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Confusions in Differential Equations

1. When confusion arises in a proof, it shows that one fails to fully understand the theorem. Consequently, confusion serves as a checkpoint to see if one truly understands a theorem. Sometimes, even if one cannot detect any flaws in the author's formulations and is certain that one understands every logical step of the proof, these endeavors may still not suffice to dispel the confusion. It may take some time to clarify the confusion.
       In [Zyg, vol.1, p.52, Lemma 6.4], c(t) = f(x+t)g(t) ® 0 uniformly in xÎ[0, 2p].
   In [Zyg, vol.1, p.53, l.8], S_n*(x) ® 0 uniformly in x Î I '.
   One may wonder why we must restrict x from [0, 2p] to I '.
   Indeed, the last term of [Zyg, vol.1, p.53, (6.5)] converges to zero uniformly in xÎ[0, 2p].
   However, this term is not equal to S_n*(x) unless x Î I '.
   
   
2. A cause of confusion: We may mistake one parameter for another because they are similar.
       Suppose we try to analyze the meaning of the statement given in [Zyg, vol.1, p.56, l.-8-l.-7]. According to the definition given in [Zyg, vol.1, p.55, l.-3], we should consider the integral
   ò_[-T,T] (sin lt)/t dt = ò c _[-T,T] (t) (sin lt)/t dt, where l is a parameter and T is a variable of the domain to be integrated. We may treat T as another parameter. l can approach +¥, so can T. Since l and T share similar properties, we may mistakenly mix up their roles. For example, we may easily mistake "outside an arbitrarily small neighborhood of l = 0" for "outside an arbitrarily small neighborhood of T = 0". This is because T and t are closely related and we are often concerned with the neighborhood of t = 0, a singularity of (sin lt)/t.
   
   
3. If one just reads one method and if follows the argument closely, one will not get confused. However, if one reads many methods without careful comparison, one may easily get confused. Deep understanding can only be reached through clarification of confusion by careful comparison.
   Example. After reading [Wat1, p.257, l.1-l.-11], one may ask why the expression given in [Hob, p.191, l.8] considers merely the argument of (t²-1)ⁿ or that of (t-m)^-n-m-1 while [Wat1, p.257, l.-12-l.-11] considers both of them. First, the path in [Wat1, p.257, l.-12-l.-11] connects two branch points, while each contour in [Hob, p.191, l.8] is around a branch point. Only the argument of (t²-1)ⁿ affects the integral around branch point t = 1. The argument of (t-m)^-n-m-1 will not affect the integration on the Riemann surface of (t-1)ⁿ along CgdC around t = 1 no matter whether the integration along CgdC is before the integration along CabC or after the integration along CabC. This is because for the Riemann surface of (t-1)ⁿ, the argument of C returns to its original value after going around CabC.



4. When we use the partial derivative notation, we must indicate which variables should be treated as constants.
       The relations between the rectangular and spherical coordinates are
   x = r sin q cos f, y = r sin q sin f, z = r cos q.
   (¶z/¶r)_q = cos q = z/r.
   r² = x²+y²+z² Þ (¶r/¶z)_x,y = z/r.
   Without subscripts to indicate which variables should be treated as constants, one may wonder why ¶z/¶r = ¶r/¶z.



5. [Wae, vol.1, p.181, l.-4-l.-2] claims, "Each single Q is fixed by 8 permutations; the three of them together remain fixed only under B_4."
One may say, "An automorphism s fixing Q₁ will fix Q₂ because Q₂ ÎD(D^1/2)[Q₁] = D(D^1/2)(Q₁, Q₂, Q₃). This would contradict the first part of the claim."
Clarification. s has the above property if s belongs to the Galois group of D(D^1/2)[Q₁] /D(D^1/2). However, the 8 permutations belong to the Galois group of S/D. If t belongs to the Galois group of S/D, t is determined only if its value at the the generator of S/D is determined. Q₁ is the generator of D(D^1/2)[Q₁] /D(D^1/2), but is not the generator of S/D.


6. The area of surface of revolution is given by [Cou2, vol.II, p.738, l.-3,(2b)] or [Cou2, vol.II, p.429, (31a)].
   Note that the above two formulas are similar but different. The former is derived from the expression given in [O'N, p.281, l.11], while the latter is derived from [Johnson, p.532, 13.29]. u and x are related by the first formula given in [Cou2, vol.II, p.429, l.-11]