A New Formulation of Maxwell's Equations - MDPI

symmetryS S

Article

A New Formulation of Maxwellrsquos Equations

Simona Fialovaacute and František Pochylyacute

Citation Fialovaacute S Pochylyacute F A

New Formulation of Maxwellrsquos

Equations Symmetry 2021 13 868

httpsdoiorg103390sym13050868

Academic Editor Radu Abrudan

Received 1 April 2021

Accepted 7 May 2021

Published 12 May 2021

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40)

Victor Kaplan Department of Fluids Engineering Brno University of Technology Technickaacute 2896261669 Brno Czech Republic pochylyfmevutbrcz Correspondence fialovafmevutbrcz

Abstract In this paper new forms of Maxwellrsquos equations in vector and scalar variants are presentedThe new forms are based on the use of Gaussrsquos theorem for magnetic induction and electricalinduction The equations are formulated in both differential and integral forms In particular thenew forms of the equations relate to the non-stationary expressions and their integral identities Theindicated methodology enables a thorough analysis of non-stationary boundary conditions on thebehavior of electromagnetic fields in multiple continuous regions It can be used both for qualitativeanalysis and in numerical methods (control volume method) and optimization The last Sectionintroduces an application to equations of magnetic fluid in both differential and integral forms

Keywords Maxwellrsquos equations divergence theorem integral form magnetism optimizationanalysis

1 Introduction

Magnetic and magnetorheological fluids have been increasingly used in recent years [1ndash3]Their properties especially viscosity can be significantly changed by the effects of themagnetic field [4ndash10] The mathematical model is in such cases composed of Maxwellrsquosand NavierndashStokes equations [11] which are mostly solved by numerical methods [12ndash15]One of these models is presented at the end of this paper The solution behavior of thementioned models can be assessed on the basis of the symmetry or asymmetry of theoperators forming the mathematical model [1617] Let us write for example Maxwellrsquosequations in the following operator form

∥∥∥∥ minus partpartt curl

div 0

∥∥∥∥middot∣∣∣∣∣∣

D

H

∣∣∣∣∣∣ =

∣∣∣∣∣∣j

ρ

∣∣∣∣∣∣ (1)

∥∥∥∥ partpartt curl

div 0

∥∥∥∥middot∣∣∣∣∣∣

B

E

∣∣∣∣∣∣ = 0 (2)

From here the symmetry of the problem for a nonconductive environment wherej = 0 ρ = 0 is quite obvious Furthermore it follows that the non-stationary field isrotational because it is not possible to set E = gradφ It is therefore necessary to use newmethods for non-stationary problems solution The operator equations written in thesummation symbolics also show that

(curlE)i = εijkpartEkpartxj

(3)

The expression partEkpartxj

can be decomposed into

a symmetric Ekj =12

(partEkpartxj

+partEjpartxk

)and

Symmetry 2021 13 868 httpsdoiorg103390sym13050868 httpswwwmdpicomjournalsymmetry

Symmetry 2021 13 868 2 of 12

an antisymmetric part Ekj =12

(partEkpartxjminus partEj

partxk

)

While the LevindashCivit tensor εijk is antisymmetric only the antisymmetric part Ekj isused in Expression (3) Thus

(curlE)i = εijkEkj (4)

These modifications could be deepened by a more detailed study of the symmetryof the mathematical model operators depending on the boundary conditions A newformulation of Maxwellrsquos equations which is essentially based on Gaussrsquos divergencetheorem [17] can also contribute to this analysis

According to the authors the mathematical formulation of Gaussrsquos divergence the-orem is underestimated in terms of the physical substantiality of the problem Gaussrsquosdivergence theorem explains the following important finding every spatial change in thephysical variable f(x t) regardless of its tensor character has its response at the boundaryof a closed region int

V

partf(x t)partxi

dV =intS

f(x t)nidS (5)

where f can be understood as a symbolic variable Of course the term (5) has the oppositemeaning If we act on the boundary of the region S with the effect of the variable f thisvariable changes within the volume V In brief every change within the volume V can bemeasured at the boundary S where it is reflected in its functional value see Figure 1

Figure 1 General volume of the liquid V surrounded by continuous closed area S that consists ofopen areas S (outlined by closed curve k) Volume V can be split into control volumes ∆V closed bycontrol areas ∆S

This principle can be used in qualitative analysis numerical methods and optimiza-tion of the electromagnetic field The main reason is that this principle gives a very goodidea both qualitative and quantitative of the influence of boundary conditions on theproblem which assignment is often in hands of the researcher

After all in electromagnetic fields solution the engineering practice successfully usesintegral identities both on the principle of Gaussian divergence theorem for the closed areaS and Stokes theorem for the open area S bounded by a curve k [15ndash19]

For example

curlE + partBpartt = 0

intk

Edk +intS

partBpartt middotndS = 0 (6)

Symmetry 2021 13 868 3 of 12

and others [20] Since this paper is focused on solving non-stationary problems it isappropriate to modify Maxwellrsquos equations into a more suitable form for optimizationqualitative analysis and numerical methods This can be achieved by using Gaussrsquosdivergence theorem and the symmetry of the Kronecker delta operator

(δij = δji

) The

adjustment applies to all non-stationary terms of Maxwellrsquos equations for non-conductivemedia and partB

partt for conductive media where j 6= 0 ρ 6= 0 The modification the proof ofwhich is given in Section 2 can be expressed by a non-stationary term in a more suitableform for the application of Gaussrsquos divergence theorem

partBipartt

=part

partxj

(partBipartt

xi

)(7)

The scalar variant of Maxwellrsquos equations (Section 4) is also presented in the workwhere new functions are introduced

Modified intensity of imprinted forces

E[Vmiddotmminus1

]=

partBpartt

x (8)

Modified stress of printed forces

U[V] =partBpartt|x|2 (9)

Based on Gaussrsquos divergence theorem they are again modified to a form suitablefor analysis

partBipartt

xi =part

partxj

(partBipartt

xixi

)(10)

Based on (7) by integrating (6) over the domain V surrounded by the surface S newintegral identities can be found this time over the closed surface where all boundaryconditions appear int

S

(ntimes E)dS +intS

(partBparttmiddotn)

xdS = 0 (11)

The proof is given in Section 2 It follows from the above-mentioned equations that thearticle focuses on the appropriate modification of Maxwellrsquos equations so that the influenceof non-stationary terms in the field V is expressed by their values on the boundary S ofthe closed region The solution is based on the use of symmetry conditions and Gaussrsquosdivergence theorem The use of this procedure for qualitative analysis numerical methodsand optimization are presented in the individual Sections for both the vector variantand the scalar variant of Maxwellrsquos equations Both conductive and non-conductiveenvironments are considered in the solution The last Section presents a mathematicalmodel of the interaction of a magnetorheological fluid with a magnetic field Even forthis interdisciplinary problem Gaussrsquos divergence theorem can be used to redefine themathematical model of NavierndashStokes equations An example is given in Section 6

A special part is devoted to the finite volume method for non-stationary problems [13ndash22]In the classical method a non-stationary term is identified through the control volume interms of the mean values of the integral calculus This method does not allow the use ofthe finite volume method while the new variant will allow it see Section 2

2 Symmetry in Principles of the Solution

In the technical sciences symmetry conditions play a special role especially in thestability conditions of the system [131423] They have the same importance in the solution

Symmetry 2021 13 868 4 of 12

of electromagnetism tasks where we can find in the term partBipartxj

and partDipartxj

These terms concludethe symmetric Sij and antisymmetric Aij part It can be written

partBipartxj

= Sij + Aij where (12)

Sij =12

(partBipartxj

+partBj

partxi

)and Aij =

12

(partBipartxjminus

partBj

partxi

)Based on the symmetry principles it is possible to find new shapes of Maxwellrsquos

equations using the Gauss divergence theorem The principle can be explained for exampleon magnetic induction Let us consider the following equations

curlE = minuspartBpartt

divB = 0 (13)

The same formulated in the index symbolic (Einstein summation symbolics)

εijkpartEkpartxj

= minuspartBipartt

partBipartxi

= 0 (14)

Einsteinrsquos summation symbolic is used in the mentioned relation and the followingtext We note that these relations depend on the antisymmetric operator εijk and the

expression partEkpartxj

which can be decomposed into symmetric and antisymmetric parts (seeSection 1) Therefore in the left part of Equation (14) only the antisymmetric part of theexpression partEk

partxjmanifests

After these remarks let us proceed to a derivation of a new variant of the equationεijk

partEkpartxj

= minus partBipartt by modifying its right part containing a non-stationary term We start from

the validity of the equation partBipartxi

= 0 For this purpose let us put

partBjpartxj

xi = 0intV

partBjpartxj

xidV =intS

BjxinjdSminusintV

Bjpartxipartxj

dV = 0(15)

The relationship uses per partes integration in 3D space After using a Kronecker deltasymmetry adjustment

partxipartxj

= δij = δji (16)

Thus Bjpartxipartxj

= Bi It is possible to write Equation (15) in the shape

intV

BidV =intS

BjxinjdS or in the vector form (17)

intV

BdV =intS

(Bmiddotn)xdS (18)

Because it also holds that partpartxj

(partBjpartt

)= 0 it can be derived by analogy

intV

partBipartt

dV =intS

partBj

parttxinjdS or in the vector form (19)

Symmetry 2021 13 868 5 of 12

intV

partBpartt

dV =intS

(partBparttmiddotn)

xdS (20)

Expressions (19) and (20) are very important because they point out the fact thatunsteady states of the magnetic flux density are generated on the borders of the areaand vice versa From the Expression (17) follows the next important result based on thedivergence theorem

partBipartt

=part

partxj

(partBj

parttxi

)(21)

Using Expression (21) it is possible to correct the form of Equation (14) on the principleof symmetry After the substitution of the above-mentioned criteria we obtain a new shapeof Maxwell equations

part

partxj

(εijkEk +

partBj

parttxi

)= 0 (22)

In Equation (22) the first term in bracelets represents the antisymmetric tensor ofsecond grade and in the second term it is possible to decompose into the symmetric andantisymmetric parts

Equation (19) can be used to modify the control volume methodIn the current control volumes method the integration of the non-stationary term

given in Maxwellrsquos equations is expressed on the basis of the mean value of the integralcalculus [511ndash14202224] Thus in the form (see Figure 1)

intV

partBpartt

d(∆V) =partBc

partt∆V (23)

where

∆V =13

int∆S

(xmiddotn)d(∆S) (24)

In the newly proposed method the integration is performed directly by usingRelation (21) as followsint

∆V

partBipartt

d(∆V) =int

∆V

part

partxj

(partBj

parttxi

)d(∆V) =

int∆S

(partBj

parttnj

)xid(∆S) (25)

The same in the vector formint∆V

partBpartt

d(∆V) =int

∆S

(partBpartt

n)

xd(∆S) (26)

After the integration over the volume V (22) can be easily written in the vector variant

intS

[(ntimes E) +

(partBparttmiddotn)

x]

dS = 0 (27)

Note that from the obtained results (20) it is observable that the non-stationary changein magnetic induction within the field V can be determined by integration only at thesystem boundary Conversely time changes in the magnetic induction in the field V aregenerated at the system boundary This fact can be advantageously used for both qualitativeanalysis of non-stationary boundary conditions and optimization of non-stationary tasksby selecting a suitable target function depending on the boundary conditions For thenon-conductive space the derivations are described in Section 3

Symmetry 2021 13 868 6 of 12

3 A Non-Conductive Environment

We assume σ = 0 ρe = 0 In this case Maxwellrsquos equations will have the indexsymbolic form [13]

εijkpartHkpartxj

= partDipartt partDi

partxi= 0 (28)

εijkpartEkpartxj

= minus partBipartt partBi

partxi= 0 (29)

Due to the validity of Equations (17) and (18) using transform (21) new forms ofMaxwellrsquos equations can be written without evidence since

partDipartt = part

partxj

(partDjpartt xi

)partBipartt = part

partxj

(partBjpartt xi

) (30)

intV

partDipartt dV =

intS

partDjpartt xinjdSint

V

partDpartt dV =

intS

(partDpartt middotn

)xdS

(31)

intV

partBipartt dV =

intS

partBjpartt xinjdSint

V

partBpartt dV =

intS

(partBpartt middotn

)xdS

(32)

partpartxj

(partDjpartt xi minus εijk Hk

)= 0

partpartxj

(partBjpartt xi + εijkEk

)= 0

(33)

intS

[(partDpartt middotn

)xminus ntimesH

]dS = 0int

S

[(partBpartt middotn

)x + ntimes E

]dS = 0

(34)

From the above it is visible that in the case of a non-conductive environment it ispossible to derive a new variant for all Maxwellrsquos equations All the conclusions given inSection 3 remain valid including the control volumes method

All these results can be used to solve the interdisciplinary problem of the motion of anincompressible fluid with the effects of a nonconductive magnetic field using the Maxwellstress tensor for this case see Section 6

4 The Scalar Variants of Maxwellrsquos Equations

By the scalar variant of Maxwellrsquos equations [13] we mean the product of the multi-plication of Equation (13) and the position vector x

curlEmiddotx = minus partBpartt middotx

εijkpartEkpartxj

xi = minus partBipartt xi

(35)

The results of the solution of the scalar variant can be used again for the boundary con-ditions analysis and in the optimization area For the solution the divergence theorem (13)is beneficially used

divB = 0 rArrpartBj

partxjxixi = 0 (36)

By analogy to (15) we apply the multiplication

partBjpartxj

xixi = 0intV

partBjpartxj

xixidV =intS

BjxixinjdSminus 2intV

Bjpartxipartxj

xidV(37)

Symmetry 2021 13 868 7 of 12

From (37) follows the important knowledge

intV

BjδijxidV =intV

BixidV =12

intS

BjxixinjdS (38)

In the vector form written asintV

BmiddotxdV =12

intS

(Bmiddotn)(xmiddotx)dS (39)

Considering the divergence theorem in the shape div partBpartt = 0 Equations (38) and (39)

can be written intV

partBipartt

xidV =12

intS

partBj

parttxixinjdS (40)

intV

partBparttmiddotxdV =

12

intS

(partBparttmiddotn)(xmiddotx) dS (41)

where (xmiddotx) = xixi = |x|2

From Equation (40) and using the divergence theorem it is possible to derive thefollowing important dependence that allows one to reformulate Maxwellrsquos Equation (35)

partBipartt

xi =12

part

partxj(

partBj

parttxixi) (42)

When we implement (42) into (35) we obtain

part

partxj

(εijkEkxi +

12

partBj

parttxixi

)= 0 (43)

The term in the bracelet can again be divided into the symmetric and antisymmetricparts as well as in (22) In the integral form

intS

[12

partBj

parttxixi + εijkEkxi

]njdS = 0 (44)

For the vector form it holdsintS

[(partBparttmiddotn)(xmiddotx) + 2(ntimes E)x

]dS = 0 (45)

One of the scalar variants of Maxwellrsquos equations was again derived under the as-sumption of Gaussrsquos divergence theorem validity Derived relationships can be used toevaluate the results obtained by numerical methods Even in this case non-stationarychanges in magnetic induction are reflected at the boundary of the region and here it ispossible to determine their values as a function of time The resulting equations can also beeasily used for optimization because the target function is scalar in this case

5 A Non-Conductive EnvironmentmdashScalar Variant

If we return to the problem of a non-conductive environment we can rewriteEquations (28) and (29) in a differential form

Original equationspartDipartt xi minus εijk

partHkpartxj

xi = 0partBipartt xi + εijk

partEkpartxj

xi = 0(46)

Symmetry 2021 13 868 8 of 12

partBipartxi

= 0 (47)

New variantpart

partxj

(partDjpartt xixi minus 2εijk

partHkpartxj

xi

)= 0

partpartxj

(partBjpartt xixi + 2εijk

partEkpartxj

xi

)= 0

(48)

Integral formOriginal equations int

V

partDpartt middotxdV =

intS(ntimesH)middotxdS int

V

partBpartt middotxdV = minus

intS(ntimes E)middotxdS

(49)

New variant intS

(partDpartt middotn

)(xmiddotx)dS = 2

intS(ntimesH)middotxdSint

S

(partBpartt middotn

)(xmiddotx)dS = minus2

intS(ntimes E)middotxdS

(50)

The new form of Maxwellrsquos equations is useful whether for analysis or numericalsolution Comparing the left sides of Equations (49) and (50) it is obvious that non-stationary variables D(x t) B(x t) are generated only at the boundary of the system andtherefore it is possible to influence their process within the volume V This can be essentialin optimizing the non-stationary problems of electromagnetism

Here scalar variants of Maxwellrsquos equations were also derived under the assumptionof Gaussrsquos divergence theorem validity Derived relations can be used to evaluate theresults obtained by numerical methods Even in this case non-stationary changes inmagnetic induction and electrical induction are reflected at the boundary of the region andhere their values can be determined as a function of time The resulting equations can alsobe easily used for optimization because the target function is scalar in this case

6 An Interaction of a Non-Conductive Magnetic Liquid with a Magnetic Field

In Section 3 it is shown that in the case of a non-conductive environment it is possibleto derive a new variant for all Maxwellrsquos equations All the conclusions given in the Sectionremain valid including the control volumes method All these results can be used insolving the interdisciplinary problem of the motion of an incompressible fluid with theeffects of a non-conductive magnetic field using the Maxwell stress tensor for this case [3]In the presented case we assume

ρe = 0 B = micro0H + M M = χH (51)

The density of the volumetric magnetic force that acts on the elementary volume canbe written in the form [8]

f = 12 χ gradH2 H2 = HmiddotH (52)

NavierndashStokes equations of the magnetic liquid in the presented case are in theform [21125ndash27]

ρpartvipartt

+part

partxj

(vivj minus σij

)= ρgi +

12

χpart

partxi

(H2)

(53)

Considering

gi =part

partxi(gkxk) (54)

Symmetry 2021 13 868 9 of 12

then Equation (53) can be written in a more transparent form

ρpartvipartt

+part

partxj

(ρvivj minus σij minus ρδijgkxk minus

12

χδijH2)= 0 (55)

Because the liquid is considered to be incompressible the continuity equation is inthe form

partvipartxi

= divv = 0 (56)

Now if we consider Equation (21) the NavierndashStokes equation for the incompressiblemagnetic liquid can be written in a new form

part

partxj

[ρ

partvj

parttxi + ρvivj minus σij minus ρδij

(gkxk minus

12

χH2)]

= 0 (57)

This equation can be using the Divergence theorem [1617] rewritten in the newintegral form

intS

[ρ

(partvparttmiddotn)

x + ρ(vmiddotn)vminus σ minus(

gmiddotxminus 12

χ (HmiddotH)

)middotn]

dS = 0 (58)

σ = (σ1 σ2 σ3) σi = σij nj (59)

By comparing the original equation Equation (53) and the new equation Equation (57)the advantage of the new variant is evident both for the numerical solution by the finitevolume method and for the analysis of the influence of boundary conditions Since in theabove-mentioned case assuming diva = 0 with respect to (20) it holds

intV

partvpartt

dV =intS

(partvparttmiddotn)

xdS (60)

intV

partBpartt

dV =intS

(partBparttmiddotn)

xdS (61)

and concurrently for the result of the continuity equation

intS

partvparttmiddotndS = 0

intS

partBparttmiddotndS = 0

7 Discussion

A new formulation of Maxwellrsquos equations was derived both in differential andintegral variants The basis for the derivation was Gaussrsquos divergence theorem used formagnetic flux density B and electric flux density D By the use of Gaussrsquos divergencetheorem Maxwellrsquos equations were transformed This resulted in a tool that can be used inthe numerical finite volume method and optimization The obtained equations will alsoallow the qualitative analysis of the influence of boundary conditions The mentionedchanges concern the non-stationary terms of the type partB

partt resp partDpartt This resulted in a new

form of Maxwellrsquos equations that can be used in solving the interdisciplinary problem ofthe motion of an incompressible fluid with the effects of a non-conductive magnetic fieldusing the Maxwell stress tensor for this case For example

Symmetry 2021 13 868 10 of 12

Differential formsOriginal Maxwellacutes equation

partBipartt

= minusεijkpartEkpartxj

New variantpart

partxj

(partBj

parttxi

)= minusεijk

partEkpartxj

Integral formsOriginal variant int

V

partBpartt

dV = minusintS

ntimes E dS

New variant intS

(partBparttmiddotn)

xdS = minusintS

ntimes E dS

In these relations it is interesting that the effect of non-stationary members partBpartt within

the region V is reflected at the system boundary only by its normal component(

partBpartt middotn

)

8 Conclusions

The work was focused on the analysis of non-stationary Maxwell equations A newshape of non-stationary magnetic flux density was derived This made the analyses ofMaxwellrsquos equations possible by using the Gaussian divergence theorem Maxwellrsquosequations were defined in both vector and scalar variants The new shape of the Maxwellequations simplifies the analyses of the solution quality depending on the boundaryconditions considering the non-stationary magnetic induction It also allows the numericalsolution of Maxwellrsquos equations to be extended to the large control volume method Usingthe Gaussian divergence theorem the new method allows the region to be optimizeddepending on the non-stationary field of magnetic induction

A special part was devoted to the finite volume method for non-stationary problemsIn the classical method a non-stationary term is identified through the control volume interms of the mean values of the integral calculus This method does not allow the use oflarge control volumes while the new variant allows it

Both conductive and non-conductive environments were considered in the solutionThe last Section presents a mathematical model of the interaction of a magnetorheologicalfluid with a magnetic field Even for this interdisciplinary problem Gaussrsquos divergencetheorem can be used to redefine the mathematical model of NavierndashStokes equations

Author Contributions Conceptualization SF and FP validation SF formal analysis FP writingmdashoriginal draft preparation SF and FP writingmdashreview and editing SF visualization SF supervi-sion FP project administration SF funding acquisition SF according to the CRediT taxonomyAll authors have read and agreed to the published version of the manuscript

Funding This paper was supported by the projects ldquoComputer Simulations for Effective Low-Emission Energyrdquo funded as project No CZ02101000016_0260008392 by the OperationalProgram Research Development and Education Priority axis 1 Strengthening capacity for high-quality research and ldquoResearch of the flow and interaction of two-component liquids with solidsand external magnetic fieldrdquo funded as project No GA10119-06666S by the Grant Agency ofCzech Republic

Conflicts of Interest The authors declare no conflict of interest

Notes Einstein summation convention is used in the article

Symmetry 2021 13 868 11 of 12

Nomenclature

xi Cartesian coordinatest timeV volume∆V control volume∆S control surfaceS closed surfaceS open surfacex = (x1 x2 x3) spatial vectorn = (n1 n2 n3) unit normal vectorE electric field intensityD electric flux densityj current densityρe charge densityB magnetic flux densityH magnetic field intensityσ conductivityε permittivityM magnetizationρ fluid densityv fluid velocity v = (v1 v2 v3)g gravity accelerationymiddotz = yizi scalar product of two vectors y bσi stress vectorσij stress tensorδij Kronecker deltaεijk LevindashCivit tensorχ magnetic susceptibilitymicro0 surroundings permeability

References1 Jiles D Introduction to Magnetism and Magnetic Materials CRC Press New York NY USA 20162 Odenbach S Ferrofluids Lecture Notes in Physics Available online httpwwwspringerdephys (accessed on

26 November 2002)3 Guru BS Hiziroglu HR Electromagnetic Field Theory Fundamentals Cambridge University Press Cambridge MA USA 2004

ISBN 0-521-8301684 Hammond P Electromagnetism for Engineers Oxford University Press New York NY USA 1997 ISBN 0-19-856299-35 Ida N Bastos JPA Electromagnetics and Calculation of Fields Springer BerlinHeidelberg Germany 1992 ISBN 0-387-97852-66 Kroumlger R Unbehauen R Elektrodynamik BG Teuhner Stuttgart Germany 1993 ISBN 3-319-23031-37 Liao S Dourmashkin P Belcher JW MIT Electricity and Magnetism-Physics 802 Massachusetts Institute of Technology

Cambridge MA USA 20068 Marinescu M Elektrische und Magnetische Felder Springer Berlin Germany 2009 ISBN 978-3-540-89696-89 Plonus MA Applied Electromagnetics Mc-Graw Hill Book Co New York NY USA 1978 ISBN 0-07-050345-110 Zangwill A Modern Electrodynamics Cambridge University Press Cambridge UK 2013 ISBN 978-05-21896-97-911 Pochylyacute F Fialovaacute S Krausovaacute H Variants of Navier-Stokes Equations In Proceedings of the 18th International Conference

Engineering Mechanics 2012 Svratka Czech Republic 14ndash17 May 2012 pp 1011ndash1016 ISBN 978-80-86246-40-612 Chari MVK Salon SJ Numerical Methods in Electromagnetism Academic Press San Diego CA USA London UK 2000

ISBN 0-12-615760-X13 Kost A Numerische Methoden in der Berechnung elektrischer Felder Springer Berlin Germany 1994 ISBN 3-540-55005-414 Mayer D Ulrych B Simulation and Design of Induction Heating J Electr Eng 1997 48 48ndash5215 Eymard R Galloueumlt T Herbin R Handbook of Numerical Analysis Elsevier Amsterdam The Netherlands 200016 Ženiacutešek A Surface Integral and Gauss- Ostrogradsky Theorem from the View Point of Applications In Applications of Mathematics

Springer Berlin Germany 1999 Volume 44 pp 169ndash24117 Pfeffer WF The Divergence Theorem and Sets of Finite Perimeter Chapman and HallCRC London UK 201218 Reineker P Schulz M Schulz M Theoretische Physik IImdashElektrodynamik J Wiley Darmstadt Germany 2006 ISBN 3-527-40450-319 Vanderlinde J Classical Electromagnetics Theory Springer Dordrecht The Netherland 2004 ISBN 10-1-4020-2699-420 Smirnov VI A Course of Higher Mathematics Elsevier Amsterdam The Netherlands 1964 Volume 2

Symmetry 2021 13 868 12 of 12

21 Humphries S Jr Field Solutions on Computers CRC Press LLC Boca Raton FL USA 1998 ISBN 0-8493-1668-522 Chari MVK Silvestr PP Finite Elements in Electrical and Magnetic Field Probleme J Wiley and Sons Chichester UK 1980

ISBN 0-471-27578-623 Kim J Kim D Choi H Journal of Computational Physics Elsevier Amsterdam The Netherlands 200124 Ida N Engineering Electromagnetics Springer New York NY USA 2004 ISBN 0-387-20156-425 Katz J Introductory Fluid Mechanics Cambridge University Press Cambridge MA USA 2010 ISBN 978-1-107-6171326 de Grott SR Mazur P Non Equlibrium Thermodynamics Courier Corporation Amsterdam The Netherlands 196227 Pochylyacute F Fialovaacute S Krutil J New Mathematical Model of Certain Class of Continuum Mechanics Problems Eng Mech 2014

21 61ndash66