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On Helmholtz's Theorem and its Interpretations X. L. Zhou Published online: 03 Apr 2012.

To cite this article: X. L. Zhou (2007) On Helmholtz's Theorem and its Interpretations, Journal of Electromagnetic Waves and Applications, 21:4, 471-483, DOI: 10.1163/156939307779367314 To link to this article: http://dx.doi.org/10.1163/156939307779367314

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J. of Electromagn. Waves and Appl., Vol. 21, No. 4, 471–483, 2007

ON HELMHOLTZ’S THEOREM AND ITS INTERPRETATIONS X. L. Zhou

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156 Wyndham Drive Allentown, PA, 18104, USA Abstract—This paper presents a direct, natural proof of a generalized Helmholtz’s theorem for piecewise continuously differentiable vector functions in vector analysis and mathematical physics and its precise statement. Based on the generalized Helmholtz’s identity, it is pointed out that Helmholtz’s theorem is an operator-based decomposition theorem of a vector function. As a mathematical identity, although it is compatible with some uniqueness theorems (especially those in electromagnetics), it does not indicate directly any uniqueness theorems for boundary value problems. Most existing versions of Helmholtz’s theorem are commented. As an important application of the generalized Helmholtz’s identity, the definitions of irrotational and solenoidal vector functions are revisited and complete definitions are proposed as a result. The generalized Helmholtz’s theorem and the present conclusions should have important indications in vector analysis related disciplines such as electromagnetics.

1. INTRODUCTION Helmholtz’s theorem in vector analysis and mathematical physics is a critical mathematical theorem for applications involving general vector fields such as electromagnetics, gravity theory, elasticity and hydrodynamics etc. Although named after a great German scientist Hermann von Helmholtz, it was essentially used first by a British mathematician G. Stokes in 1849, almost nine years before Helmholtz’s paper in 1858. This theorem is so old that its history has not been mentioned in many modern books and articles except for few of them such as [1] and [2]. J. Carvallo wrote in 1922, “Of all the results of mathematical physics Vaschy’s theorem is that which has most †

The author is currently with Agere Systems Inc.

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practical consequences and most philosophical importance” [1, p. 14]. Here Vaschy’s theorem refers to Helmholtz’s theorem. Because of its important applications in electromagnetics, it has been related very closely to the divergence and curl of a vector field. There are many versions of statements and proofs in books and literatures [1–15]. In most statements of Helmholtz’s theorem, scalar and vector potentials are introduced even they were not used in G. Stokes’ original work. Furthermore, Helmholtz’s theorem has been widely accepted as the uniqueness theorem of a vector field. [4] interpreted Coulomb’s law based on Helmholtz’s theorem. It has also been generalized to fourvector fields in theoretical physics [5, 15]. Helmholtzs theorem is also the rudiment of a more general decomposition — Hodge decomposition [25]. This paper presents a direct, natural proof of Helmholtz’s theorem in Section 2, its precise statement, and generalizes the Helmholtz’s identity to piecewise continuously differentiable vector functions, then concludes that Helmholtz’s theorem is essentially an operator-based decomposition theorem of a vector function, it does not indicate directly any uniqueness theorems. Some discussions and comments on existing statements and proofs of Helmholtz’s theorem are presented in Section 3 and Section 4. As an important application of Helmholtz’s identity, the definitions of irrotational and solenoidal vector functions are revisited and complete definitions are proposed as a result in Section 5. 2. DIRECT PROOF OF A GENERALIZED HELMHOLTZ’S THEOREM We will give the proof and then propose the precise statements. Suppose a piecewise differentiable vector function F(r) is known in a space V0 . Our purpose is to expand it in terms of a sum of only an irrotational part and a solenoidal part. This attempt is obviously inspired by the well-developed potential theory about irrotational and solenoidal fields and the corresponding applications in electrostatics, magnetostatics etc. Is it possible for any given piecewise differentiable F(r)? We will find the requirements on F(r) in the proof. The proof is motivated by the vector identity ∇2 L = ∇(∇ · L) − ∇ × (∇ × L)

(1)

The above is always true for any second order differentiable L(r). The first part is irrotational because ∇×∇ ≡ 0; the second part is solenoidal because ∇·∇× ≡ 0. This is different from the definitions of irrotational and solenoidal vectors that will be revisited later. It is easy to verify by

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looking into any handbooks of mathematics [14] that (1) is the simplest vector identity that includes only an irrotational part (the first term) and a solenoidal part (the second term). If we can find the relation between L(r) and F(r), the expansion is completed. Obviously, the only way to establish a relation between L(r) and F(r) is to link them by an equation. Let us define F = −∇(∇ · L) + ∇ × (∇ × L)

(2)

This satisfies our purpose if and only if L(r) can be determined in terms of F(r) only. From (1) and (2), we get

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∇2 L(r) = −F(r)

(3)

Fortunately, (3) is a vector Poisson equation from which L(r) can be solved based on potential theory. (3) is only a differential equation, boundary conditions must be specified. Note that (3) is just a constructed problem of L(r) based on a given F(r). L(r) can be continuously differentiable. F(r) is considered as mathematical source of L(r) only, different boundary conditions can be used while (2) and (3) can be always true. Let us assume that the constructed problem is defined in a volume V enclosed by a surface S. V must be equal to or greater than V0 . If V is larger than V0 , let F(r) ≡ 0 in V − V0 . If the corresponding Green’s function is G(r, r ) for a given boundary condition, then the solution to (3) is  G(r, r )F(r )d3 r (4) L(r) = V

By substituting (4) into (2), our purpose of decomposing F(r) into an irrotational part and a solenoidal part is achieved. The irrotational part is      3  (5) Fi (r) = −∇ ∇ · G(r, r )F(r )d r V

and the solenoidal part is





Fs (r) = ∇ × ∇ ×





3 

G(r, r )F(r )d r

 (6)

V

Then F(r) = Fi (r) + Fs (r)

(7)

Since the solution to (3) for a given V and boundary condition is unique, the decomposition is unique. Note that the uniqueness theorem

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of Poisson equation is proven based on the maximum principle of harmonic functions in potential theory [16]. The proof is logically independent of the current reasoning. Since (3) is a constructed problem, we can pose Dirichlet or Neumann or even mixed boundary conditions to the vector Poisson equation (3). They correspond to different Green’s functions [17], then different L. This arbitrariness leads to different decompositions of the same vector function in different V . This property can be employed to develop techniques for non-uniform structures. Although other choices are possible, let’s discuss the simplest and most popular one. V is chosen to be the whole Euclidean free space with Dirichlet condition, which is the simplest. Then G(r, r ) is the Green’s function of a free space, 1 4π|r − r |

(8)

F(r ) d3 r 4π|r − r |

(9)

G(r, r ) = Thus, (4) becomes  L(r) = V

Thus property of L is determined completely by that of F and so does the decomposition. From the existence theorem of solutions to Poisson equation [16, p. 246][18, Chap. III], the requirement on F as mathematical source of L, is that F is bounded, integrable and piecewise continuously differentiable. Therefore, the expansion (2) is valid at all points, including the discontinuities. If F is given in the whole Euclidean space (9) requires that |r|2 |F| is bounded at infinity, (5) and (6) become    F(r ) 3  (10) d r Fi (r) = −∇ ∇ ·  V 4π|r − r |    F(r ) 3  Fs (r) = ∇ × ∇ × r d (11)  V 4π|r − r | where V is the entire Euclidean space. If F is defined in a finite volume, it just implies mathematically that F ≡ 0 outside the volume. F is not required to be continuously differentiable in V . Note that the above procedure seems similar to the one given in [12, pp. 799–800], where the property of δ function is used. However, the present derivation exhibits other possible decompositions and is logically straightforward and general. The motivation is also different.

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Following the same procedures given in [9, pp. 29–36][12] and [13, p. 53], we can change the forms of (10) and (11) to obtain the socalled Helmholtz’s identity. The existing Helmholtz’s identity for a continuously differentiable F [12] can be obtained from (10) and (11)    ∇ · F(r ) 3  F(r ) · n 2  d r − d r F(r) = −∇   V 4π|r − r | S 4π|r − r |    ∇ × F(r ) 3  F(r ) × n 2  +∇ × d r + d r (12)   V 4π|r − r | S 4π|r − r | where the property of Green’s function ∇G(r, r ) = −∇ G(r, r ) is used, and ∇ operates on the primed variables only. When S goes to infinity, (12) reduces to the one originally given by Stokes. We will show that the Stokes’ version of Helmholtz’s identity is valid only for a continuously differentiable vector with |r|2 |F| bounded at infinity, and (12) is for a continuously differentiable vector defined in a space with finite boundary surface S. For a general piecewise continuously differentiable vector F(r), assume  that it is continuously differentiable in each partial region Vl ( Vl = V ) bounded by Sl , then   F(r ) Fl (r ) 3  3  ∇· r = ∇ · d d r   V 4π|r − r | Vl 4π|r − r | l    ∇ · Fl (r ) Fl (r ) · n 2  3  = d r + d r   Vl 4π|r − r | Sl 4π|r − r | l    ∇ · F(r ) 3  Fl (r ) · n 2  = d r + d r   V 4π|r − r | Sl 4π|r − r | l (13) where n is the outward unit vector normal to Sl . Theoretically, there is no requirement on the number of Sl although l is finite in many practical cases. Similarly,     F(r ) ∇ × F(r ) 3  Fl (r ) × n 2  3  ∇× d r = d r + d r   4π|r − r | V 4π|r − r | V 4π|r − r | S l l (14) It is not difficult to show that if F(r) is continuously differentiable in a single volume V bounded by S, and F(r) = 0 outside V , (13) and (14) lead to the Helmholtz’s identity (12) given in [9, p. 31][12, p. 800]

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[15], which is of fundamental importance in the theory of uniform electromagnetic cavities and resonators. Furthermore, if S goes to infinity Stokes’ version without any surface integrals is obtained. (9) requires |r|2 |F| be bounded at infinity. Equivalently, if |r|2 |∇ · F|, |r|2 |∇ × F|, |r||F · n|, |r||F × n|, and |r||F| approach zero at infinity, the infinite surface integrals in (12) are zero vectors. Consequently,  ∇ · F(r ) 3  Fi (r) = −∇ (15) d r  V 4π|r − r |  ∇ × F(r ) 3  (16) Fs (r) = ∇ × d r  V 4π|r − r | The decomposition is unique in the entire Euclidean space since the solution (9) to the vector Poisson equation (3) is unique (not simply because F is given). The decomposition is complete since there are no other operator-based components. The two parts (components) Fi and Fs are independent since any one of the three items in (1) can not be expressed in terms of just another one. Note that the above decomposition has nothing to do with the physical significance of F as noticed by O’Rahilly [1, p.14]. Furthermore, introduction of scalar and vector potentials is not necessary. Stokes himself did not introduce any potentials in his original big paper [2, p. 147]. Based on the above derivations, we propose the precise statement of a generalized Helmholtz’s decomposition theorem and two corollaries. Theorem. Any finite and piecewise continuously differentiable vector function F(r) given in the entire Euclidean space, with |r|2 |F| bounded at infinity, can be completely and uniquely decomposed into a sum of an irrotational part and a solenoidal part. The two parts are independent and determined by (10)and (11) or through (13)and (14) respectively. Corollary 1 Any finite and piecewise continuously differentiable vector function F(r) given in a region bounded by a finite surface S can be completely and uniquely decomposed into a sum of an irrotational part and a solenoidal part. The two parts are independent and determined by (10)and (11) or through (13)and (14) respectively. Corollary 2 Any continuously differentiable vector function F(r) given in an infinite space with |r|2 |F| bounded at infinity can be completely and uniquely decomposed into a sum of an irrotational part and a solenoidal part. The two parts are independent and determined by (15)and (16) respectively.

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The theorem describes the exact behavior of F(r) when r → ∞. Since F(r) = −∇2 L ≡ 0 outside the finite region V then |r|2 |F| = 0 at infinity, corollary1 is included in the theorem. Corollary 2 is also a reduced case of the theorem. The theorem claims not only the existence of the decomposition but also the exact way in which the decomposition is achieved.

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3. OTHER POSSIBLE OPERATOR-BASED EXPANSIONS As we can see in the above section, Helmholtz’s decomposition theorem is an operator-based expansion of a vector function. It is not difficult to imagine that a vector function can be decomposed in terms of other operators. In fact, a solenoidal vector can be decomposed into toroidal and poloidal parts [19]. A direct example is the following. If we want to decompose F in terms of ∇ and ∇·, regardless of the physical meaning, we just define F = ∇(∇ · L) − ∇2 L

(17)

Notice that ∇2 = ∇ · ∇. Then solve ∇ × (∇ × L) = F

(18)

for L. The integration of (18) is discussed in [18, p. 250]. Of course the operators are not necessarily irrotational or solenoidal. Another example is to use the vector identity [14]

 ∇4 L = ∇2 [∇(∇ · L)] − ∇ × ∇2 (∇ × L) (19) to expand a vector function in terms of other operators. The discussion suggests that at least other possible operator-based expansions of a vector function with probably different explanations may exist. This is similar to decomposing a vector in different coordinate systems. We can expand a vector function in terms of different sets of operators according to applications. However, up to now, only the decomposition discussed in Section 2 has found most important and practical applications in physics such as electrostatics, magnetism etc., because theories about an irrotational and a solenoidal fields have been well-developed. A more abstract decomposition is Hodge decomposition in modern differential forms of manifolds [25]. This may lead to a more general topic. Although the introduction of scalar potential φ and vector potential A is not necessary in the proof of Helmholtz’s theorem, they are often used in real computations and theoretical research. Then we have the following corollary.

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Corollary 3 In Euclidean space, any piecewise continuously differentiable vector function F(r) with |r|2 |F| bounded at infinity, can be expressed as F = −∇φ + ∇ × A

(20)

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where φ is a scalar potential, A a vector potential, ∇φ and ∇ × A are unique. This corollary is almost the description given in literatures [2, p. 147][3, p. 163] and O’Rahilly’s version [1, p. 14]. However, conditions and emphases are different. For example, in [2, p. 147], Sommerfeld introduced vectorial constants which do not appear in the present proof. The uniqueness of the decomposition needs to be proven separately. Similar troubles may occur in King’s version and others if we introduce φ from ∇ × Fi = 0 and A from ∇ · Fs = 0. Or, if we start from (20) to show the Helmholtz’s theorem [6][10, p. 93][20, p. 326], the proof is not complete since vectorial constants may appear. In order to resolve the non-uniqueness, Sommerfeld wrote “· · · the constants in (2) and (5) must cancel each other, since the sum V = V1 + V2 is supposed to vanish at infinity · · · ” [2, p. 147]. His assumption is over-constrained. We already show that the necessary and sufficient condition is |r|2 |V| bounded at infinity. More importantly, it will be shown in Section 5 that the introduction of φ should depend on the condition of Fs = 0; the introduction of A should depend on Fi = 0. The theorem, corollaries 1 and 2 show the possibility, uniqueness and completeness of the decomposition and how to do it. This can be important in theoretical research. In practical applications, we seldom use the decomposition theorem to decompose a known vector function, even it allows us to do so. In contrary to this fact, the corollary 3 with ∇φ and ∇ × A expressions does not emphasize how to decompose a vector function. Instead, it emphasizes the existence and uniqueness of the decomposition in Euclidean space. This form has important practical and theoretical applications. Corollary 3 can be employed to solve actual boundary value problems in disciplines involving vector fields. 4. UNIQUENESS OF A VECTOR FIELD AND HELMHOLTZ’S THEOREM In many literatures, the uniqueness theorem of a vector field is commonly stated as part of Helmholtz’s theorem [4, 5][7, p. 166][9, p. 29][11, p. 63][21, pp. 187–190][14]. For example, in [9, p. 29], the authors claimed, “Another way of stating the Helmholtz’s theorem is

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that a vector field completely specified by its divergence and curl”. In the proof of O’Rahilly’s version, the uniqueness theorem of a divergence and curl problem of a vector has to be used. When the author constructs (1.204b), he used the condition ∇ · A = 0 implicitly which is also based on the same uniqueness theorem. Similar procedure is used in [13]. It is a logic circle. The attempt to prove uniqueness theorem from decomposition theorems has been tried for other problems [5, 15]. When time domain Maxwell’s equations are dealt with, Shadowitz wrote, ”It is seen that they really do not satisfy the requirements of Helmholtz’s theorem · · · This is too bad insofar · · · .” [21, p. 412]. Obviously, the requirements of his Helmholtz’s theorem refer to that the values of divergence and curl are specified [21, pp. 187–190]. Another interesting attempt is to construct (compute) a vector from the decomposition theorem [5, 12]. All above statements and attempts are obviously based on observations about the Stokes’ version of Helmholtz’s identity from (15) and (16)     ∇ · F(r ) 3  ∇ × F(r ) 3  d r +∇× d r (21) F = −∇   V 4π|r − r | V 4π|r − r | (21) includes explicitly the divergence and curl of F only. However, it does not apply for uniform regions bounded by finite surfaces, and the boundary condition at infinity is assumed. From (13) and (14), for general cases, the generalized Helmholtz’s identity is F(r) = Fi (r) + Fs (r)    ∇ · F(r ) 3  Fl (r ) · n 2  = −∇ d r + d r   V 4π|r − r | Sl 4π|r − r | l    ∇ × F(r ) 3  Fl (r ) × n 2  (22) +∇× d r + d r   V 4π|r − r | Sl 4π|r − r | l

It is easy to understand that for a given F, its divergence, curl and other terms (in general, any proper operations on F) in (22) are determined. But a proper boundary value problem of F cannot be deduced from (22) itself. Of course, a solution to a physical problem satisfies (22). But the original boundary problem cannot be deduced/discovered reversely from it. (22) includes not only the divergence and curl of F but also all the tangential and normal components on all discontinuities of F. If interpreted as potentials, the sources ∇ · F, n · F, etc. include not only free sources but also induced ones [18, Sec. 3.13 and Sec. 4.10]. However, as it is well-known

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that in electromagnetics, in order to solve fields, it is not necessary (even impossible) to specify induced (polarized) sources in V and on internal surfaces. For general constitutive relations, the divergence and curl of a quantity (for example, electric field) are not specified explicitly. Although F is continuously defined in the entire Euclidean space, specifying its divergence ∇ · F and curl ∇ × F is only one of many mathematical possibilities of determining the vector F. As a mathematical identity, (22) does not exclude other possibilities. The vector Poisson equations (3) is a good example for existence of other types of vector differential equations that can determine a vector function. Other examples can also be found in the theory of elasticity [22]. Furthermore, for the same set of differential equations, different boundary conditions can be imposed [23]. In fact, the precondition in the proof in Section 2 is nothing more than that F is given and satisfies the conditions in the theorem or its corollaries. F may be solutions to any suitably posed physical or mathematical problems. It does not matter whether the solutions to the problems are unique or not because any one of the solutions can be decomposed in the same way. Non-unique examples can be found in bifurcation theory [24]. Of course, since specifying the divergence and curl of a vector is so important in science, it is reasonable to pay a lot attention to this case. Unfortunately (22) does not tell us how to properly pose boundary value problems with necessary and sufficient equations and boundary conditions. Of course, if all the terms on the right of (22) are given, F is uniquely determined. Unfortunately, we cannot tell if the problem is over-constrained. Although (22) includes boundary information, it is impossible to know if the boundary information is over much or insufficient. As an example, (22) can not tell us how to pose boundary conditions for a vector Poisson equation. In fact, as we pointed out that it is not necessary to specify all internal boundary information. At most (22) could be a revelation of posing some kinds of boundary value problems. Usually, boundary value problems should be imposed first based on physical laws and the cases under study, then their existence and uniqueness theorems have to be studied in other ways in which the Helmholtz’s theorem and its corollaries may be used. Of course, it has to be compatible with the uniqueness of any vector boundary problem. In conclusion, Helmholtz’s theorem is a pure mathematical identity or decomposition theorem about vector functions, rather than a uniqueness theorem. We cannot deduce any physics laws or boundary value problems from it although some laws and boundary problems in electromagnetics are compatible with (22).

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5. DEFINITIONS OF IRROTATIONAL AND SOLENOIDAL VECTOR FUNCTIONS In all literatures, it is widely accepted that if the divergence of a vector is zero, the vector function is solenoidal; in parallel, if the curl is zero, the vector function is irrotational [2, 10, 12, 20] etc. The introduction of potentials is based on this statement only. Some authors even use it to prove Helmholtz’s theorem as an obvious fact. Of course, those are the necessary conditions. Are they sufficient? Unfortunately, based on the generalized theorem, we can easily show that the existing definitions are incomplete. In fact, it is found this is one of the most important applications of the theorem. The expansion (22) is valid in all regular regions in which F is differentiable. The question is: for a given vector function F to satisfy ∇ · F = 0 in regular regions, is there any requirement on the values of F on its discontinuities or on the boundaries? (22) reveals the secret. Since (22) is valid at all points, a more accurate condition for a piecewise differentiable vector to be solenoidal is that the irrotational part is a zero vector, Fi ≡ 0, only solenoidal part is non-zero. From (22), Fi ≡ 0 is always true if and only if both ∇ · F(r ) = 0 and F(r ) · n = 0 hold true. That is, in term of divergence, a solenoidal vector function must satisfy, ∇ · F(r) = 0 in Vl ∈ V F(r) · n = 0 on S

(23a) (23b)

(23b) is required on S only since the surface divergence concept must lead to cancellation of normal components on interior surfaces [18]. Without (23b), (22) can not be consistent with (23a) even at regular points. If (23b) is not true, the volume integral is identically zero because of (23a), then its divergence; but the surface integrals could possibly lead to non-zero divergence for a given F. This is contradictory to (23a). Is (23) a proper boundary value problem of F? This is a different topic. (23b) cannot be deduced from (23a) in general. It is interesting to notice that the definition of solenoidal vectors is related to the irrotational part of (22). Similarly, from the second term of (22) Fs ≡ 0, in term of curl, an irrotational vector function must satisfy, ∇ × F(r) = 0 in Vl ∈ V F(r) × n = 0 on S

(24a) (24b)

In fact, R.E. Collin already recognized the above conditions in [12, p. 800]. Unfortunately, it is not incorporated into the definitions of

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irrotational and solenoidal vector functions explicitly. His discussion is limited to continuously differentiable vectors. Obviously, it is expected that the new definitions have not only literal meanings but also physical significances about our knowledge of irrotational and solenoidal fields. Consequently, the introduction of the scalar potential φ depends on (24a) and (24b) and the introduction of the vector potential A relies on (23a) and (23b). The traditional introduction of potentials based on zero divergence and zero curl is then incomplete for piecewise continuously differentiable vector functions. In fact, the generalized Helmholtzs theorem and the new definitions are employed in [26] to prove a complete uniqueness theorem of a vector function. REFERENCES 1. O’Rahilly, A., Electromagnetics, A Discussion of Fundamentals, Cork University Press, London, 1938. 2. Sommerfeld, A., Mechanics of Deformable Bodies, Translated from the 2d German ed. by G. Kuerti, Vol. 2, Academic Press, New York, 1950. 3. King, R. W. P., Fundamental Electromagnetic Fields, Dover Publications, New York, 1963. 4. Miller, B. P., “Interpretations from Helmholtz’ theorem in classical electromagnetism,” American Journal of Physics, Vol. 52, No. 10, 948–950, Oct. 1984. 5. Kobe, D. H., “Helmholtz’s theorem for antisymmetric second-rank tensor fields and electromagnetism with magnetic monopoles,” American Journal of Physics, Vol. 52, No. 4, 354–358, April 1984. 6. Kobe, D. H., “Helmholtz’s theorem revisited,” American Journal of Physics, Vol. 54, No. 6, 552–554, June 1986. 7. Korn, G. A. and T. M. Korn, Mathematical Handbook for Scientists and Engineering (Definition, formulas, references and reviews), McGraw-Hill, New York, 1968. 8. Hauser, W., “On the fundamental equations of electromagnetics,” American Journl of Physics, Vol. 38, No. 1, 80–85, Jan. 1970. 9. Plonsey, R. and R. E. Collin, Principles and Applications of Electromagnetic Fields, McGraw-Hill, New York, 1961. 10. Arfken, G. B. and H. J. Weber, Mathematical Methods for Physicists, Academic Press, San Diego, 1995.

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11. Cheng, D. K., Field and Wave Electromagnetics, Addison-Wesley Pub. Co., New York, 1989. 12. Collin, R. E., Field Theory of Guided Waves, IEEE Press, New York, 1991. 13. Morse, P. M. and H. Feshbach, Methods of Theoretical Physics, Vol. 1, McGraw-Hill Book Company, Inc., New York, 1953. 14. Weisstein, E. W., CRC Concise Encyclopedia of Mathematics, CRC Press, Boca Raton, Fla, 1999. 15. Woodside, D. A., “Uniqueness theorems for classical fourvector fields in Euclidean and Minkowski spaces,” Journal of Mathematical Physics, Vol. 40, No. 10, 4911–4943, Oct. 1999. 16. Courant, R., Methods of Mathematical Physics, Vol. 2, Interscience Publisher, New York, 1962. 17. Courant, R., Methods of Mathematical Physics, Vol. 1, Interscience Publisher, New York, 1953. 18. Stratton, J. A., Electromagnetic Theory, John Wiley & Sons, New York, 1941. 19. Lamb, H., Hydrodynamics, 6th edition, Cambridge University Press, New York, 1993. 20. Schwartz, M., S. Green, and W. A. Rutledge, Vector Analysis, with Applications to Geometry and Physics, Harper, New York, 1960. 21. Shadowitz, A., The Electromagnetic Field, McGraw-Hill, New York, 1975. 22. Landau, L. D. and E. M. Lifshitz, Theory of Elasticity, Translated from Russian by J. B. Sykes and W. H. Reid, Pergamon Press, New York, 1970. 23. Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity: Fundamental Equations, Plane Theory of Elasticity, Torsion, and Bending, Translated from Russian, J. R. M. Radok (ed.), P. Noordhoff, Groningen, 1953. 24. Marsden, J. E. and T. J. R. Hughes, Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs, N. J., 1983. 25. Schwarz, G., Hodge Decomposition — A Method for Solving Boundary Value Problems, Springer, Berlin, 1995. 26. Zhou, X. L., “On uniqueness theorem of a vector function,” Accepted for publication in Progress In Electromagnetics Research, PIER 65, 93–102, 2006.