Table Of ContentBernard Jancewicz
Directed
Quantities in
Electrodynamics
Bernard Jancewicz
Directed Quantities
in Electrodynamics
BernardJancewicz
InstituteofTheoreticalPhysics
UniversityofWrocław
Wrocław,Poland
ISBN978-3-030-90470-8 ISBN978-3-030-90471-5 (eBook)
https://doi.org/10.1007/978-3-030-90471-5
MathematicsSubjectClassification:15A75,53Z05,78A25,78A40,83A05
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Preface
The first extensive description of electromagnetism using the notion of a field—
introduced by Michael Faraday—was in A Treatise on Electricity and Magnetism
byJamesClerkMaxwell,publishedin1873.Sincethen,Maxwell’sequationshave
constitutedthecoreofelectrodynamics—theclassicaltheoryoftheelectromagnetic
field. The traditional mathematical tools for the presentation of Maxwell’s theory
were created by Josiah Willard Gibbs in his vector calculus, a blend of algebraic
and geometric notions. A typical example is the noncommutative vector product.
SuchaproductwasintroducedbyHermannGuenterGrassmanninabout1840and
called the exterior product. However, the result of this product was not a vector,
but a bivector—an element of an exterior algebra of multivectors, later called a
Grassmann algebra. It consists of scalars, vectors, bivectors, trivectors (generally
k-vectors)andtheirlinearcombinations.
Grassmanndidnotdevelopfullyalgebraicstructurescorrespondingtogeometric
notions, because he lacked the idea of duality, that is, a notion of linear form
over given linear space. Duality allows one to replace the scalar product with one
factor fixed by a linear form. (The reader will see this in the example of the wave
vectorreplacedbyalinearformofwavefronts.)Moreover,thescalarproductwith
two arbitrary factors can be replaced by a symmetric bilinear form. An object
dual to a kvector is kform, which is an exterior form of kth order. A set of such
forms constitutes an exterior algebra, also called a Grassmann algebra. Exterior
forms, depending on points in space, called differential forms, are appropriate to
integrateovercurves,surfacesorthree-dimensionalregions,withoutreferringtoa
scalar product. They were introduced by Gregorio Ricci-Cubastro in about 1880
and developed by Elie Joseph Cartan at the turn of the nineteenth and twentieth
centuries.
The presentation of electrodynamics with a wide use of differential forms
has become quite popular; see Refs. [1–9, 11–14, 52]. Some authors address
electromagnetism through differential forms in relation to electrical engineering
[15–21, 23, 25]. Such a formulation ensures a deep synthesis of formulae and
simplifiesmanydeductions.Usually,geometricimagingdoesnotfollowtheformal
v
vi Preface
mathematical definitions of the notions, which could be done by analogy with
vectorsandcanbeeasilydepicted.
Other mathematical structures have been applied to physics, namely Clifford
algebras, see [22, 24, 26, 27], where basic ingredients are multivectors as a
generalization of vectors. The present book is intended to keep a balance between
multivectorsandexteriorformsastwomathematicaltools.Asbothphysicistsand
engineersknow,avectorcanbeshownasadirectedsegmentwhichhasadirection
(a straight line with an orientation) and the magnitude (length). The linear form,
as the dual object, also has a direction and magnitude. In addition to vectors and
linearforms,wemayintroducealso multivectorsandtheirduals,namelyexterior
forms of higher grade (known also as multilinear forms when acting on vectors).
Here, I call these objects directed quantities because each of them has a direction
and magnitude. Schouten [46] was probably the first author who considered the
directions of multivectors and exterior forms. They were attractively illustrated
in the book of Misner, Thorne, and Wheeler [40]. Similar pictures can be seen
in publications of Burke [6–8] and Schutz [47], in an article by Warnick et al.
[54]. Hehl and Obukhov illustrate them richly in their book [20]. However, the
most careful presentation of the directed quantities in many domains of physics
is contained in the book by Tonti [51] under the name “physical variables”. The
discussion of the directed nature of physical quantities is not covered in such
depth here as in Tonti’s book. Nevertheless, the present book is focused on the
elaboration of the graphical presentation of directed quantities in mathematics,
thereby deepening the understanding of their properties and the role played in
physics. I have used as many diagrams as possible with the hope that this helps
withvisualization.
Not all publications using differential forms mention odd forms. These were
introduced by Weyl [55] and developed by Schouten [46], who called them W-p-
vectors (for Weyl). De Rham [10] called them odd forms, as did Ingarden with
Jamiołkowski[28]afterhim.ThetermtwistedformswasusedbyFrankel[17]and
Burke[7,8].Edelen[15],andlaterFrankel[18],usedthetermpseudo-forms.
AsitisshowninChap.1,sixteendirectedquantitiesareconnectedwiththethree-
dimensional vector space. We demonstrate in Chap.2, that when a scalar product
is present, the whole variety of directed quantities becomes unnecessary. There
thenappearsanaturalwaytoreplacethemwithvectors,pseudovectors, scalarsor
pseudoscalars—inthiswayonecanreturntothedescriptionknownfromthetime
ofGibbsandemployedintraditionaltextbooks.
Anaturalquestionarises:whydoweintroducesixteendirectedquantitieswhen
four of them can also do the job in three dimensions with the scalar product?
In addition to the above mentioned simplicity of deductions, there are two extra
reasons for this, which are discussed in Chap.4. First, the authors of textbooks
on electrodynamics, using exterior forms, have not stressed sufficiently strongly
that to some extent one can proceed without the scalar product, i.e. without
the metric. Van Dantzig [52] and Post [44] could be mentioned among authors
treating a metric-free approach to electromagnetism, and more recent is the book
by Hehl and Obukhov [20]. One may pose the question: how far can one go in
Preface vii
electrodynamicswithoutthemetric?Myansweristhatwecanonlydefinephysical
quantities and formulate principal equations (Maxwell’s equations, Lorentz force,
continuity equation, gauge transformation, continuity conditions on interfaces and
electromagnetic stress tensor). The part of it that can be formulated in a scalar
product-independent way has been called premetric electrodynamics. The best
advocates of this attitude are Hehl and Obukhov [20, 23], who also build up an
axiomaticstructureofelectromagnetism.
Itshouldbementionedthatthepresentbookisanexpandedversionofmybook
[30],thePolishtitleofwhichshouldbetranslatedasDirectedQuantitiesinElectro-
dynamics.Thisisnotatextbookonclassicalelectrodynamics,andaccordingly,its
materialisneitherstandardnorcomplete.Itisintendedforreaderswhohavegained
atleastabasicknowledgeofelectrodynamicsandspecialrelativity.Italsodemands
competence in the advanced calculus, linear algebra and differential equations.
At advanced undergraduate and graduate levels in departments of mathematics
and physics, the book will be useful for those lecturers and students who want
to deepen their understanding of fundamental aspects of classical field theory
and electrodynamics. The inclusion of exercises of different levels of complexity
also contributes to the possibility of using the book as supplementary reading in
combinationwiththemorestandardandcompletetextbooks.
I want to express my gratitude to Zbigniew Oziewicz who, when living in
Wrocław,gavemanylecturesandseminarsaboutdifferentialgeometry.Theyhave
revealedthebeautyofthisbranchofmathematicsanditsusefulnesswhenapplied
to physics. I am also grateful to Friedrich Hehl for plenty of valuable discussions
on the subjects present in this book. However, not all his suggestions have been
fulfilled.ThanksareduetoArkadiuszJadczykforcooperationwhenpreparingthe
Appendix.
Wrocław,Poland BernardJancewicz
April2021
Introduction
BydirectedquantitiesImeanmultivectorsand,dualtothem,exteriorforms.They
maybedividedintotwogroups.Thequantitiesofthefirstgroup,namelyordinary
(even) multivectors, have internal orientations, and the ordinary (even) forms
have external orientations, whereas the quantities of the second group are defined
conversely: odd multivectors have external orientations odd forms have internal
orientations. A detailed discussion of the adjectives “internal” and “external” is
containedinthebookbyTonti[51].Icanillustratethisdistinctionwiththeexample
of vectors: an even (ordinary) vector has an orientation chosen on its straight
segment;anodd(pseudo-)vectorhasitsorientationasadirectedcirclearoundthe
straightsegment.Thenames“even”and“odd”arenotgoodbecausetheyhavetoo
many meanings and connotations. For instance, the expression “even form” may
equallywelldenoteaformofevengrade.Ifyoucalltheoddobjectspseudo-forms
or pseudo-multivectors, there is no natural word to describe objects that are not
“pseudo”.Thename“twistedquantities”isnotappropriateforallquantitiesofthe
second group: for instance, an odd vector has its orientation marked as a directed
ringsurroundingastraightsegment,andhencethiscanbeseenastwisteddirection,
but the odd two-form has its orientation placed on a straight line, so this does not
fit the intuitive meaning of the word “twist”. Nevertheless, I accept the opinion of
Burkein[8,p.183],that“thelanguageisforcedonusbyhistory”,andIshalluse
theadjective“nontwisted”forthefirstgroup,and“twisted”forthesecondgroupof
quantities.
Sixteendirectedquantitiesofthethree-dimensionalvectorspaceareintroduced
in Chap.1 with careful definitions of their directions and magnitudes. In this
mathematical chapter, Sect.1.4 is an exception because it gives examples of
physicalquantities,showingthatallthesixteenquantitieshaveatleastonephysical
counterpart.Whatisstrikingisthateachofthefourphysicalquantitiesdescribing
the electromagnetic field, E—electric field strength, D—electric induction, H—
magneticfieldstrengthandB—magneticinduction,isanexteriorformofadifferent
directed nature. Some examples of physical quantities are also placed in other
sections of the two mathematical chapters when the mathematical tools necessary
fortheirintroductionbecomeavailable.
ix
x Introduction
Chapter 2 describes linear spaces of directed quantities, their transformation
propertiesandthestatusofthescalarproduct.Franklyspeaking,thewholevarietyof
directedquantitiesbecomesunnecessarywhenascalarproductisgiveninthelinear
space of vectors and, therefore, also a metric defined by it. There then appears a
naturalwaytoreplacethemwithvectors,pseudovectors,scalarsorpseudoscalars—
in this way one can return to the description known from the times of Gibbs and
employedintraditionaltextbooks.
Chapter 3 is devoted to algebra and analysis of the directed quantities: exterior
multiplication,contractions,linearoperatorsastensors,differentialforms,exterior
derivativeandintegration.AmoststrikingfeatureisvisibleinSect.3.5,namelythat
thedifferentialequationscanbewritteninthesameshapeinanarbitrarysystemof
curvilinear coordinates. This property goes under the name of general covariance.
A well-recognized fact is that the calculus of differential forms was developed
specificallytounifyandsimplifytheintegrationtheory.Wepresentitratherbriefly,
onlytoprovidetoolsforapplicationsinelectromagnetism.
Relevantphysicalquantities,theprincipalrelationsbetweenthem,anddifferen-
tialequationsarepresentedinChap.4,inmetric-freemanner.However,inorderto
writedownthesolutionsofpreviouslyintroducedequations,ametricisnecessary,
which enters the constitutive relations D = ε(E), B = μ(H). Then the Coulomb
field, the Biot-Savart law, the field of rectilinear current, and plane waves can
be obtained. The Laplace and d’Alembert equations also belong to this part of
electrodynamics.Itisusefultomentionalessondrawnfromtheseconsiderations:
the constitutive relations are needed to get solutions to Maxwell’s equations. The
authors who have stressed this role are, among others, van Dantzig [52], Post
[44] and Edelen [15], who wrote on page 372: “...wave properties of solutions
arise from the electromagnetic constitutive relations and the resulting differential
relationsbetweenquantities...”.
The second reason is that in anisotropic media, it is worth introducing another
metric, dependent on the electric permittivity or magnetic permeability. Then the
prescription for reducing forms to vectors or pseudovectors can be arranged so
that the medium looks like an isotropic one. Thus, the typical solutions of the
electrostaticandmagnetostaticproblemscanbeeasilytransposedfromthoseknown
inisotropicmedia.Chapter4isdevotedtothis,sotheCoulombfieldandBiot-Savart
lawarediscussedthere.
Electromagnetic waves are treated in Chap.5, and not only plane ones. An
attempt is made to find waves with planar wave fronts, but not constant fields
on them. For this purpose, cylindrical and spherical coordinates are used and so-
calledsemiplanewavesareintroduced.Planewavesareconsideredinananisotropic
medium.Inthiscase,thetraditionalwavevectordoesnotdescribethepropagation
directionofthewave:thisroleistakenbyaPoyntingtwo-form,whereasthewave
vectorisreplacedbyaone-formofphasedensity,wellsuitedtosurfacesofconstant
phase.So-calledeigenwavesareusefulinsuchamedium,asthephasevelocityhas
adefinitevalueforthem.
Considerations of interfaces between two anisotropic media are placed in
Chap.6. This starts with the mathematical Sect.6.1, which was not included in
Introduction xi
earlier chapters because it is not necessary for understanding Chaps.4 and 5.
Directed quantities of two-dimensional space (i.e. a plane) are introduced—only
twelve of them are possible—and a transition from two- to three-dimensional
directed quantities is discussed. The continuity conditions for the electromagnetic
field on interfaces are derived. They follow from (integral) Maxwell’s equations
only,andthereforetheybelongtothepremetricelectrodynamics.Theseconditions
help to obtain electromagnetic fields generated by infinite plates with uniform
charges or currents, and fields of ideal electric and magnetic capacitors. The
electromagneticstresstensorisintroduced.Reflectionandrefractionofplanewaves
areconclusionsfromtheinterfacialconditions.
The described transition from three-dimensional space to a plane helps us
to move one step higher and predict what kinds of directed quantities can be
present in four dimensions. There are twenty of them—they are introduced in
Chap.7. In the existing literature, the prolongation of the exterior forms from
three dimensions to four was considered to be automatic, whereas—as I show—
a distinguished time-like vector is necessary for this purpose, and without it, the
directions of the prolongated forms cannot be defined. This knowledge has been
used in Sects.7.2 and 7.3 to present physical quantities appropriate in the space-
time and the four-dimensional formulation of electrodynamics. After introducing
a Faraday two-form F=B+E∧dt and a Gauss two-form G=D−H∧dt, it
is sufficient to write down the two Maxwell equations in very compact shape.
Theuseofdifferentialformsandexteriorderivativeensuresthattheyareinvariant
underarbitrarycoordinateandframetransformationsandindependentofthemetric
of spacetime see [20]. Since the metric of spacetime serves as the gravitational
potential in general relativity, it is useful to know that there is a gravity-free way
of formulating Maxwell’s equations. Accordingly, the equations are valid in a flat
space-timeandcanbetransferredtoacurvedone.
The plane electromagnetic wave in vacuum is discussed for four-dimensional
space-time. The distinguished role of its phase velocity is used to find the
Minkowskiscalarproduct.Areasonisshownwhyametrictensorwithoneminus
and three pluses is better than the alternative with opposite signs on the diagonal.
TheconstitutiverelationbetweenFandGisfound.ItcontainsaHodgemapwith
themetricincluded.ThisrelationallowsonetoexpresstwoMaxwellequationsby
the coordinates of the Faraday field only. The homogeneous equation is generally
covariant,whereastheinhomogeneousone(withthefour-currentontheright-hand
side)isonlyLorentzcovariant.
