The
Poincaré phase plane is designed to illustrate the following:
How to describe stability [Arn, p.118, l.-9].
That zeros of the solution of
a Sturm-Liouville system are isolated [Arn1, p.253, l.3] and oscillate [Bir,
p.266, l.-10; Arn1, p.253, Fig. 201].
Independence of coordinate systems.
When we say that a definition or a theorem is independent of coordinate
systems, we refer to one of the following two cases:
(Intrinsic) We may define the term or prove the theorem by using a
coordinate system. However, the use of coordinate system can be avoided.
Examples. Linear transformations;
L_{X}Y = [X, Y] [Spi, vol.1, p.213,
l.6-l.10]. Remark. The coordinate-free proof of [Spi, vol.1, p.214, Theorem
1] is based on the fact that the trajectories of the integral curves are
independent of coordinates. The passage given in [Spi, vol.1, p.213, l.6-l.10]
reveals that Spivak does not understand tensors. The main function of tensors is
that they are covariant with coordinate systems. Vector fields are tensors [Spi,
vol.1, p.212, l.8]. If a
tensor equality is valid in one coordinate system, it is automatically valid in
all coordinate systems.
We must use the concept of coordinates. However, no matter which
coordinate systems we choose, as long as they are compatible, the result will be
the same.
Examples. det A= det A'; the
differential structure of a finite-dim vector space using the dual basis as
coordinate functions [War, p.7, l.5]; manifold [Olv, p.2, l.-11-p.3,l.5].
Remark 1. In case II, we would like to make the role that the coordinates play as
trivial as possible. For tangent vectors, please compare [War, p.12, Definition
1.14] with [Olv, p.24, l.21].
Remark 2. The method of characterizing the differential of a form in [Spi,
vol.1, p.289, Theorem 13] is essentially coordinate-free. However, if we use
two coordinate systems to define the differential of a form [Spi, vol.1, p.286,
l.5], we would like to determine their consistency by general properties [Spi,
vol.1, p.288, Proposition 11] rather than a brute-force computation [Spi, vol.1,
p.286, l.8].
Remark 3. When we say that property P is independent of coordinates, in a
narrow sense, coordinates refer to the bases in a finite-dimensional vector
space. In a broader sense [War, p.7, (b)], coordinates can be interpreted as the
compatible coordinate systems in a manifold.
Remark 4. We prefer the Case I-type of coordinate-free definition or theorem because the essence of
geometry (e.g. plane geometry [Boo, p.5, l.14] & topology) does not involve
coordinates. [Boo, p.5, l.21-l.24] gives us a right attitude toward coordinates in
a manifold.
Suppose a curve (x(t),y(t)) rotates in the counterclockwise direction. After
the transformation (x,y)®(x,-y),
the curve will rotate in the clockwise direction [Coh, p.838, l.-9].
Whether a curve is an extremal is independent of the choice of the coordinate system [Fomi, p.30, l.-7].
Identities in vector calculus and geometric properties [Cou2, vol.2, p.196,
l.-13] do not depend on the coordinate system we choose.
The area of a curved surface is independent of the coordinate patch we choose [Cou2, vol.2, p.428,
l.2-l.5].
The way that modern textbooks introduce the definition of the moving
trihedron [Kre, p.33, l.26] of a curve only tells the readers what it is: it fails to tell the readers how useful the device is. We take its
benefit for
granted just as we take the benefit of electricity for granted. We cannot appreciate the use of electricity
unless we have an outage for a week. Similarly, we cannot appreciate the use of
the moving trihedron unless we carefully compare a solution using one fixed Cartesian
coordinate system with the solution using the moving trihedron. For example,
by comparing [For, pp.13-15, §14] with
[Wea1, pp.46-48, §20], we find that we will save tremendous calculations if we choose to use the
moving trihedron. At the same time, we will gain great insight about essential
ideas and internal structures because our
analysis follows the texture of the intrinsic geometry more closely.
Parameterize an ellipsoid
In [Wea1, vol. 1, §62], Weatherburn uses
the concept of confocal conicoids [Fin, §340]
to parameterize an ellipsoid. If you see the figure in [Fin,
§167], you will agree that the
parameterization gives a natural coordinate patch [O'N, p.124, Definition 1.1]
for an ellipsoid. Furthermore, the same parameterization is also good for an
hyperboloid of one sheet or two sheets. Therefore, it is universal: the same
parameterization applies to three types of surfaces. In contrast, in [Kre, p.63, l.5], Kreyszig uses polar coordinates
to parameterize an ellipsoid. The parameterization not only destroys the
coordinates' symmetry [If you know one case, you can predict the form of other
cases. See the following discussion D] but also requires a different parameterization for a
hyperboloid of two sheets [Kre, p.63, l.8]. Kreyszig's choice reminds me of a
children's story. A fox invites a crane to dinner. He treats her to a plate of
soup. The crane can eat nothing. To take revenge, the crane also invites the fox
to dinner. She tucks all the food in a long-neck bottle and the fox gets
nothing. To fetch round beans, use a spoon instead of chopsticks. In my opinion,
differential geometricians nowadays fail to adopt the former parameterization
because it is troublesome to deal with the sign of a square root when one tries
to calculate the second fundamental form of an ellipsoid. However, once you know
a few tricks, the
calculations will no longer be difficult. There are
several tricks can be used to overcome
the difficulties when one tries to prove [Wea1, vol. 1, p.126, (6)]. Without
loss of generality, we may assume x, y, z > 0. We take only the positive value
of a square root and allow only a positive quantity
inside the sign of the square root. The square root of a negative quantity is
meaningless in this context. If there is a negative factor inside the sign of
a square root, try to pair it with another negative factor and treat them as a
positive unit. Or we may change a pair of negative quantities into a pair of
positive quantities by changing both signs.
z_{11} = -(1/4)c^{1/2}
[(c-a)(c-b)]^{-1/2}
(c+v)^{2} [(c+u)(c+v)]^{-3/2}.
Do not try to combine (c+v)^{2} and [(c+u)(c+v)]^{-3/2}
because you must be careful when dealing with a negative quantity such as c+v,
and because you will ruin a good chance for cancellation when you calculate the
product of z_{11} and the z-component of n.
Express similar quantities with the same form.
x_{12} = (1/4) a^{1/2}[(a-b)(a-c)(a+u)(a+v)]^{-1/2}.
z_{12} = (1/4) c^{1/2}[(c-a)(c-b)(c+u)(c+v)]^{-1/2}.
Do not leave y_{12} in the form y_{12}
= (1/4) b^{1/2}[(b+u)(b-c)]^{-1/2}
[(b+v)/(b-a)]^{-1/2}
(b-a)^{-1}.
In order to express y_{12} in the same form of x_{12}
and z_{12}, we try to combine
[(b+v)/(b-a)]^{-1/2}
with
(b-a)^{-1}.
Only a positive quantity can be moved into the sign of the square root from outside.
For example, (-5)(2)^{1/2}
= -5(2)^{1/2} =
-(50)^{1/2}. Otherwise, it will contradict the definition of a square root.
(-5)(2)^{1/2}
¹ (50)^{1/2}. Since b-a<0, [(b+v)/(b-a)]^{-1/2}
(b-a)^{-1}
= -[(b+v)/(b-a)]^{-1/2}
(a-b)^{-1}
= -[(b+v)(b-a)]^{-1/2}.
When you calculate the product of y_{12} and the
y-component of n, note that
[(b-c)^{2}(b-a)^{2}]^{-1/2} = [(a-b)(b-c)]^{-1} because a>b.
The proof of [Wea1, vol. 1, p.126, (6)(iii)] is essentially the same as that
of [Wea1, vol. 1, p.126, (6)(i)] except we have to replace 1 by 2, u by v and v by u. For
extra tricks, see E and F.
When you try to calculate the product of y_{22} and the
y-component of n, note that the y-component of n equals [(ab)/(uv)]^{1/2}
[(b+u)/(b-c)]^{1/2} (a-b)^{-1}
[(b+v)(b-a)]^{1/2}.
Parameterize a hyperboloid of one sheet
In [Wea1, vol. 1, §62], Weatherburn uses
the concept of confocal conicoids [Fin, §340]
to parameterize a hyperboloid of one sheet. The unit normal given in [Wea1, vol.
1, p.126, l.3-l.5] is incorrect. The z-component of n should have been
-(ab/u)^{1/2} [(c+v)/v]^{1/2}
(c+u)^{1/2} [(c-a)(c-b)]^{-1/2},
a negative number, since pz/c<0. This fact is also obvious from the shape of the
surface (see the photo pages at the back of the book, [Fin, Figure 2]). The
calculations of the surface's second fundamental form is
essentially the
same as those for an ellipsoid. The only difference is how to pair negative factors inside the sign of
a square root. Therefore, in the following I mention only a few
tricks that address this difference.
Without loss of generality, we may assume x, y, z > 0.
When you calculate the product of z_{11} and
the z-component of n, note that [(c+v)^{2}]^{1/2}
= -(c+v).
When you calculate z_{12}, note that c[c(c+v)]^{-1/2}
= -[(c/(c+v)]^{1/2}.
When you calculate z_{22}, note that [c/(c+v)]^{-1/2}
[-c(c+v)^{-2}]
= c(c+v)^{-1}[c(c+v)]^{-1/2}.
When you calculate the product of z_{22} and the
z-component of n, note that c(c+v)^{-1} [c(c+v)]
^{-1/2} =
-(c+v)^{-1}
[c/(c+v)]^{1/2}.
Parameterize a hyperboloid of two sheets
In [Wea1, vol. 1, §62], Weatherburn uses
the concept of confocal conicoids [Fin, §340]
to parameterize a hyperboloid of two sheets. The unit normal given in [Wea1, vol.
1, p.126, l.3-l.5] is incorrect. The y-component of n should have been
-(a/uv)^{1/2} [c/(b-a)]^{1/2}
(b-c)^{-1/2}
[(b+u)(b+v)]^{1/2},
a negative number, since py/b<0. The z-component of n should have been
-(a/uv)^{1/2} [b(c+v)]^{1/2}
(c+u)^{1/2} [(c-a)(c-b)]^{-1/2},
a negative number, since pz/c<0. Note that the unit normal points toward the
inner side of the two cups rather than the outer side. His
choice is counter to current practice. The calculations of the surface's second
fundamental form is
essentially the
same as those for an
ellipsoid. The only difference is how to pair negative factors inside the
sign of a square root. Therefore, in the following I mention only a few tricks
that address this difference.
Without loss of generality, we may assume x, y, z > 0.
When you calculate the product of y_{11} and the
y-component of n, note that [(b-a)^{2}]^{-1/2}
= -(b-a)^{-1}.
When you calculate the product of z_{11} and the
z-component of n, note that [(c+v)^{2}]^{1/2}
= -(c+v).
When you calculate z_{12}, note that c[c(c+v)]^{-1/2}
= -[(c/(c+v)]^{1/2}.
When you calculate the product of y_{22} and the
y-component of n, note that [(b-a)^{2}]^{-1/2}
= -(b-a)^{-1}.
When you calculate z_{22}, note that [c/(c+v)]^{-1/2}
[-c(c+v)^{-2}]
= [c(c+v)]^{1/2}(c+v)^{-2}.
When you calculate the product of z_{22} and the
z-component of n, note that [(c+v)^{2}]^{1/2}
= -(c+v).
Parameterize an elliptic paraboloid
In [Wea1, vol. 1, §66], Weatherburn uses
the concept of confocal conicoids [Fin, §340]
to parameterize an elliptic paraboloid. Note that the unit normal [Wea1, vol. 1,
p.132, l.4] points toward the inner side of the surface rather than the outer
side. His choice is opposite to the current practice. The calculations of
the surface's second fundamental form is
essentially the
same as those for an ellipsoid. The only difference is how
to pair negative factors inside the sign of a square root. Therefore, in the
following I mention only a few tricks that address this difference.
Without loss of generality, we may assume x, y, z > 0.
2xx_{1} = -4a(a-v)/(b-a)^{-1}
>0 ̃ x_{1 }>0
̃ x_{1} = a^{1/2}(b-a)^{-1/2} [(v-a)/(u-a)]^{1/2}.
If you write x_{1} = -a^{1/2}(b-a)^{-1/2} [(a-v)/(a-u)]^{1/2},
it would be incorrect because x_{1 }>0.
When you calculate the product of y_{11} and the
y-component of n, note that [(b-u)/(a-b)]^{1/2}[(a-b)/(b-u)]^{-1/2}
= (b-u)/(a-b).
When you calculate x_{22}, note that [(u-a)/(v-a)]^{-1/2}
[-(u-a)(v-a)^{-2}]
= [(u-a)(v-a)]^{-1/2}[-(u-a)(v-a)^{-1}].
Parameterize a hyperbolic paraboloid
In [Wea1, vol. 1, §66], Weatherburn uses
the concept of confocal conicoids [Fin, §340]
to parameterize a hyperbolic paraboloid. The unit normal given in [Wea1, vol.
1, p.132, l.4] is incorrect. The x-component of n should have been [(a-u)/v]^{1/2} [b(a-v)]^{1/2}
[(b-a)u]^{-1/2},
a positive number, since -x/a>0 [Wea1, vol. 1, p.132, l.2]. The calculations of the surface's second
fundamental form is
essentially the
same as those for an
ellipsoid. The only difference is how to pair up negative factors inside the
sign of square root. Therefore, in the following I mention only such tricks.
Without loss of generality, we may assume x, y, z > 0.
When you calculate the product of x_{11} and the
x-component of n, note that
b^{1/2} [(u-a)/(-v)]^{1/2}
[(a-v)/u]^{1/2} (-a)^{1/2}
(a-v)^{1/2} (u-a)^{-3/2}
= [ab/(uv)]^{1/2} (a-v)(u-a)^{-1}.
When you calculate the product of x_{12} and the
x-component of n, note
ab^{1/2} (auv)^{-1/2} =
-(a^{2}b)^{1/2} (auv)^{-1/2} =
-[ab/(uv)]^{1/2}.
Remark. When an author mentions a theorem in his book but fails to provide the proof, he must
have proven the theorem himself at least once. He should not make a general
conclusion simply because one specific case is proved. Even if the cases are
quite similar, he must have hands-on experience with each case individually.
Otherwise, he will not formulate the
theorem correctly because any small thing can go wrong. The resulting misguidance
will be an academic crime. It wastes readers precious time in correcting mistakes.
For careful readers, if they find the result of their calculations does not
match that of a textbook and check their computations three times without
detecting an error, then it is the time to suspect that the author made a mistake.
How the selection of coordinate axes affects calculations
Read [Mari, p.213, l.-9-p.216, l.20].
Remark. It is more difficult to guess the solution given in [Fomi, p.26, l.$-$1] from the Euler equations than to guess [Mari, p.212, (6.23)] from [Mari, p.212, (6.22)]. Consequently, it is important to choose the right coordinate system.