba(x0) U is also Ck." Then he claims that this is a very hard theorem [Spi, vol.1, p.200, l.-2]. In [Arn1, p.285, l.12], Arnold proves it using plain language [1]. In [Spi, vol.1, p.200, l.-2-p.201, l.8], Spivak promotes Lang's work because Spivak pretty much uses Lang's language and follows Lang's scheme and style. The purpose of mathematics is to use a familiar language to write about content with depth. Shallow content even if written with difficult language simply cannot last no matter how hard it is promoted.

  11. A reference should provide complete information.
    The reference given in [Dug, p.144, l.7] is inadequate. Dugundji should have added [Dug, p.224, Theorem 1.4(4); p.233, Theorem 1.2] to his reference.

  12. (Indexing)
        Since the advent of the Internet era, search engines have played an important role in finding what we want. However, there is a lot of junk in cyberspace which prevent us finding the right stuff. Idealistically, future books will be in the CD form which saves space in a library. However, for the present, textbooks in book form still play the dominant role in study. Consequently, it is important for an author to design an index in a textbook to enable readers to use it as a reference. Many word processors have features such as word search or word count. I wonder if any current word processor has the feature of automatic or manual indexing. Suppose we choose a term as an index entry. If the computer will automatically record the pages on which the term appears as one writes a book, then the indexing is called automatic indexing. If one has to manually record its occurrence at selected spots, then the indexing is called manual indexing.
        An index is arranged in alphabetical order using the key word. If an entry contains many words, then we have to ask ourselves which words are key words, the noun or adjectives. For the entry "simple order", [Mun00] uses the adjective as the key order. If one wants to find "simple order" in the index of [Mun00], one has to look under "S". If one looks under "O", one will not find it. The most closely related entry one can find there is "order relation". For the entry "non-degenerate bilinear forms", [Jaco, vol. 2] uses the second adjective as the key word. Obviously, Jacobson considers "bilinear" to have more weight than the other two words. For the entry "Euclidean geometry", [Jaco, vol. 2] uses the noun as the key word. For important entries such as "differential forms" or "differentiable manifold", both the adjective and the noun are considered key words in [Spi, vol. 1].
        We should avoid the use of the secondary index. For example, if we want to find the meaning of H1(X) under "H", the index of [Jack] tells us to look for another entry: first homology group.
        [Spi, vol. 1] contains a notation index in addition to the main index. The notation index in some books is inadequate. For example, the notation index given in [Dug, p.xvi] is too selective. A list of important symbols with their meanings attached is concealed in the middle of the book [Dug, p.3]. The readers often meet a notation that they have seen before but cannot find what meaning it represents in [Dug]. [Zyg] mingles its notation index with its main index. How about Greek letters? [Zyg] uses the equivalence between Roman letters and Greek letters for indexing, while [Mun00] uses the English spelling of a Greek letter for indexing. How about Arabic numerals or Hebrew letters? Again, we use the English spelling. See "4-vector" in the index of [Rob].
        In the index of [Spi, vol. 1], the notations in each chapter seem to be grouped in the following order: alphabetical order, from capital letters to small letters, Greek letters, and then miscellaneous. In [Spi, vol. 1], if an entry appears in chapter 7 for the first time, then one has to look for the entry under "Chapter 7". [Spi, vol. 1] only records the pages on which it appears in chapter 7. Beyond chapter 7, the notation index of [Spi, vol. 1] does not record its occurrences. One interpretation is that after one has served his apprenticeship, he should be on his own. The following story might provide another interpretation. In one's hometown, one writes down the phone numbers and addresses of one's friends for mutual help. However, if one goes to a foreign country, one won't do so because one does not trust foreigners.

  13. Mutual references
        [Inc1, 7.231] uses [Wat1, 14.11] and [Wat1, p.197, l.1-l.3] uses [Har, p.70, Theorem 10.1]. Thus, a book on differential equations can use a calculus book as a reference book [Inc1, p.160, l.-1] and a calculus book can use a book on differential equations as a reference book [Wat1, p.194, l.-5-l.-3]. There is no conflict in mutual references. If Book B is a reference book of Book A, it does not mean that Book B is a prerequisite for reading Book A. All we need to do is read a small part of Book B in order to understand the point in discussion in Book A.

  14. In order to fully describe the ideas involved and leave no room for doubt or confusion, a reference should provide a complete list of quoted theorems.
        The statement in [Wat1, p.368, l.13-l.16] is given without indicating what theorems to use in order to prove its truth. In contrast, [Guo, p.378, l.-5] provides a quoted theorem. However, the quotation is not complete. A complete quotation should include two theorems: [Guo, p.307, (2)] and [Wat1, p.345, l.-8]. For a complete illustration, read [1].


  15. Links {1, 2}.