Editing Maxwell's equations (section)

== Overdetermination of Maxwell's equations ==
Maxwell's equations ''seem'' [[Overdetermined system|overdetermined]], in that they involve six unknowns (the three components of {{math|'''E'''}} and {{math|'''B'''}}) but eight equations (one for each of the two Gauss's laws, three vector components each for Faraday's and Ampère's circuital laws). (The currents and charges are not unknowns, being freely specifiable subject to [[charge conservation]].) This is related to a certain limited kind of redundancy in Maxwell's equations: It can be proven that any system satisfying Faraday's law and Ampère's circuital law ''automatically'' also satisfies the two Gauss's laws, as long as the system's initial condition does, and assuming conservation of charge and the nonexistence of magnetic monopoles.<ref>{{cite book|author=H Freistühler & G Warnecke |title=Hyperbolic Problems: Theory, Numerics, Applications |year=2001 |page=605 |publisher=Springer |url=https://books.google.com/books?id=XXX_mG0vneMC&pg=PA605|isbn=9783764367107 }}</ref><ref>{{cite journal |title=Redundancy and superfluity for electromagnetic fields and potentials |journal=American Journal of Physics |author=J Rosen |volume=48 |issue=12 |page=1071 |doi=10.1119/1.12289|bibcode = 1980AmJPh..48.1071R |year=1980 }}</ref> 
This explanation was first introduced by [[Julius Adams Stratton]] in 1941.<ref>{{cite book|author=J. A. Stratton|title=Electromagnetic Theory |url=https://books.google.com/books?id=zFeWdS2luE4C |year=1941 |publisher=McGraw-Hill Book Company |pages=1–6|isbn=9780470131534 }}</ref>

Although it is possible to simply ignore the two Gauss's laws in a numerical algorithm (apart from the initial conditions), the imperfect precision of the calculations can lead to ever-increasing violations of those laws. By introducing dummy variables characterizing these violations, the four equations become not overdetermined after all. The resulting formulation can lead to more accurate algorithms that take all four laws into account.<ref>{{cite journal |title=The Origin of Spurious Solutions in Computational Electromagnetics |author=B Jiang & J Wu & L. A. Povinelli |doi=10.1006/jcph.1996.0082 |year=1996 |journal=Journal of Computational Physics |volume=125 |issue=1 |page=104|bibcode = 1996JCoPh.125..104J |hdl=2060/19950021305 |hdl-access=free }}</ref>

Both identities <math>\nabla\cdot \nabla\times \mathbf{B} \equiv 0, \nabla\cdot \nabla\times \mathbf{E} \equiv 0</math>, which reduce eight equations to six independent ones, are the true reason of overdetermination.<ref>{{cite book | first = Steven | last = Weinberg | title = Gravitation and Cosmology | publisher = John Wiley | date = 1972 | isbn = 978-0-471-92567-5 | pages = [https://archive.org/details/gravitationcosmo00stev_0/page/161 161–162] | url = https://archive.org/details/gravitationcosmo00stev_0/page/161 }}</ref><ref>{{Citation |first1=R. |last1=Courant|author-link=Richard Courant|name-list-style=amp |first2=D. |last2=Hilbert|author2-link=David Hilbert|title=Methods of Mathematical Physics: Partial Differential Equations |volume=II |publisher=Wiley-Interscience |location=New York |year=1962 |pages=15–18 |isbn=9783527617241| url=https://books.google.com/books?id=fcZV4ohrerwC}}</ref>

Equivalently, the overdetermination can be viewed as implying conservation of electric and magnetic charge, as they are required in the derivation described above but implied by the two Gauss's laws.

For linear algebraic equations, one can make 'nice' rules to rewrite the equations and unknowns. The equations can be linearly dependent. But in differential equations, and especially partial differential equations (PDEs), one needs appropriate boundary conditions, which depend in not so obvious ways on the equations. Even more, if one rewrites them in terms of vector and scalar potential, then the equations are underdetermined because of [[gauge fixing]].