This paper develops a generalized Helmholtz theorem for an arbitrary localized time-varying vector function F(r,t) and shows that the mathematical relations commonly referred to as Maxwell’s equations can be derived from that theorem alone.

1.
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Kapuścik
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Generalized Helmholtz theorem and gauge invariance of classical field theories
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Jośe A.
Heras
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Comment on ‘Alternate ‘Derivation’ of Maxwell’s source equations from gauge invariance of classical mechanics,’ by James S. Marsh [Am. J. Phys. 61, 177–178 (1993)]
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4.
By “Maxwell’s equations” we mean the mathematical relations commonly referred to by this term—and not the corpus of empirical laws and results for which these equations constitute a shorthand resume. See also the note in conjunction with Eqs. (37) and (38).
5.
David J.
Griffiths
and
Mark A.
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Time-dependent generalizations of the Biot-Savart and Coulomb laws
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J. D.
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Historical roots of gauge invariance
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680
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2001
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7.
Conditions for uniqueness are discussed in the appendix.
8.
Often mistakenly referred to as the Lorentz gauge condition. See Jackson and Okun (Ref. 6).
9.
We have used the symbol ρa to stand for “abstract charge.” Conditions for uniqueness of ρa are discussed in the Appendix.
10.
Paul
Nahin
,
Oliver Heaviside, Sage in Solitude
(
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, New York,
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and
M.
Phillips
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2
7
.
12.
Our definitions here are purely mathematical. We have used the symbols E and B for obvious purposes of correlation with the physical fields of a current distribution; however, it is only in their interpretation in terms of forces on stationary and moving charges that they achieve physical significance—and this identification is, of course, subject to experiment for validation.
13.
Oleg
Jefimenko
,
Electricity and Magnetism
(
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, New York,
1966
), pp.
515
517
.
14.
Alan M.
Portis
,
Electromagnetic Fields: Sources and Media
(
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, New York,
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), p.
3
and pp.
191
197
.
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