Confusions in Differential Equations
- 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], Sn*(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
However, this term is not equal to Sn*(x)
unless x Î I '.
- A cause of confusion: We may mistake one parameter for another because they
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
(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
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.
- 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.
- 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.
- [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
B4." One may say, "An automorphism s fixing Q1 will fix Q2 because Q2 ÎD(D1/2)[Q1] = D(D1/2)(Q1, Q2, Q3).
This would contradict the first part of the claim."
has the above property if s belongs to the Galois group of D(D1/2)[Q1]
/D(D1/2). However, the 8 permutations belong to the
Galois group of S/D. If
t belongs to the Galois group of
t is determined only if its value at the the
generator of S/D is determined. Q1 is the
generator of D(D1/2)[Q1]
/D(D1/2), but is not the generator of S/D.
- 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]