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## Good Illustrations in Mechanics

1. Boundary conditions [Sad, p.331, (8.41); p.332, (8.45)].

• Electromagnetic wave Propagation
1. (General case) In lossy dielectrics [Sad, pp.417-422, §10.3].
2. (Special cases)
1. Plane waves in lossless dielectrics [Sad, p.423, §10.4].
2. Plane waves in good conductors [Sad, pp.425-428; §10.6].

• Determination of crystal structures by X-ray diffraction
1. The Laue condition: Constructive interference occurs if and only if Dk is a reciprocal lattice vector.
Proof. Þ: [Kit2, p.35, l.-6].
Ü: [Ashc, p.98, (6.4)-(6.7)].
2. The Ewald construction of diffraction peaks [Ashc, pp.101-104].
1. The Laue method using a range of wave lengths.
2. The rotating-crystal method.
3. The powder (randomly oriented grains) method.

• Derive Lagrange's equations using calculus of variations
1. From the viewpoint of equilibrium [Go2, pp.16-21, §1-4]
The key idea: (D'Alembert's principle) The infinitesimal work [F-(dP/dt)dr is zero [Go2, p.17, (1-44)] when the system is nearly in equilibrium.
The procedure:
1. Eliminate the appearance of the forces of constraints [Go2, p.18, l.8].
2. Transform the constraint coordinates to the generalized coordinates [Go2, p.18, l.10].
2. From the viewpoint of action [Fom, §9 & §21].
The key idea: (Hamilton's principle) The infinitesimal action change is zero near the actual path. In other words, the action integral along the actual path [Go2, p.36, (2-1)] is stationary.
Remark. Method A is more difficult than Method B (compare [Fom, p.46, Theorem 2] with [Fom, p.35, Theorem]) because the former involves constraints [Fom, p.48, footnote 9]. The procedure A(b) complicates the problem even more because it fails to use Lagrange multipliers to exploit the symmetry [Rei, p.621, l.16-l.17].

• Normal coordinates in a lattice.
The construction of normal coordinates in [Kit2, p.639, (5)] is much simpler than that in [Sym, pp.469-471, §12.3] or that in [Lan1, p.68]. I would like to try to explain why Kittel's construction is a natural way to decouple the equation of motion [Kit2, p.641, (19)] and the total energy [Kit2, p.641, (21)] even though my explanation is not precise and complete.
1. The reciprocal lattice corresponds to the basis of the Fourier expansion of a periodic function [Kit2, p.32, (5)].
2. There is a one-to-one correspondence between the lattice points and the reciprocal lattice points [Ashc, p.87, l.-18].
3. The energy of a harmonic oscillator is quantized by its frequency [Coh, p.494, (B-34)].
4. The direct enumeration of all the wavelike solutions [Hoo, p.37, (2.8)] can be viewed as a method of decoupling [Hoo, p.38, l.17; p.40, l.11-l.16; Kit2, p.101, Fig. 5] even though decoupling the equation of motion is traditionally considered the first step toward finding the solutions.
5. The dispersion relation [Hoo, p.38, (2.9); Kit2, p.640, (15)] is the major link between [Hoo, p.37, (2.8)] and [Kit2, p.639, (5)].
6. The uncertainty principle implies that the Fourier transforms of two strongly-coupled, broad wave packets in position space are two distantly-separated, narrow wave packets in momentum space. Thus, the use of Fourier transforms facilitates the decoupling process of equations of motion in a lattice.
Remark 1. [Coh, p.591, (19) & (21)] motivate us to define the phonon coordinates as [Kit2, p.639, (5)].
Remark 2. [Mari, p.501, (12.142)] is an extremely powerful device for decoupling (see [Mari, p.501, l.-1]).
Remark 3. In [Rei, p.408, l.22], Reif says that the concept of normal variables reduces the complicated problem of interacting atoms to the equivalent problem of noninteracting harmonic oscillators. However, he only justifies his statement from the viewpoint of energy [Rei, p.408, (10.1.8)].  In fact, whenever we call certain variables normal variables we must routinely test whether they satisfy the following requirements in classical and quantum mechanics.
1. The requirements in classical mechanics.
1. Decouple the total energy [Coh, p.580, (20)]: Express the total energy in terms of the energies which can be associated with each of the modes.
Examples. [Coh, p.596, (47) & (50)] (when x and p are considered as normal variables); [Coh, p.597, (52) & (53)] (when a is considered as a normal variable).
2. Decouple the equations of motion [Coh, p.577, (11)].
2. The requirements in quantum mechanics.
1. Decouple the total energy [Coh, p.600, (76) & (77)].
2. Decouple the equations of motion [Kit2, p.641, (19)].
3. Redecompose the state space as the tensor product of eigenspaces [Coh, pp.583-584, §c; especially, p.583, l.2-l.3]: The old component eigenspace is not invariant under the coupling operator [Coh, p.598, l.-6-l.-5]. We must redecompose the state space as a tensor product of new eigenspaces ([Coh, p.600, (79)] define the new ground state. The new tensor product can be generated by creation operators [Coh, p.600, l.-11].)
4. The uncertainty principle [Coh, p.597, (54)].
5. Any pair of component operators corresponding to different modes commute: Position and momentum operators [Coh, pp.581-582, (26)-(31); p.597, (54)]; annihilation and creation operators [Coh, p.600, (72-a)]; the total Hamiltonian [Coh, p.600, (79)].

• Standing-wave ratio <http://hep.ph.liv.ac.uk/~hutchcroft/Phys258/CN8StandingWaveRatio.pdf>.

• Liénard-Wiechert potentials and fields for a moving charge [Jack, pp.661-665, §14.1].

• A potential well of arbitrary shape.
1. bound states: the energies are bounded [Coh, p.357, l.3] and discrete [Coh, p.354, l.-10].

• ### Energy bands for a periodic potential [Coh, p.372, Fig. 2 & p.379, Fig. 4]. Remark 1. Remark 2. At this general stage, we can only have a qualitative (i. e. geometric) analysis for the big picture. For example, we can discuss the structure of its reciprocal lattice. As the case becomes more specific, more physical meanings can be precisely associated with [Coh, p.379, Fig. 4]. Remark 3. How an energy gap arises (the mathematical (quantum) explanation (Energy gaps must exist somewhere, but we cannot pinpoint their locations.): matching conditions [Coh, p.369. l.-8; Eis, p.p.458, l.-15]; the physical explanations (At a zone boundary, the symmetric wavefuncton and the antisymmetric have different energies.):  qualitative [Eis, p.459, l.-6-p.460, l.18]; quantitative [Kit2, p.179, (6)]). The general theory of an electron in a solid: The main feature of this approach is that the Hamiltonian is not specified [Coh, pp.1161-1168, Complement FXI, §2]. The Hamiltonian can refer to a free electron or an electron bound to an atom. The allowed energy band: [Coh, p.1163, (9) & Fig. 5]. Stationary states: Bloch functions [Coh, p.1164, (14), (15), and (16)]. Remark. The delocalization of the electron: [Coh, p.410, (C-20)] ® [Coh, p.1159, l.2] ® [Coh, p.1164, (17)]. Nearly free electron theory [Hoo, pp.100-104, §4.1].Remarks. We can only focus on one theory at a time. Aiming at too many goals will lead nowhere. [Ashc, chap. 9] and [Kit2, chap. 7] assume that their readers do not have a background in perturbation theory, so they try to develop both perturbation theory and nearly free electron theory at the same time. It turns out that both approaches fail to provide a clear picture of energy bands. For example, n in [Hoo, p.102, (4.4)] is associated with the standing wavefunction sin(npx/a) [Hoo, p.102, l.14] and the n-th term of the Fourier expansion of the potential. In contrast, the meaning of n in [Kit2, p.187, l.-18] and [Ashc, p.162, l.17] is not as specific enough as it could be. I like Hook's approach because it focuses on nearly free electron theory and assumes that his readers have a background in perturbation theory. Furthermore, I prefer having a complete understanding of a 1-dim lattice to having a vague picture of a 3-dim lattice [Ashc, pp.152-166].Hook's approach shows insight. Although his approach is not perfect, it is amenable to improvements. We may use [Kit2, p.183, l.13] to prove that the only important term in the lattice potential of [Hoo, p.101, (4.2)] is V1cos(2px/a) [Hoo, p.129, l.11].We may use [Ashc, pp.155-156, Case 2] to prove that y has the form aeikx+bei(k-2p/a)x [Hoo, p.129, l.13; Ashc, p.156, (9.22)]. [Kit2, p.179, (6)] is not as good as [Hoo, p.102, (4.4) & (4.5)] because the former only calculates the energy gap of the first band, while the latter calculates the energy gap of the n-th band for every n. The tight binding approximation [Iba, pp.137-142, §7.3; Ashc, pp.176-184, §General formulation; §Application to an s-band from a single atomic s-level]. Remark 1. The caption of [Coh, p.1161, Fig.4] gives a more fundamental reason than that given in [Iba, p.141, ii)] to explain why a deep lying band is narrower than the shadow lying band. Remark 2. Both [Abr, p.10, (1.21)] and [Iba, 139, (7.31)] are based on the Ritz method [Iba, p.139, l.13]. Although the latter formula is more intuitive, the former formula has the following advantages: (1). The inversion theorem [Ru2, p.199, Theorem 9.11]; (2). The complicated computation given in [Abr, p.10, (1.23)] is actually a simple consequence of [Ru2, p.202, (13)], a fact that Abrikosov probably did not recognize. [Kit2, pp.245-248, §Tight binding method for energy bands] fails to fully use these advantages of Fourier analysis. Note that [Abr, p.10, (1.21)] is based on [Ru2, p.192, (4)], but the physical interpretations of the two formulas can be different: The domain of wn is cleverly preserved as the position space, while the domain of f^ is often interpreted as the momentum space (The position variable disappears because it becomes the dummy variable of integration). Remark 3. A scholar should not just discuss trivialities and avoid discussing difficult issues by pretending not to see them. Most textbooks in solid state physics fail to explain why g(R) in [Ashc, p.182, l.-9] is the same constant for each of the atom's 12 nearest neighbors. Some of the above books still lack any improvement on this point even after many editions. [Ashc] is one exception. However, Ashcroft gives only a vague hint [Ashc, p.182, l.-12-l.-9]. Ashcroft's argument would be clarified if he were to add that he uses the formula [Ru2, p.186, §8.27, (1)]. The H2+ ion-covalent bonding [Hoo, pp.111-115, §4.3.2]. The physical meaning of the limits of the first allowed band is given by [Hoo, p.114, Fig. 4.7(a); Coh, p.1159, Fig.2 & p.1179, (48)]. Quantum resonance: [Coh, p.1177, l.-15-l.-9]. The origin and stability of the chemical bond [Coh, p.1179, l.-15-l.-6]. The way to improve the result of the variational method is to enlarge the family of trial kets [Coh, p.1182, l.-9; p.1183, Table I; p.1173, Fig. 2].A 1-dim chain.     The physical meaning of the n-th band refers to the n-th principal quantum number for a single atom [Ashc, p.183, Fig. 10.4]. Remark 4. The physical meaning of [Ashc, p.141, (8.50)] is given by [Hoo, p.116, l.-6-p.117, l.6]. Remark 5. Both nearly free electron theory and the tight binding approach have similar dispersion relations [Hoo, p.119, l.-17-p.120, l.-8]. [Iba, p.106, Fig. 6.1] explains why the results derived from the two theories are consistent. Remark 6. A scholar should be brave enough to face a challenge  and should not sweep what he does not understand under the rug. [Cra, p.8, l.9-l.17; Ashc, p.140, footnote 17] explain why E(k) is a continuous function of k. In contrast, [Kit2, chap.7 & chap.9] and [Abr, chap. 1] do not even mention such a problem. Unless he or she is extremely careful, an average reader will not be able to detect these authors have left out something important.

• Faraday's law of induction [Wangs, chap. 17].
1. Faraday's observations [Jack, p.208, l.-16-l.-6].
2. For a static situation, there is no connection between the electric field and the magnetic field. Faraday's law of induction establishes their connection only for a nonstatic situation [Wangs, p.263, l.-27-l.-23].
3. Sometimes we define flux as the product of density and velocity [Wangs, p.393, l.3-l.6]; sometimes we define flux as the dot product of a vector field and an area (e.g, the magnetic flux [Wangs, p.251, (16.6)]). What is the relationship between these two ideas? Answer: [Coh, p.238, l.20-21]. Thus, a generalized concept keeps only a small number of the properties of its original concept.
4. Electromotive force [Hall, pp.518-519, §29-1]; Lenz's law [Hall. pp.577-579, §32-3; p.580, l.25-l.30; Wangs, p.264, l.-17-p.265, l.-10].
5. Faraday's law written in the form of [Wangs, p.272, (17-30)] is independent of the motion of the medium [Wangs, p.272, l.9].
6. [Jack, p.210, (5.137)] is [Chou, p.251, (6.18)]. Its proof is given in [Chou, §6.3]. d/dt in [Chou, p.251, l.5] refers to a fixed charge (particle) in the moving circuit (fluid).  For its physical meaning, see [Lan6, p.3, l.1-l.3]. /t refers to a fixed point in space. For its physical meaning, see [Sym, p.313, l.25-l.26; l.-3-l.-1]. The argument given in [Chou, §6.3] follows closely the formalism given in [Chou, Appendix I] which has well established physical interpretations. The proof given in [Wangs, §17-3] is well tailored to this particular problem, and is simple, direct, and clear in the mathematical sense. However, there is a gap in the deduction from [Wangs, p.271, (17-25)] to [Wangs, (17-26)]. The gap can be filled using [Jack, p.209, l.-15-l.-5] ([Wangs, p.266, (17-8)] is invariant under the Galilean transformation when v <<c) or using the argument in [Cor, §23.2 & §23.7] ([Cor, (23-28)] is invariant under the Lorentz transformation [Cor, (23-61)] when v's magnitude is comparable to that of c).

• Infinitely long ideal solenoid.
1. A has only a jˆ component [Wangs, p.260, Fig. 16-6]. Inside the solenoid, A is given by [Wangs, p.259, (16-49)]. Outside the solenoid, A is given by [Wangs, p.259, (16-50)].
Remark. [Cor, p.350, l.5] gives a direct physical reason why A¹0 outside the solenoid.
2. By symmetry, B is independent of z and of j [Cor, p.355, l.9]. Br= 0 [Cor, p.355, l.14]. Inside the solenoid, Bj is given by [Cor, p.356, l.15] and Bz is given by [Wangs, p.260, l.-5]. Outside the solenoid, Bj is given by [Cor, p.356, l.11] and Bz is given by [Wangs, p.260, l.-4].
Remark. The direct physical reason why Bz=0 outside the solenoid can be attributed to the following two facts: 1. [Wangs, p.226, l.23]. 2. The denominator of the integrand in [Wang, p.227, (14-11)] is large.

• Lightning rods (Background material: <http://www.glenbrook.k12.il.us/gbssci/phys/Class/estatics/u8l4e.html>; basic principles: [Jack, p.78, l.28-p.79, l.6; pp.104-107, §3.4] & <http://www.saburchill.com/physics/chapters/0032.html>).

• Bloch's theorem.
1. Proofs
1. First proof [Ashc, pp.133-135; Kit2, pp.179-180].
1. The Hamiltonian is periodic [Ashc, p.134, l.16]. (Proof. Let y=x+R. Then /y=/x.)
2. Considering the lattice symmetry [Hoo, p.328, l.-9-l.-4], we must require that the wave function satisfy [Kit2, p.160, (8)]. [Ashc, p.134, l.26-l.30] shows that this physical requirement is theoretically feasible. If the wave function is degenerate, there will be some  difficulty in proving [Ashc, p.134, (8.12)], but this difficulty can be overcome by the method indicated in [Coh, pp.141-142, (ii)]. Thus, the first proof is still valid even without Kittel's extra assumption that yk is nondegenerate [Kit2, p.179, l.-3-l.-2].
3. The proof given in [Tin, p.38, (3-26)] is also based on the idea of diagonalization, but its discussion is limited to a special case [Tin, p.38, l.11-l.14. Here, the group of the Schrödinger equation [Tin, p.33, l.-11] must be cyclic] and uses the language of the group representation theory.
2. Second proof [Ashc, pp.137-139; Kit2, pp.183-185].
1. The Fourier series expansion of the wave function: [Ashc, p.137, (8.30)], where {q}=all the values of the wavevector permitted by the Born-Von Karman boundary conditions (see [Ashc, p.136, (8.27)]; [Kit2, p.183, (25)]).
Remark. For the physical origin of the Born-Von Karman boundary conditions, see [Iba, p.83, l.-8-p.84, l.9]. The advantage of the Born-Von Karman boundary conditions is that we can base our discussion of a finite crystal on the model of an infinite crystal [Ashc, p.136, (8.27)] rather than restart the discussion from scratch ([Coh, Complement OIII partially repeats the discussion of [Coh, Complement FXI])
2. Decouple the family of {cq} in [Ashc, p.137, (8.30) into subfamilies ck+K [Ashc, p.138, l.-11], where k's are defined by [Ashc, p.136, (8.27)]. We label the decoupled wave function y as yk [Kit2, p.184, l.7-l.10]. [Kit2, p.235, l.-9-l.-3] and [Hoo, p.330, l.-13-l.-7; p.331, l.-7-p.332, l.-1]-332, §11.4.1] say the same thing, but Kittel's formulation is more concise and precise.
3. Third proof [Coh, p.1162,l.1-p.1164, l.-4].
2. The dashed curves in [Ashc, p.133, Fig. 8.1] are derived from [Coh, p.790, (C-4)].
3. The crystal momentum is not the electronic momentum [Ashc, p.139, (8.45); Kit2, p.205, (11)].
4. [Ashc, p.141, (8.50)] is clearly explained by the labeling scheme in A.b.ii. The steps in A.b.ii pinpoint the reason why the second proof allows the yk to be degenerate [Kit2, p.185, l.-1].
5. The origin of the set {enk | where k is fixed and n is any integer} in [Ashc, p.141, (8.50)].
1. From the viewpoint of eigenvalues: the roots of the determinant of [Kit2, p.187, (32)].
2. From the viewpoint of eigenfunctions: [Cra, p.13, l.5-l.7].
6. The essential ideas of the above three proofs are the same (decoupling). The second proof uses Fourier analysis to convert the equation of motion into decoupled linear systems of algebraic equations. The first proof uses linear algebra to find a basis to simultaneously diagonalize the Hamiltonian and translation operators.  The third proof specifies the wavefunction [Coh, p.1164, (13)] and is a special case of the first proof [Coh, p.1164, l.-2-p.1165, l.19].
7. Empty lattice approximation [Kit2, p.188, l.-4-p.189, l.-7]: let the potential functions Un(x) ® 0 uniformly in x as n®+¥; the displacements indicated in the caption of [Kit2, p.236, Fig. 3] are justified by [Kit2, p.237, (2)].

• Energy levels near a single Bragg plane [Ashc, pp.152-159].
1. In the case of no near degeneracy, by [Ashc, p.155, (9.13)], the shift in energy from the free electron value is second order in U [Ashc, p.155, l.9].
2. In the nearly degenerate case, by [Ashc, p.156, (9.19)], the shift in energy from the free electron value is linear in U [Ashc, p.155, l.10].
3. Through the careful estimation from [Ashc, p.155, (9.16)], we shift our attention from [Ashc, p.152, (9.2)] to [Ashc, p.156, (9.19)]. Kittel jumps from [Kit2, p.186, (31)] to [Kit2, p.191, l.20-l.21] by observing the superficial similarity between [Kit2, p.177, (5)] and [Kit2, p.191, (49)]. Thus, Kittel's argument is not as rigorous and careful as Ashcroft's. The argument in [Iba, p.135, l.-2-p.136, l.19] is also better than Kittel's.
The reason given in [Hoo, p.101, l.-4] why we should give up the method used for the nondegenerate case is inadequate because the fact that the first-order energy correction=0 should not stop us from pursuing the second-order energy correction. In contrast, [Abr, p.14, l.1-l.8] gives a good reason why we should switch to nondegenerate case. Furthermore, [Abr, p.14, l.9] gives more choices than those given in [Hoo, p.101, l.-2-p.102, l.5]
4. The caption of [Coh, p.409, Fig. 11] says that the two perturbed levels "repel each other". The meaning of this statement is not clear. In contrast, [Ashc, p.155, l.4-l.7] defines the phrase "two energy levels repel each other" clearly and mathematically.

• Semiconductor crystals.
1. The equation of motion in k space of an electron in an energy band.
1. in a uniform electronic field: [Kit2, p.204, (4)].
2. in a uniform magnetic field: [Kit2, p.204, (7)].
Remark 1. [Kit2, p.204, l.-3-p.205. l.3] illustrates [Ashc, p.229, Fig. 12.6].
Remark 2. The projection of a real space orbit in a plane perpendicular to the field is an orbit of the same shape and rotation direction as the k-space orbit, but rotated 90° around the field direction [Ashc, p.230, l.10-l.13; Hoo, p.375, Fig. 13.7].
2. A hole.
1. wavevector: [Kit2, p.206, (17)].
2. energy: [Kit2, p.207, (18)].
3. velocity: [Kit2, p. 208, (19)].
4. effective mass: [Kit2, p.208, (20)].
5. equation of motion: [[Kit2, p.208, (21)].
Remark. [Kit2, p.209, Fig. 9] is derived from [Kit2, p.204, (4)].

• Anharmonic effects.
1. Thermal expansion [Iba, pp.91-94, §5.5; Hoo, pp.63-66, §2.7.1].
2. Heat conduction by phonons [Iba, pp.94-99, 5.6; Hoo, pp.67-74].
Remark 1. [Hoo, p.69, (2.73)] is the mathematical proof of the physical formulas [Hoo, p.67, (2.68) & (2.69].
Remark 2. [Hoo, p.70, (2.75)] and [Iba, p.96, (5.43)] are the same. The former formula is derived from elementary kinetic theory, while the latter formula is derived from the consideration of the canonical distribution [Rei, p.205, l.10] .
Remark 3. Normal processes versus umklapp processes [Hoo, p.67, Fig. 2.17(b) versus Fig. 2,17(c); Iba, p.97, Fig.5.6(a) versus Fig. 5.6(b); Kit2, pp.134-135, Fig. 16a,c versus Fig. 16b,d].

• The heat capacity of electrons in metal.
1. The tangents in [Iba, p.114, Fig. 6.6 & p.115, Fig. 6.7] help clarify the procedure for estimating the small  fraction of the free elections that can absorb thermal energy.
2. The estimate in [Hoo, p.82, (3.16)] is better than that of [Iba, p.115, (6.36)]. See [Iba, p.117, (6.46)].

• General relativity.
Remark. The tensor design serves to keep the measurement of physical quantities covariant with coordinate transformations so that physical laws will retain the same form.
1. The strong equivalence principle [Ken, p.11, l.-12-l.-10; p.12, l.7].
Remark 1.  The strong equivalence principle based on the weak equivalence principle [Ken, p.10, l.-15] is an extension of the first postulate of special relativity [Ken, p.11, l.-6-l.-3].
Remark 2. A frame in free fall can cover the space-time manifold locally but not globally [Ken, p.12, l.3; p.40, l.7-l.-9; Pee, pp.231-233, § The Metric Tensor].
Remark 3. The principle of generalized covariance [Ken, p.63, §6.4] can be considered the tensor version of the strong equivalence principle.
2. If we express the physical laws in special relativity in terms of tensors, they will retain the same form in any other accelerated frame. In particular, if a formula involves derivatives, the derivatives in the corresponding formula under a coordinate transformation should be replaced by covariant derivatives [Ken, p.81, l.-15-l.-10].
1. Mass affects the metric of the space-time manifold: The Schwarzschild metric equation [Ken, p.44, (4.10)] reduces to the Minkowski metric equation [Ken, p.44, l.8] in the limit of zero mass.
Remark 1. Geodesics in (space ® Minkowski space ® curved space-time) [Ken, p.41, l.10-l.18]. The length of a geodesic in space is a minimum, while the length of a geodesic in the space-time of special relativity is a maximum [Ber, p.56, l.4-p.57, l.-5]. This is because the metric tensor in space is positive definite, while the metric tensor in space-time is indefinite.
Remark 2. The Schwarzschild metric tensor can be derived from [Pee, p.271, (10.84); p.273, (10.92)]. Another proof can be found in [Ken, p.15, l.-3-p.17, l.15; Ber, p.75, l.1-l.8]. The first equality in [Ken, p.16, l.12] can be derived from [Rin, p.40, (17.1)]. The second derivation of the Schwarzschild metric tensor can be made rigorous by using Einstein's field equation [Ken, Appendix D, pp.195-196].
2. Newton's second law [Ken, p.63, (6.11)].
3. The conservation laws of the four vector momentum [Ken, p.81, (7.13)].
4. In the Newtonian limit, Einstein's equation [Ken, p.83, (7.19)] will reduce to Newton's law of gravitation [Ken, p.85, l.7].
3. The tangential acceleration vs. the normal acceleration [Cou2, vol. 1, p.396, (41) & (42)].

• Electromagnetic properties of matter [Fur, §2.4; Wangs, pp. 546-568]
1. It is easier to recognize the outline of electromagnetic properties of matter in [Fur, §2.4] than in [Wangs, pp. 546-568]. Furthermore, few books explain [Fur, p.104, Fig. 2.18] as clearly as [Fur]. However, it is better to study [Kit2, pp.380-392] before one reads [Fur, §2.4]. This is because [Kit2, pp.380-392] provides rigorous definitions of applied electric field, the macroscopic electric field, and the local electric field. The prerequisite to understanding [Kit2, pp.380-392] is [Wangs, chap. 10 and chap. 23].
2. For clarity, [Wangs, p.548, Fig. B-1] should be supplemented with [Hall, p.472, Fig. 26-12].

• Hysteresis [Cor, pp.375-377, §21.2 & p.422, Example].
Remark. [Wangs, p.338, l.22-l.32] illustrates the theoretical advantage of using a Rowland ring, while [Cor, p.375, footnote] explains why Rowland rings are no longer used in practice.

• The Lorentz condition [Wangs, p.365, l.12-l.16].

• The basis of the Debye interpolation scheme.
1. [Ashc, p.466, l.3] To be consistent with the Dulong and Petit law at high temperatures, the area under the theoretical curve gD(w) [Ashc, p.466, l.1] must be the same as that under the experimental curve [Rei, p.410, Fig. 10.1.1].
2. [Ashc, p.466, l.4] To obtain the correct specific heat law at low temperatures, the theoretical curve must agree with the experimental curve in the neighborhood of w = 0. Thus, the Debye scheme should adopt the simplification given in [Ashc, p.456, l.1-l.10; Fig. 23.1].

• Noether's theorem conservation laws [Sag, pp118-123, §A2.16].
1. The energy [Lan1, p.14, l.10], momentum [Lan1, p.17, l.-14-l.-12], and angular momentum [Lan1, p.20, l.-8-p.21, l.9] of a closed system.
2. Conservation laws for the energy-momentum tensor of the electromagnetic field:
1. Special relativity: [Lan2, p.82, (33.6)].
2. General relativity: [Ken, p.81, l.16].
3. Conservation of crystal momentum [Ashc, p.786, (M.7)]:
1. Isolated insulator [Ashc, p.787, (M.18)].
2. Scattering of a neutron by an insulator [Ashc, p.788, (M.22)].
3. Isolated metal [Ashc, p.788, l.-7-l.-5].
4. Scattering of a neutron by metal [Ashc, p.788, l.-4-p.789, l.2].
Remark 1. In terms of reduced symmetry, the conservation of crystal momentum in Case C.a is similar to the the conservation of angular momentum in the examples given in [Lan1, p.21, l.1-l.9].
Remark 2. In order to understand the precise meaning of Noether's theorem, one needs an elaborate analysis like Sagan's. Compare [Sag, p.120, Definition A2.16] with [Fomi, p.80, Definition].

• Thermal conductivity.
1. The formula for thermal conductivity [Ashc, p.500, (25.30) & (25.31)]. The derivation of the formula can be found in [Rei, p.479, Fig. 12.4.2 (1-dim) ® Ashc, p.500, Fig. 25.3 (3-dim)]. The arrow implies that the basic idea in the two cases is the same.
2. The reasons why a perfectly harmonic crystal would have an infinite thermal conductivity.
1. The phonon states are stationary states [Ashc, p.496, l.-11] Û There are no collisions between different phonons [Kit2, p.133, l.23] (i.e. there is no thermal resistivity).
Remark. The scattering of phonons means that the wave functions of phonons evolve with time.
2. A nonvanishing mean velocity is given by [Ashc, p.141, (8.51)] (see [Ashc, p.497, footnote 15]). This mean velocity is not driven by a temperature gradient [Kit2, p.134, Fig. 16a, l.5].
3. [Pei, p.40, (2.56); l.10-l.11].
3.
1. At high temperatures (T >> QD): [Ashc, p.501, (25.32) & (25.33)].
2. At low temperatures (T << QD): As the temperature decreases, the conductivity will increase [Ashc, p.504, (25.40) & p.500, (25.31]. The phonon mean free path will increase up to the length limit imposed by lattice imperfections, impurities, or size. Hence, the phonon mean free path will become independent of temperature. Thus, the temperature dependence of the conductivity is determined by the specific heat. Specifically, the conductivity will rise as T3. The rise will reach a maximum at a temperature where umclapp processes [Ashc, p.502, l.1] become frequent [Ashc, p.504, (25.39)] enough to yield a mean free path shorter [Ashc, p.504, (25.40)] than the temperature-independent one. Beyond this temperature, the conductivity continues to decline as exp(T0/T) [Ashc, p.504, (25.40)] up to temperatures well above QD. After this the exponential decline is quickly replaced by a slow power law [Ashc, p.501, (25.33)].
Remark. When studying thermal conductivity, I had a hard time understanding both [Ashc] and [Kit2]. Ashcroft repeats the same word "the mean free path" five times in a single paragraph (see [Ashc, p.504, l.-19-l.-10]). His act of repeating the same words as though they were an incantation and his consideration on the impacts on the mean free path due to an overwhelming number of factors only obscure, rather than clarify, the key point. [Kit2, pp134-135, Fig. 16] and its illustrations occupies almost half of the space of the entire section [Kit2, pp133-135, § Thermal resistivity of phonon gas]. However, this figure is only a minor point in understanding thermal conductivity. Thus, Kittel's emphasis is misplaced.

• The Hartree-Fock approximation.
1. The expectation value of the Hamiltonian: [Ashc, p.333, (17.14) or Ost, p.111, (3.2)].
2. The Hartree-Fock equations: [Ashc, p.333, (17.15) or Ost, p.114, (3.14)].
Remark. For the purpose of the derivation of the above equalities, Ostlund's simplified notions are more appropriate.

• Rayleigh scattering [Jack, p.466, (10.35)] explains why the sky is blue, why the sunset is red, why it is easy to get a sunburn at midday, and why infrared is good for seeing distant stars through the dust in the Milky Way.

• The dispersion relation in a plasma [Kit2, p.274, (15)] explains the transparency of alkali metals in the ultraviolet and the reflection of radio waves from the ionosphere [Kit2, p.274, l.-3-l.-2].

• The Robertson-Walker metric [Ber, p.105, (6.1.3)].
Remark. Peebles' oversimplified introduction to this metric [Pee, p.54, (5.9)] fails to stress its insight: the Gaussian curvature is invariant. The proof of Theorema egregium in [Ber, Appendix B, pp.160-163] is awkward. A better proof can be found in [Lau, p.65, Theorem 5.5.1].

• Special relativity.
1. How we synchronize clocks [Rin, p.9, l.-13-l.-9].
2. Why the transformation from one inertial frame to another is linear [Rin, p.11, l.17-l.27].
3. The analogy between a rotation and a Lorentz transformation [Cou2, vol. I, §4.1.j].

• Transport theory.
1. Reif's approach goes from the easy to the complicated: using average velocity v to express the collision frequency [Reif, p.470, (12.2.7)] ® using the distribution function f(r,v,t) to formulate the Boltzmann equation [Reif, p.525, (14.3.8)]. In contrast, Huang's approach jumps to the complicated directly [Hua, chap.3]. Thus, Huang leaves out the following two important turning points of the theory's development:
Flux: [Reif, p.470, (12.2.6)] ® [Reif, p.497, (13.1.3)].
The Boltzmann equation: [Reif, p.509, (13.6.2)] ® [Reif, p.525, (14.3.7)]. The equivalence [Reif, p.510, l.1] of the two formulations enables us to jump from a crude approach [Reif, p.504, l.3-l.6] to an more accurate approach.
2. Huang fails to prove du=du' [Hua, p.60, l.-13], while Reif gives a rigorous proof [Reif, p.521, l.15].
3. Reif establishes a relationship between s(v1,v2ïv1',v2') and s(W) [Reif, p.520, (14.2.4)], while Huang does not. Therefore, the statement in [Hua, p.69, l.-8] is not clear. Similarly, [Reif, p.524, (14.3.3) is clear, while [Hua, p.66, (3.29)] is not.
4. The generalization from an inversion [Reif, p.522, l.1-l.2] to a rotation or reflection [Hua, p.63, l.9-l.10] is unnecessary because it does not have any other useful application than the inverse collision.
5. [Reif, p.523, l.-18-l.-15] imposes an essential assumption on f(r,v,t) to justify the format of the mathematical expression in [Hua, p.56, (3.2)]. The reasons given in Huang's justification [Hua, p.56, l.8-l.14] are related, but are not essential.
6. The equality in [Hua, p.96, l.8] is derived from [Reif, p.529, (14.4.20)] and its corresponding formula for the inverse collision.
7. The Collision frequency: [Reif, p.470, (12.2.7)] is too crude and [Reif, p.470, (12.2.8)] is too sophisticated. [Reic, p.660, (11.14)] gives an appropriate interpretation of the Collision frequency.
8. [Hua, p.106, (5.72)] is correct, but Huang's argument for its derivation is incorrect. It would be better to use brute force to calculate each coefficient of Lkl in the summation on the right-hand side of [Hua, p.106, (5.71)].
9. (Unifications) The conservation theorem [Hua, p.96, (5.14)] unifies the conservation laws of mass, momentum, and energy [Hua, p.98, (5.21)-(5.23)]. Huang derives the conservation theorem from the Boltzmann transport equation [Hua, p.67, (3.36)], which involves the concept of differential cross section. In contrast, [Reic, pp.534-537, §10.B.1] derives the conservation laws of mass, momentum, and energy without using the concept of differential cross section. The entropy source [Reic, p.539, (10.26)] helps define the generalized currents and forces [Reic, p.539, l.-11-l.-5]. Thus, the discussion in [Reic, pp.537-541, §10.B.2] is an indispensable step toward recognizing that transport coefficients are the generalized conductivities of a hydrodynamic system [Reic, p.539, l.-4-p.540, l.1;p.541, l.5-l.10; p.543, (10.29)-(10.31)]. Putting transport coefficients and conductivity into the same category is a kind of unification that stride across different fields.

• Classical statistical mechanics
1. Ensemble [Hua, p.141, l.8].
Remark. If we can prove a statement directly, we should not take a detour. [Hua, p.141, (7.6)] can be directly derived from the definition of a partial derivative. Huang's detour approach [Hua, p.141, l.-13-l.-12] indicates that he does not understand the definition of a partial derivative very well.
2. The density of states [Man, pp.324-335, Appendix B].
Remark. [Man, p.334, (B-38)] can be proved using [Kit2, p.87, Fig. 18].
3. A system in a heat bath [Man, pp.52-64, §2.5].
Remark 1. The proof of [Man, p.57, (2.29)] can be found in [Reif, p.213, (6.5.8)].
Remark 2. We need not repeat the historical approach. The classical method of counting states must be fully justified in terms of quantum mechanics. Compare [Reif, p.51, l.8-l.16] with [Man, p.174, l.-7-l.-1].
4. Fluctuations
1. Energy [Reif, p.110, (3.7.14); p.213, (6.5.8) or Man, p.58, (2.31) (the canonical ensemble)].
2. Occupation numbers [Hua, p.82, (4.54) (the ideal gas)].
5. The canonical ensemble evolves from the microcanonical ensemble:
1. The drawback of the microcanonical ensemble with respect to calculations [Hua, p.153, l.7-l.12].
2. The new constraint imposed on the canonical ensemble [Hua, p.157, l.2-l.4].
6. The ideal gas.
Remark 1. When considering ideal gases, the first thing one has to do is to throw away all the world's documents about ideal gases. One should study only [Reif, §9.1, §9.2, §9.6, §9.7, §9.8, §9.10], except for any discussions about Maxwell-Boltzmann statistics they contain. If one needs the required background on quantum mechanics, one should read only [Coh, §XIV,C.3.d]. This approach relieves one of the burden of studying a tremendous amount of incorrect physics. To stop the practice of torturing physics students, the future authors of physics textbooks should follow my advice.
Remark 2. Mandl points out that [Hua, p.146, (7.22)] is based on [Hua, p.152, (7.52)]. However, Mandl's strategy to prove [Man, p.188, (7.70)] does not work. It is better to follow Huang's calculation scheme [Hua, p.152, l.9-l.10].
Remark 3. In classical mechanics, a rigorous definition of physical states [Coh, p.1392, (C-9)] for a system of identical particles does not exist. Therefore, for the partition function of the ideal gas, the classical method of counting states requires a correction when compared with the correct quantum result. The only book in classical mechanics that contains a clear definition of a macrostate is [Zem, p.279, l.11], but the concept is borrowed from quantum mechanics. The goal of counting states in classical mechanics is to lead to a rigorous definition of physical states. A good classical method of counting states should facilitate accomplishment of this goal. For example, the first term of [Man, p.168, (7.9)] corresponds to [Coh, p.1390, (C-11)] and the second term of [Man, p.168, (7.8)] corresponds to [Coh, p.1390, (C-10)] (up to the normalization factor). Even so, the classical derivation of [Man, p.169, (7.10)] is not as rigorous as the quantum mechanical derivation of [Reif, p.361, (0.10.3)].
7. (Paramagnetism) The discrepancy between [Reif, p.208, (6.3.7)] and [Pat, p.81, (14)] is due to different averaging methods. The former averages the magnetic susceptibility over the two spin states [Reif, p.207, (6.3.3)], while the latter averages the magnetic susceptibility over all solid angles [Pat, p.80, (7)].
8. (Microcanonical ensemble) Reichl fails to explain why CN = N! h3N in [Reic, p.348, (7.16)]. N! can be explained using the strategy given in [Reic, p.359, l.8]. h3N can be explained by [Man, p.174, l.-1]. Similarly, Huang fails to explain the Gibbs correction factor in [Hua, p.195, (9.42)]. Intrinsically, Microcannonical ensemble is a classical design. Its shortcomings are discussed in [Man, p.182, l.9-l.18]. The problem with [Hua, §9.5] is that Huang throws quantum particles into a classical design without explaining why it is justifiable to do so. A theory should be built with its essential features. Building a theory should not be like making a pizza with every topping on it. it should not involve throwing all the knowledge into one pot. A cumbersome theory that has no application or that is designed to solve every problem is trash. Even just looking at [Hua, p.194, Fig. 9.1] makes one dizzy. The discussion that goes with this figure is even more confusing. One should apply a method in a flexible manner instead of being entrapped in its mathematical structures. Furthermore, a model should be as simple as possible.  Consequently, the discussion of an ideal gas in [Reic, p.348, Exercise 7.2] is better than that in [Hua, p.196, l.-3-p.197, l.16]. The same remark applies to [Pat, §6.1], Cliff's notes of [Hua].
Remark. Landau uses the fact that levels broaden into bands [Lan5, p.15, l.12; p.22, l.-12-l.-9] to explain why the microcanonical ensemble, a classical design, still applies to quantum statistics.
9. The grand partition function
1. The justification of the definition given in [Hua, p.190, (9.27)]: [Reif, p.347, l.1-l.18].
2. Its simplified relationship with the partition function [Reif, p.347, (9.6.6)].
3. Equalities [Hua, p.198, (9.61) & l.-3-l.-1].
Remark. Reichl derives [Reic, p.382, (7.121) & (7.123)] using terminology that is less intuitive but conveys the same idea (i.e., step a and step c). However, the advantage of the Lagrange multipliers, trace, and the number representation allows Reichl's argument to go directly from [Reic, p.378, (7.109)] to [Reic, p.382, (7.121) & (7.123)]. It is unnecessary to pass through the middle stage given in [Hua, p.190, (9.27)], and then change the basis (see step c) to 1obtain the desired result.

• Generalizations of field equations.
1. Einstein's gravitational field equation [Pee, p.268, (10.65)] ® Gravitational field equations for nonrelativistic material [Pee, p.269, (10.69); Rin, p.103, l.12-l.14].
2. Electromagnetic field equations [Rin, p.103, (38.3); p.104, (38.5) & (38.7)] Û the Maxwell equations [Rin, p.107, (38.20), (38.21) & (38.23)].

• Polarizability.
1. The Lorentz relation [Chou, p.76, (2.92)].
2. The Clausius Mossotti relation [Chou, p.76, (2.95)].
3. Electronic polarization (induced dipoles) [Chou, p.77, (a); p.78, (2.96)].
4. Orientation polarization (permanent dipoles) [Chou, p.77, (b); p.79, (2.101)].
Remark. For details of this topic, consult [Wangs, pp.546-554, Appendix B-1].

• Boundary-value problems in electrostatics.
1. Formal solutions of the Poisson equation [Chou, p.31, l.-2-p.32, l.9].
2. The existence of solutions of the Poisson equation with Dirichlet or Neumann boundary conditions [Chou, §3.2].
3. The uniqueness of solutions of the Poisson equation with Dirichlet or Neumann boundary conditions [Chou, §3.2].
Remark 1. The assumptions of this problem are carefully written in [Chou, §3.1]. The argument in [Jack, §1.9] cannot be considered rigorous because Jackson fails to state these assumptions clearly.
Remark 2. All of the following statements are justified by the uniqueness theorem.
1. [Chou, p.115, l.-10-l.-8].
2. [Chou, p.117, l.2-l.5].
3. [Chou, p.120, l.1-l.2].
4. The method of images [Chou, §3.4; Sad, §6.6; Jack, §2.1-§2.5]].
1. The image charge must be external to the region of interest [Jack, p.57, l.-9-l.-8].
2. The solution of the Poisson equation is provided by the sum of the potentials of the charges inside the region of interest [Jack, p.57, l.-6-l.-5].
Remark 1. [Wangs, p.175, l.18-l.6] gives three methods to find the force on q. By the uniqueness theorem, all we need is the method given in [Chou, p.117, l.2-l.5]. Other methods are unnecessarily complicated. Even trying to find a method other than the method of images is meaningless in the first place because anyone who understands the uniqueness theorem thoroughly should not raise such a question. [Jack, p.60, l.-20-l.-13], produced by a Berkley professor, is also of no value.
Remark 2. [Sad, p.240, l.9-p.241, l.6] provides a piece of general guidance that helps to solve boundary-value problems using the method of images. This guidance, based on experiences, can only be described by guidelines rather than specific details. The success of applying these guidelines relies on one's skill and experiences.
Although guidelines are valuable advice, they do not guarantee the success of problem solving. Driving is an example. Driver A with ten year experience is accident free now. Driver B just received his driver license for the first time. Even if Driver A gives Driver B good guidelines about safe driving, the latter still has to go through many accidents during the first year to learn to become a safe driver. Similarly, to become a skilful problem solver, one has to practice constantly and compile new guidelines from one's own experiences. [Lau, p.67, l.12-l.16] provides another interesting example.
5. Complex-variable methods [Chou, §3.5].
6. Conformal representation [Chou, §3.6].
7. Solutions for the spherical boundary conditions: [Jack, §2.6].
8. Boundary-value problems with azimuthal symmetry
1. Dielectric sphere in a previously uniform electric field: [Cor, pp.231-233, Example], [Wangs, pp.194-197, Example] and [Chou, pp.148-150, (ii)] all give the formula of the electrostatic potential. Of the three proofs of this formula, Corson's proof is the best. His concept is clear and his analysis is rigorous. In [Wangs, p.197, Fig. 11-13], Wangsness says the lines belong to the E field. In fact, they belong to the D field (see [Cor, p.232, Fig. 12-2]).
2. A useful device: [Jack, p.101, l.-2-p.104, l.4].
Remark 1. Although Jackson elegantly uses the uniqueness theorem to prove [Jack, p.102, (3.38)], the proof of [Wangs, p.112, (8-12)] is a natural approach which links Legendre polynomials more closely to their generating function [Guo, §5.3].
Remark 2. Although Jackson applies the device only to the boundary-value problems, the key idea of this device is actually based on [Ru2, p.226, Corollary].
9. Mixed boundary conditions (e.g., conducting plane with a circular hole) [Jack, §3.13}
Remark. The equations of [Jack, p.132, (3.179)] can be solved using [Guo, p.406, (8) & (9)].
10. Boundary-value problems with dielectrics [Jack, §4.4].
Remark. For a plane, we use the method of images [Jack, p.154, l.-2-p.157, l.9];  for a sphere or spherical cavity, we use separation of variables in spherical coordinates and expand the solution in a series using the basis of the Legendre polynomials. These methods are essentially the same as those of finding the green functions .

• Electric property of dielectrics
1. The Ewald-Oseen extinction theorem [Born, p.101, l.4-l.7]
The dipole field is the sum of two terms [Born, p.102, (21)]: one cancels out the incident wave [Born, p.102, (23)], whereas the other satisfies the wave equation with velocity c/n [Born, p.100, (10)].
Remark 1. (Internal references) The validity of the statement in [Born, p.101, l.4-l.7] is not well documented, so one may not catch its meanings immediately until one finishes reading [Born, §2.4.2]. However, if Born had pointed out where the readers can find the proof for each phrase of the statement in this long section [Born, §2.4.2] as I did above, the readers would catch the meaning at the first reading and would have a clearer picture for understanding the rest of material in [Born, §2.4.2]. The mathematics textbooks written by N. Bourbaki are famous for their internal references: The validity of almost every statement is well documented whether the proof is given before the statement or after.

Remark 2. For various electric fields in dielectrics only [Kit2, pp.380-392] provides clear definitions. Therefore, it is important to identify the effective field E' in [Born, p.85, l.23] with the local field Elocal in [Kit2, p.386, (14)] or the polarizing field Ep (producing the displacement of charges) in [Wangs, p.547, l.-14] and identify the mean field E in [Born, p.85, l.24] with the total macroscopic electric field E in [Kit2, p.384, (7)].
Remark 3. The assumption [Born, p.104, (32)] is not used in [Born, p.104, l.8-p.107, l.12]. [Born, p.104, (32)] is proved by [Born, p.107, (49); p.105, (41)] with the assumption [Born, p.104, (33)]. The purpose of presenting [Born, p.104, (32)] before its proof is to help create [Born, p.104, Fig. 2.4] so that we know what is going on.
Remark 4. The extinction theorem provides the insightful relationship between the incident field and the dipole field. This relationship based on the microscopic viewpoint (dipoles) is so powerful that it can be used to derive both the law of refraction [Born, p.107, (52)] and the Fresnel formulae [Born, p.107, (55a) & (55b)].
2. Molecular polarizability and electric susceptibility [Jack, §4.5]
1. The Clausius-Mossotti equation [Jack, p.162, (4.70)].
2. The Lorentz-Lorenz equation [Jack, p.162, l.-1].
Remark. [Born, p.87, (17)] serves to link the microscopic quantity a [Born, p.92, (30)] to the macroscopic phenomena (e and n). The Lorentz-Lorenz equation implies that the refractive index depends on frequency [Born, p.92, (31)]. The proof given in [Born, §2.3.3] is valid only for the first approximation [Born, p.85, l.12]. In contrast, the proof given in [Born, §2.4.2] is rigorously derived from an integral equation concerning polarization [Born, p.100, (4)].
3. Electrostatic Energy in dielectric media [Jack, §4.7].
Remark. The material in [Jack, p.166, l.-2-p.167, l.17] is treated by Jackson as part of the content of his advanced textbook. However, in [Wangs, p.164, l.1-l.4], Wangsness treats the same material as an exercise of his elementary textbook. It is too difficult for the reader of an elementary textbook to encounter an exercise that is accorded extensive coverage in an advanced textbook. There is difference between writing a paper and doing an exercise. Similarly, it is not proper to put an exercise from an elementary textbook into the content of an advanced textbook. Considering the intended reader, it is clear that one of the above two authors must be seriously wrong.
Remark. The concept of various fields in the first paragraph of [Jack, p.160] is clear. In comparison with [Jack, (4.71) & (4.95)], the signs of forces are carefully explained in [Wangs, (7-36), (10-99) and (B-7)]. In contrast with the abstract theory given in [Jack, §4.5, §4.7], [Wangs, p.166, Fig. 10-18] gives a concrete example. To reap benefits from both books, it is important to the differences in terminology they employ. Let us attach a subscript J to a notation if the notation is used in [Jack] and a subscript W to a notation if the notation is used in [Wangs]. By comparing [Jack, p.161, (4.67)] with [Wangs, p.548, (B-9)], we see Ep;W = E;J+Ei;J and the sum in [Wangs, p.163, (4.73)] = a in [Wangs, p.548, (B-9)]. The following identities show that there are no inconsistencies in concepts between [Jack, §4.5] and [Wangs, Appendix §B-1] even though the same notation or terminology in the two books may mean different things:
E I;W = E near;J and
E;J = E p;W - E i;J
= (E ;W + E O;W + E I;W) - (E near;J - E P;J)
= E ;W + E O;W + E P;J .
The above comparison is only a temporary remedy. In the future, we must unify the terminology in this area so that physicists can speak the same language.

• Force on a localized current distribution in an external magnetic induction [Jack, p.188, l.-12-p.189, l.-7; Wangs, pp.531-538, §A-2]
The proof of [Jack, p.189, (5.69)] is incorrect. The notation m´Ñ in [Jack, p.189, (5.67)] is problematic. This notation is not defined in any math textbook because multiplication and differentiation are not commutative: m(f/x)¹¶(mf)/x. Furthermore, [Jack, p.189, (5.68)] is incorrect. For a correct formula of Ñ(m×B), see [Wangs, p.34, (1-112)]. Thus, [Jack, p.189, (5.67)] should have been
Fi = S eijk[m´ÑBk(0)]j; [Jack, p.189, (5.68)] should have been
F = m´(Ñ´B) + (m×Ñ)B = Ñ(m×B).
Remark 1. Jackson's serious mistakes reveal the urgent need to strengthen the teaching staff in today's institution of American higher education.
Remark 2. Using [Chou, p.214, (5.62)], Choudhury gives an elegant proof of [Chou, p.214, (5.63)].
Remark 3. There is no inconsistency between [Jack, p.189, (5.69)] and [Wangs, p.537, (A-35]. We can use [Wangs, p.533, (A-20)] to explain why the two formulae look different.

• The Magnetic Hyperfine Hamiltonian
1. A classical treatment [Jack, p.190, (5.73)].
2. A quantum mechanical treatment [Coh, pp.1247-1256, Complement A XII].
Remark 1. [Coh, p.1251, Fig. 2; p.1252, l.1-l.19; p.1253, l.-17-l.-4] provide a better explanation of the second term of [Jack, p.188, (5.64)] and of the contact term of [Jack, p.190, (5.73)].
Remark 2. [Jack, p.145, l.-7-p.146, l.-5] gives a rigorous proof of the statement in [Coh, p.1066, l.6].
Remark 3. The formula given in [Coh, p.1249, l.-2] and the formula given in [Jack, p.190, l.-7] are the same. The latter formula is derived from [Jack, p.176, (5.5)] by replacing the x in [Jack, p.175, Fig. 5.1] with -x (where x is the position of the electron).
Remark 4. [Jack, p.190, l.-13-l.-11] tells us what the hyperfine interaction is, while [Coh, p.1248, (5)] traces to the origin of the hyperfine interaction.

• Magnetization
1. ### A substance with permanent magnetization: B and H [Jack, §5.10]. Without an external field [Jack, p.198, (5.105) & (5.106)]. In an external field [Jack, p.200, (5.112)]. Remark. The difficulty of the method given in [Jack, p.199, l.1-l.8] lies in calculating ò [0, a]:ò [0, a] = ò [0, r]  + ò [r, a]. The difficulty of the method using the vector potential [Jack, p.199, l.-12-p.200, l.6] lies in calculating the curl. The method given in [Jack, p.198, l.6-l.-1] does not have these difficulties, so it is the simplest. A paramagnetic or diamagnetic substance: the magnetization is the result of the application of an external field [Jack, p.200, (5.115)]. A ferromagnetic substance: the phenomenon of hysteresis allows the creation of permanent magnets [Jack, p.201, Fig. 5.12]. Remark. For the basics of hysteresis, see [Sad, p.328, l.5-p.329, l.-1].

• Magnetic shielding
1. Analogies between a conductor and a body with high permeability
1. The field lines outside and near to the surface [Jack, p.201, l.-17-l.-15].
2. Cavities [Jack, p.201, l.-15-l.-13].
2. The dipole moment and the inner field [Jack, p.202, (5.121); p.203, Fig.5.14].

• The Daniell cell: <http://www.mpoweruk.com/chemistries.htm>.

• Surface tension: <http://www3.interscience.wiley.com:8100/legacy/college/cutnell/0471151831/ste/ste.pdf > & <http://en.wikipedia.org/wiki/Surface_tension >.
Remark. The discussion of surface tension in [Zem, §2-9 & §3-8] is ambiguous. and raises many questions. The above two web sites will help answer your questions.

• Basics of thermodynamics
1. Thermodynamic equilibrium [Zem, §1-5 & §2-1]; equations of state [Zem, §2-5; §2-8-§2-12]; macroscopic states and thermodynamic variables [Zem, p.26].
2. Quasi-static transformations [Hua1, p.4, (f)]. In order to warrant the use of an equation of state, we must perform a quasi-static process. Methods of performing a quasi-static process [Zem, p.51, l.-9-l.-8; p.56, l.14-l.16; p.57, l.6-l.7; p.85, l.-14-l.-4].
Remark. Slow free expansion is quasi-static [Hua1, p.4, l.11-l.12]; fast free expansion is not quasi-static [Zem, p.113, l.11].
3. Reversible transformations
2. Reversibility involving heat transfer: reversibility in this case refers to the universe [Zem, p.85, l.-13-l.-4; §8-5-§8.6].
Remark. [Zem, §7-7] proves that the solutions of the differential equation [Zem, p.165, (7-1)] are reversible adiabatic hypersurfaces. The illustration builds a solid foundation for the following concepts: Carnot cycles [Zem, p.173, Fig. 7-8], Kelvin temperature scale [Zem, §7-10] and entropy (Compare [Zem, p.179, l.-3] with [Zem, p.174, l.14, the first equality]; [Zem, p.180, (8-3)]). Because they lack this indispensable proof, the statements about the above concepts given in both [Hua1] and [Kit] are unclear. For this reason, their foundations of thermal physics are seriously flawed.
4. Ideal-gas temperature = Kelvin temperature [Zem, §7-8-§7-11].

• Speed of a longitudinal wave [Zem, §5-7].

• The second law of thermodynamics
The following three statements are equivalent:
1. No process is possible whose sole result is the absorption of heat from a reservoir and the conversion of this heat into work.
2. No process is possible whose sole result is the transfer of heat from a cooler to hotter body.
3. Whenever an irreversible process takes place the entropy of the universe increases.
Proof. A Û B [Zem, §6-7].
A Þ C [Zem, §8-5-§8-8].
C Þ B [Man, p.115, l.-9-l.-1].
Remark 1. If you compare [Zem, p.154, 1] with [Hua1, p.10, l.-18-l.-7], one can easily find that T1 in [Hua1, p.10, l.-8] should have been T2. This error remained undetected through two editions of [Hua1] (1963 & 1987). This reveals the fact that in these forty years, either no one reads the publications of MIT professors or no one cares about the books published by MIT professors.
Remark 2. [Hua1, p,19, l.13-l.-6] discusses some subtle points that we should pay attention to when we apply the second law.

• The Clausius theorem
The most concise and insightful proof is given by [Reic, p.28, l.5-p.31, l.4]. [Reic, p.30, Fig. 2.5] clarifies the confusion contained in other proofs. The proof of [Zem, p.180, (8-3)] is based on [Zem, p.173, (7-13)]. The inexact differential in [Zem, p.173, (7-13)] has a specific form. The inequality in [Hau1, p.14, l.-8] conveys Clausius' subtle point about the inexact differential đQ. However, the proof of the Clausius theorem in [Hua1, pp.14-15], omits too much detail. A detailed proof using the same argument can be found in http://en.wikipedia.org/wiki/Clausius_theorem.

• The Clapeyron equation [Zem, p.31, l.-4-p.35, l.12]
[Zem, p.247, l.-12] mentions the fact that the vapor pressure P(T) is a function of T only, but does not provide the proof. [Hua1, p.33, (2.3)] does give the proof.

• Chemical equilibrium
Let us compare [Reif, §8.2, §8.3 and §8.10] with [Zem, §14-8]. Note that [Zem, §14-7] emphasizes the following subtle points:
1. Even though the initial states of the phases are not in chemical equilibrium, it is still possible to describe them in terms of thermodynamic coordinates. This is because these phases are in mechanical and thermal equilibrium [Zem, p.366, l.11-l.17].
2. The functions that express the properties of a phase when it is not in chemical equilibrium must  reduce to those for thermodynamic equilibrium when the equilibrium values of the n's are substituted [Zem, p.388, l.-7-l.-4]. In other words, in thermodynamic equilibrium the n's are fixed values, but when the system is not in chemical equilibrium, these n's are variables.
Remark 1. [Reif, §8.2 and §8.3] are reduced to twelve lines in [Zem, p.368, l.-6-p.369, l.6].
Remark 2. The proof of [Reif, p.314, (8.7.18)] is excellent, while the proof given in [Zem, p.372, l.-7-p.373, l.6] is very confusing.

• Degree of reaction [Zem, §14-12].

• Equation of reaction equilibrium [Zem, §14-13].

• Law of mass action [Zem, §15-1].

• Heat of reaction [Zem, §15-3].

• Affinity [Zem, §15-5].

• The phase rule
1. Without chemical reaction [Zem, §16-2 & §16-3].
2. With chemical reaction [Zem, §16-4 & §16-5].

• Displacement of equilibrium [Zem, §16-6].

• Thermocouples [Zem, §17-6-§17-10].

1. Why do we use cavity radiation to represent black-body radiation? Because
1. Cavity radiation is in thermal equilibrium so that the thermodynamic coordinates can be defined [Man, p.246, l.-15].
2. A small hole in a wall has the same absorbing and emitting power as a black-body. Key idea: [Man, p.246, l.-6-l.-3]. Proof: [Zem, p.91, (4-17)].
2. [Man, Appendix B] proves the formula for the density of states using both particle [Man, §B.2] and wave [Man, §B.3] approaches. [Man, §10.3] proves Planck's law using both the particle and wave approaches.
3. In order to prove Wien's displacement law, [Reif, §9.13] obtains the maximum by drawing the graph of the function [Reif, p.375, Fig. 9.13.1], while [Zem, §17-14] obtains the maximum using calculus.
4. Planck's radiation equation [Zem, p.446, (17-27)] reduces to
1. The Rayleigh-Jeans law [Man, p.253, (10.21)] or the equipartition theorem [Man, p.253, l.-6] in the limit of low frequencies.
2. Wien's law [Man, p.254, (10.23)] in the limit of high frequencies.
Remark. Studying the problem from the viewpoint of entropy [Wu, p.33, (1-7)], Planck originally used the method of interpolation to derive his radiation equation from the Rayleigh-Jeans law and Wien's law [Man, p.363, l.-6-p.364, l.3].
5. We may prove the formula for radiation pressure [Zem, p.451, (17-32)] using
1. The kinetic theory [Zem, p.451, l.9-l.-1].
2. The partition function [Man, p.255, l.5-p.256, l.11].
Remark 1. How is the concept of standing waves related to cavity radiation? Ans. [Eis, p.7, l.16; p.8, l.20; p.14, l.3]. [Eis, §1-1-§1-4] can serve as both a good introduction and a good summary of black-body radiation because its formulation of the theory is closely and clearly related to basic concepts. However, there is a mistake in [Eis, p.11, l.31-l.32].  N(n)dn (the number of allowed frequencies) ¹ N(r)dr (the number of quantum states [allowed k-vectors]) unless we regard a frequency as a vector. In contrast, [Man, p.328, l.3] adopts a unified and better convention.
Remark 2. [Man, Appendix B] gives a comprehensive discussion on the density of states. There are two points worth noting: First, the density of states is independent of boundary conditions [Man, p.330, l.1-l.2]. Second, the discussion of density of states naturally leads us from the wave equation [Man, p.324, (B-1)] to the Schrödinger equation [Man, p.331, l.-6-l.-2; Reif, p.353, l.-3-p.354, l.9].

• The homopolar motor and the homopolar generator [Cor, pp.399-404].
Remark. [Wangs, p.276, l.15-p.277, l.8] can be used as a stepping stone to understand the two examples in [Cor, pp.399-404].

• Quasi-static electromagnetic fields and the skin effect [Chou, §6.4].
Remark. [Wangs, p.450, l.-2-p.451, l.8] discusses the physical significance of the neglect of the displacement current from the viewpoint of energy loss and the viewpoint of the time needed for propagation of signals, while [Chou, p.255, l.15-p.256, l.2] discusses the physical significance from the latter viewpoint alone.

• Derivation of the macroscopic Maxwell equations [Chou, §7.2].
Remark 1. Both Jackson and Choudhury call the details of the proof gory [Jack, p.255, l.-10; Chou, p.304, l.-1].  If this trivial proof is considered gory, I wonder what adjective should be used to describe Tycho Brahe's or Kepler's work.
Remark 2. A formula should be written in its simplest form. [Chou, p.309, (7.44)] can still be reduced to [Jack, p.256, (6.96)]. Both [Chou, p.309, (7.44)] and [Jack, p.256, (6.96)] are incorrect as they stand. Sn should have been inserted in front of Sr,s in [Chou, p.309, l.2]. The expression inside the [  ] in [Jack, p.256, l.7] should have been (Qn')ag(vn)b-(Qn')gb(vn)a.

• Scalar and vector potentials
[Chou, p.315, l.3-l.7] can be directly derived from [Chou, p.585, l.3-l.4]. The argument in [Chou, p.314, l.6-p.315, l.1] basically repeats the argument in [Chou, p.584, l.7-p.585, l.1].

• Debye's theory of solids
1. In order to decouple the equations of motion [Hoo, p.38, l.17], we transform from the position space [Hoo, p.37, (2.7)] to the momentum space [Hoo, p.38, (2.9); Man, p.324, (B.1)] using [Hoo, p.37, (2.8); Man, p.330, (B.19)]. This method of finding normal coordinates has a physical origin [Hoo, p.39, Fig. 2.5]. [Kit, pp.102-106] fails to point out the main purpose of phonons: decoupling the equations of motion.
2. In the one dimensional case, g(w) is given by [Hoo, p.53, (2.33)]; the assumption w = vS k is equivalent [Hoo, p.59, l.20-22] to taking g(w) as given by the broken line on [Hoo, p.54, Fig. 2.11].
3. [Wu, p.45, (I-17)] is easier to derive and evaluate than [Man, p.160, (6.27)].

• Moment of inertia [Sea1, §9-6, §9-7].
Remark. Some earlier editions of this book use summation instead of integration. I never like this practice. After I read [Sea1, §9-6, §9-7], I realized that the presentation of the 6th edition using integration is much better than [Hall, §12-5] or anything about moment of inertia existing on the web.

• The energy values of the bound states of the hydrogen atom are discrete.
[Coh, chap.VII, §C.3.c] gives a detailed explanation. [Mer, p.266, l.-14 & l.-12] make a few improvements.

• The Doppler effect for electromagnetic waves
[Rob, p.21] discusses the Doppler effect from the viewpoint of time dilation. [Matv, p.33, l.1-l.36] uses the tensor approach. The tensor approach automatically shows that all the formulas on relativity are covariant with Lorentz transformations, and effectively leads to a quick answer. It also condenses three cases (The source is moving toward the observer, away from the observer, or along a line normal to the line to the observer) into one formula [Matv, p.33, (2.62)]. However, Matveev's approach is not as insightful as Bobinson's approach. For example, it is easier to see that [Matv, p.33, (2.65)] is a purely time dilation effect from the context of [Rob, p.21] than that of [Matv, p.33, l.14-l.36].

• Optical properties of metals
[Hec, §4.8; Wangs, §24-3 & §25-6; Matv, p.120, §Color of bodies, Sec. 19 & Sec. 20] discuss the optical properties of metals. All the above books facilitate our understanding in some aspects, but none of their discussion are complete. We must piece together their discussions to see the entire picture.
1. Color of gold: For a chunk of gold, we can only see the reflected light [Wangs, p.423, l.-2-l.-1; Hec, p.131, l.c., l.-14-l.-11]. By [Hec. p.129, r.c., l.-12-l.-8], the gold appears reddish yellow. When the light source is on the other side of a thin foil of gold, we can only see the transmitted light, so the gold appears greenish [Wangs, p.423, l.14-l.20].
2. Some alkali metals are transparent to ultraviolet [Hec, p.129, l.-3-p.130, l.c., l.19].
Remark. For the proof of wp = (Nqe2/e0me)1/2, see [Wangs, p.401, (24-138)].
3. A metal (s = ) is an extension of an dielectric (s = 0).
1. The dispersion equation (compare [Hec, p.71, (3.72)] with [Hec, p.129, (4.79)]).
2. For plane waves, insulators and conductors are two extreme limiting cases and have corresponding discussions [Wangs, p.387, l.10-p.388, l.-12].

• Fiberoptics (for an introduction and the history of fiberoptics, read [Hec, §5.6]; for rigorous definitions and clear relations, read http://www.njit.edu/v2/Directory/Centers/OPSE/OPSE301/Lab14.doc ).

• Geometrical optics [Fur, chap. 3]
1. Any problem in geometrical optics can be solved either using formulas or using graphs. The latter method not only has the visual advantage, but also can be used as a check for calculations from the former method. Example: [Jen, pp.86-87, Example 2].
2.
1. Virtual objects
1. Illustrated in a figure: [Hec, p.155, Fig. 5.11].
2. Described in words: An object is virtual when the rays converge toward it [Hec, p.155, r.c., l.-7-l.-6].
3. Characterized by the object distance: so < 0 [Hec, p.163, Table 5.2].
2. Virtual images
1. Illustrated in figures: [Hec, p.152, Fig. 5.5(c); p.155, Fig. 5.10].
2. Described in words: An image is virtual when the rays diverge from it [Hec, p.155, r.c., l.-8-l.-7].
3. Characterized by the image distance: si < 0 [Hec, p.163, Table 5.2].
3. Only after understanding the meaning of [Jen, p.55, Fig. 3J] may one understand the construction of [Hec, p.151, Fig.5.3(a)].
4. Comparing the proof of [Jen, p.56, (3n)] with the proof of [Hec, p.154, (5.8)]: Although the former proof is simpler, the latter proof is more methodological.
5. If m>0, the image will be virtual [Jen, p.54, l.15]. This can be seen by [Morg, p.30, (2.11)(ii); Jen, p.49, Fig. 3E].
6. The focal plane of a lens [Hec, p.160, Fig. 5.17 (where the radius of  s is determined by [p.155, (5.10)]), and Fig. 5.18].
7. Geometrical optics uses a lot of modeling. By comparing the arrangement of sections in [Jen, chap. 3] and that in [Jen, chap. 4], we see that the theory of thin lenses is parallel to the theory of refracting surfaces. The theory of spherical mirrors is a special case of the theory of lenses [Morg, p.35, l.18]. The following three theories-thin lenses, thin-lens combinations, and thick lenses- are parallel because they use the same parallel-ray method to form images (compare [Jen, p.69, Fig. 4H] with [Jen, p.80, Fig. 5B(b)]; compare [Jen, p.75, Fig. 4M] with [Jen, p.83, Fig. 5E]). Consequently, the corresponding formulas in these three theories are the same if we properly choose the reference points. For example, [Jen, p.84, (5k)] and [Morg, p.67, (5.24)] can be considered identical. A systematic approach to the problems in geometrical optics entails mastering all the above patterns.
8. A system of lenses can be treated as a thick lens [Morg, p.67, l.-7-l.-1].
9. Treating mirrors as lenses
1. Reflection is considered as refraction [Matv, p.163, l.4-l.14].
2. (Sign conventions) Identify [Hec, p.184, Table 5.4] with the combination of [Hec, p.154, Table 5.1] and [Hec,  p.183, Table 5.2].
3. (Properties) Apply the same graphical constructions used for lenses to mirrors (e.g., identify [Jen, p.101, Fig. 6E] with [Jen, p.63, 4D]; ray 8 in [Jen, p.106, Fig. 6I] is constructed based on [Jen, p.47, Fig. 3C]), apply the same formulas for lenses to mirrors, and identify [Hec, p.185, Table 5.5] with [Hec, p.163, Table 5.3].
4. Thick mirrors [Jen, §6.5] can be considered as thick lenses.
Remark. [Jen, p.107, Fig. 6J] is consistent with the convention given in [Matv, p.163, l.4-l.8], while [Jen, p.133, Fig. 8C] is consistent with the convention given in [Hec, p.252, l.c., l.-8]. In my opinion, Jenkins should have adopted the former convention as a standard and stuck to it.
10. For a detailed and systematic study of the effects of stops, see [Jen, chap. 7].
11. In [Hec, §6.2], the method of ray tracing applies only to paraxial rays. That is, it is used only for the first-order approximation. Actually, in principle, the graphical method of ray tracing [Jen, chap. 8] and the matrix method [Jen, p.143] can be exact.
12. [Matv. Sec. 21-Sec. 23] condense geometrical optics into 21 pages and are ready for practical application using computers. In addition, Matveev proves every statement that he presents in these sections. His rigorous reasoning and ability to organize are impressive. In contrast, [Hec, chap.5 & chap. 6] use 132 pages to discuss geometrical optics and leave many statements unproved (e.g., [Hec, (6.1)-(6.4); (6.34); (6.36)-(6.37)]). In one place, Hecht claims he has proved [Hec, (6.34)]. Actually, he uses unproven [Hec, (6.2)] to prove [Hec, (6.34)]. Thus, all he has done is state the formula [Hec, (6.2)] twice. For a detailed proof, see [Matv, p.167, (23.19)]. However, logic is not the only tool to facilitate our understanding. For example, the definitions of principle points in [Matv, p.166, l.-1] is not as good as the definitions given in [Hec, p.243, Fig. 6.1]. The graphical constructions given [Jen, §3.6 and §3.7] should not be deemphasized for they have an visual advantage. Matrix methods are a useful tool only for computer calculations. A tool is used when needed. If we use methods to discuss topics other than computer calculations, the tool will become a burden rather than an advantage. In view of [Fur, chap. 3], the theory of geometrical optics are indeed made more organized and compact by the matrix method. All the necessary information on rays is essentially contained in a single matrix. However, the theory's formulation given in [Fur, §3.1-§3.4] is not as well prepared for application [Fur, §3.5] as that given in [Hec, §5.1-§5.6] for application [Hec, §5.7].
13. [Hec, p.154, (5.8)] is derived from Fermat's principle, while [Fur, p.145, (3.26)] is derived from Snell's law. Although both approaches consider a bundle of rays, the latter approach is more natural and straightforward.

• The essence of the theory of wave packets can be summarized in three stages:
1. Superposition of two plane waves [Born, p.19, Fig. 1.5].
2. Superposition of oscillations with equidistant frequencies [Matv, p.97, Fig. 53].
3. Group velocity: the velocity of the maximum of the wave packet [Coh, p.30, (C-31); Fig. 6].

• Maxwell equations
1. In vacuum or microscopic fields: [Fur, p.44, l.6-l.9].
2. In matter (macroscopic fields)
1. Average over a volume that is macroscopically small but microscopically large: [Fur, p.60, l.1-l.4].
2. In terms of (controllable) free charges and free current densities: [Fur, p.65, l.14-l.17].
3. Suppose rf = 0 and Jf = 0. In terms of the material parameters: [Fur, p.68, l.3-l.6].

• Foundations of geometrical optics [Born, chap. III]
1. Geometrical light rays [Born, p.114, l.-18-l.-17].
2. I = |<S>| [Wangs, p.357, (21-58); Matv, p.61, (7.12)].
3. Proofs of the eikonal equation: The proof given in [Born, p.112, l.1-l.9] uses the first-order approximation, while the proof given in [Born p.112, l.16-p.113, l.2] uses the second-order approximation. Therefore, the latter proof is more refined.
4. Proofs of the law of refraction
The proof given in [Hec, §4.4.1] is restricted to plane waves and planar interfaces. The method lacks potential to be applied for generalization. The proof given in [Hec, p.107, l.c., l.1-l.14] uses Fermat's principle which is artificial; The proof given in [Jack, §7.3] uses the basis of a vector space which is also artificial. In addition, the way the boundary conditions are used in [Jack, (7.34)] differs from the way they are used in [Born, p.5, Fig. 1.2], which complicates matters. The proof given in [Born, §3.2.2] is the most natural proof because it is directly derived from the boundary conditions. The method meets the requirement for axiomatization: any theorem in electromagnetism should be able to be derived from Maxwell's equations. This enables us to trace the theorem to its source. In addition, the derivation of [Born, p.125, (17)] is the same as that of [Born, p.6, (23)], which is good for unification. Furthermore, the proof given in [Born, §3.2.2] applies to the general case [Born, p.125, l.-11-l.-7]. It is the only proof that establishes a strong link to electromagnetism. I wonder why other textbooks leave out such an insightful proof.
5. [Born, p.126, (23)] represents a normal congruence; [Born, p.126, (22)] represents a normal rectilinear congruence.
For the proof of the former statement, see [Wea1, vol I, §105]. [Sne, p.21, Theorem 5] provides a proof of the identity given in [Wea1, vol. I, p.202, l.-12].
Remark. In the early twentieth century, the textbooks of optics discussed the topics in differential geometry and the textbooks of differential geometry discussed the topics in optics. Each subject solidified the other's foundation and stimulated the other's growth. Now in the twenty-first century, optics and differential geometry have become mutually exclusive subjects. The textbooks in optics are devoid of questions about differential geometry and the textbooks in differential geometry are devoid of questions about optics.

• The integral given in [Jack, p.42, (1.58)] is equal to 4p [Jack, p.42, l.18-l.19].
Proof. Ñf (r,q,f) = (f/r)r^+r-1(f/¶q)q^+(r sin q)-1(f/¶f)f^ [Wangs, p.33, (1-102)].
Ñ(|r+n|-1)×(r/r) = Ñ(|r+n|-1)×r^ = (|r+n|-1)/r.
ò [0,+¥] (|r+n|-1)/r dr = |r+n|-1 úr = 0¥ = -1.¬

• Vibration of membranes
Tension per unit width = Constant T; Vertical deflection = w(x,y,t).
Consider the displacement of an element of area dxdy at time t.
x-direction: width = dx; left slope = tan q»q; right slope = q+dq.
q = w/x Þ dq = [(w/x)/x]dx = (2w/2x)dx.
Vertical component of the tension in the x-direction: left end: -T(dy) tan » = -T(dy)q; right end: T(dy) (q+dq).
Net vertical force from x-direction tension is T(dy)dq = T(2w/2x)dxdy.
Similarly, the net vertical force from y-direction tension is T(2w/2y)dxdy.
Total vertical force on dxdy is T(2w/2x+2w/2y)dxdy.
Let r be the mass per unit area. Then (rdxdy)(2w/2t) = T(2w/2x+2w/2y)dxdy.
Therefore, r(2w/2t) = T(2w/2x+2w/2y).
The following figure is viewed along the positive y-direction: 