Helmholtz’s decomposition theorem, etc. R. I. Khrapko Moscow Aviation Institute, 125993, Moscow E-mail: khrapko_ri@hotmail.com The fact is used that electromagnetic fields are covariant (antisymmetric) tensors or contravariant (antisymmetric) tensor densities, which are mutual conjugated. The conjugation allows many-fold specific differentiation of the fields and leads to field chains. An integral operation, named the generation, is considered, which is reverse to the specific differentiation. The double generation yields zero as well as the double differentiation. The Helmholtz decomposition is compared with the Poincare decomposition, and many ways of the Helmholtz decomposition are presented. Laplace operator and the inverse Laplace operator are expressed in terms of the differential and integral operations. All results are illustrated by simple examples.

1. Introduction. Helmholtz’s decomposition and Poincare’s decomposition

The Helmholtz’s theorem is familiar to physicists [1] and mathematics [2]. The essence of the theorem is as follows. A field, e.g. an electric vector field , can be written as the sum of two terms, the transverse or solenoidal field and the longitudinal or irrotational field :

E

oE

×E

0curl,0div, =×∇≡=⋅∇≡+=×E

××EEEEEE

ooo. (1.1)

Transverse or solenoidal fields are usually denoted by E or E , and longitudinal or irrotational fields are denoted by E or E , however, we use the circle o and the cross × for marking solenoidal and irrotational vector fields respectively because this notations remind pictures of field tubes (or lines) of these fields (see Fig. 1 from [3] for a solenoidal field and Fig. 2 from [4] for

irrotational fields). We are sure these visual notations are appropriate for a pedagogical paper. Moreover, in accordance with Fig. 1, we name divergence-free fields closed fields. Thus, the circle o marks a closed field.

t ⊥

l ||

Note that 0/div,curl ερ=⋅∇≡−=×∇≡

××EEBEE &

oo. (1.2)

Helmholtz’s decomposition is well known. The irrotational part of a vector field E is

∫ π⋅∇

−∇=× )',(4

')'(')(xxrdVxx EE . (1.3)

Meanwhile, two different expressions are known for the solenoidal part of E:

⎟⎟⎠

⎞⎜⎜⎝

⎛π

×∇×∇= ∫ )',(4')'()(

xxrdVxx EE

o [1, 5], (1.4)

∫ π×∇

×∇=)',(4

')'(')(xxrdVxx EE

o [2, 6 - 9]. (1.5)

Here means the coordinates x zyx ,, ; the prime marks the variables of integrating or differentiating under the integral sign; is the distance between and . Thus, a field is integrated and differentiated by turns when the field is under the Helmgoltz’s decomposition.

||)'.( xxxxr ′−= 'x x

We pay attention that a similar consecution of integrating and differentiating takes place when an exterior differential form ω (briefly, form) is decomposed into closed and not closed parts [10, 11]:

ω+ω=ω KddK+ω+ω=

o (1.6)

(we mark closed parts of forms by the circle and not closed parts by the plus sign). This formula is very important in the theory of exterior differential forms. In Eq. (1.6), means the exterior derivative, and dK is an operation which is an inverse operation to the exterior derivative in the following sense: if

oω=ω

is a closed form, i.e. 0=ωd , then

++α=ωα=ω

ooKd & entails

ooω=ωα=α

++dKKd & . (1.7)

K -operator exists in domains which are not too complicated topologically, according to the Poincare theorem.

We present here an example of K -operator using tensor indices. If ω is a 3-form, , then )(xijkω

∫ ω=ω1

0

2 )( dttxxtK ijki

ijki . (1.8)

We name K -operator the Poincare generative operator, we name the Poincare generation from

, and we name a source of . Note that the Poincare generative operator does not contain a metric tensor.

ijkiK ω

ijkω ijkω ijkiK ω

It is easy to show that a double application of the K -operator yields zero, i.e. . For example, 0=KK

0)(1

0

21

0=τ⎥⎦

⎤⎢⎣⎡ ωτ=ω

τ=∫∫ ddttxxtxKK

xxijk

ijijk

ij (1.9)

because . We say that the generation from the generation is zero, or that the generation is sterile. So, K eliminates the sterile part of a form

0=ωijkij xx

ω while d eliminates the closed part of a form:

oooω=ω+ωω=ω+ω

+++KKdd )(,)( . (1.10)

Thus, Eq. (1.6) is the decomposition of a form ω into the closed part oω and the Poincare sterile part

+ω .

A purpose of this paper is to show that the Helmholtz’s decomposition is a decomposition into closed and sterile fields as well. 2. Differential forms, tensor densities, the boundaries, the conjugation, etc

It is important to recognize that the electromagnetism involves geometrical quantities [12] of two different types [13]. These are covariant (antisymmetric) tensors ijii BAE ,,,ϕ , which are named also exterior differential forms or simply forms, and contravariant (antisymmetric) tensor densities:

. ∧∧∧∧ ρ,,, iiik EjBThe distinction between forms and tensor densities is known long since. For example, professor J. A.

Schouten delivered lectures on this subject before the war at Delft, and after the war at Amsterdam (see the classical monograph [12], which was grown from the lectures, and Fig. 3). A similar interpretation for covectors is presented in [14] (Fig. 4). Note that the magnitude of a covector is proportional to the density of sheets. Therefore, a covector fields must be depicted by a family of bisurfaces with an outer orientation and bivector densities must be depicted by a family of bisurfaces with an inner orientation rather than by lines or tubes.

Unfortunately, this distinction is ignored by most of the physicists. So, common boldfaced characters do

not represent the quantities adequately, and we are forced to use tensor indices. Besides, instead of using of Gothic characters (as in Fig. 3), it is convenient to mark the density by the symbol ‘wedge’ ∧ at the level of bottom indices for a density of weight and at the level of top indices for a density of weight . For example, volume element is a density of weight

. Also we mark pseudo forms by the asterisk * and pseudo densities by the tilde ~: , .

1+1−

∧dV1−

∗ijE ~

ijkεThe exterior derivation of the forms is

used in the electrodynamics. The exterior derivation of a scalar is the common partial derivation,

φ⋅∇=φ=⇔φ∂= gradEiiE (2.1) (we do not write minus in this formula, so we write ), but in a general case an antisymmetrization is implied. The notion “curl” can be used with a covector,

φ∇=ρ 2

AAB ×∇==⇔∂= curl2 ][ jiij AB , , (2.2) EEB ×−∇=−=⇔∂−= curl2 ][&&

jiij EB

we denote )(21

][ ijjiji AAA ∂−∂=∂ . The exterior derivation of a covariant tensor of valence 2 is a

divergence, 0div03 ][ =⋅∇=⇔=∂ BBijk B , (2.3)

we denote )(31

][ kijjkiijkijk BBBB ∂+∂+∂=∂ .

As to tensor densities, a transvection over a (last) contravariant index is used in the electrodynamics when the specific derivation is performed. The derivation of a vector density is named divergence:

Ediv=ρ⇔∂=ρ ∧∧i

i E , (2.4) but the derivation of a bivector density is denote by curl,

Bj curl=⇔∂= ∧∧ik

ki Bj . (2.3)

The derivation of a scalar density is zero,

0≡ρ∂ ∧i , (2.4) because has no contravariant indexes. We emphasize that all presented differential operations are covariant operations. Their writing is valid no matter what coordinates are in use, Euclidean or curvilinear. The Cristoffel symbols are not needed.

∧ρ

For short, we will designate the derivations of the both types by the symbol , curly d, without indices. We name a derived field a boundary, and we name the field under derivation the filling of the boundary, i.e. (boundary) = (filling), for example,

∂

∂

ijiki

jijii BBjEBEE ∂=∂=−∂=φ∂=∂=ρ ∧∧∧∧ 0,,,, & . (2.5)

The term “boundary” is justified, for example, by the fact that lines (or tubes) of force of -field are bounded by a charge density , according to . This example is depicted in Fig. 2 where the electric charge bounds the electric vector field E. Thus, the symbol

iE∧

∧ρiE∧∧ ∂=ρ

∂ expresses the relation between a boundary and its filling, i.e. is a boundary operator. ∂

Our symbol ∂ , instead of , means the exterior derivation when it is applied to a form. We are convinced it is anti-pedagogical to use the symbol as a designation of the exterior derivation. The symbol is used for infinitesimal quantities in physics and mathematics, not for a derivation. For example, denotes the charge of an infinitesimal volume . Another example: one can write where is a velocity,

dd

ddVx)(dq ρ= dVrrvr ddt +=+ v dzdydxd kjir ++= is an infinitesimal increment of the

vector , and are the infinitesimal increments of the coordinates. Also , and we can write or even .

r dzdydx ,, dxxfxdf )(')( =22 )('')( dxffddfd == xdfdxffd 222 ')('' +=

Contrary to this, is used as an operator which takes each p-form d ω to a -form )1( +p ωd , and forever [10, 11]. Accordingly, the expressions are known as a nonindex notation

for the coordinate 1-forms, i.e., covectors, rather than as the components of the infinitesimal vector , i.e. are known as . This mishmash is inadmissible.

0)( =ωdd dzdydx ,,rd

dzdydx ,, 321 ,, iiiiii zdzydyxdx δ=∂=δ=∂=δ=∂=On the other side, the symbol means “boundary” in the theory of sets. And this is the very

meaning that our symbol ∂

∂ has. If the boundary of a field is zero, we say that the field is closed, for example, is closed:

. An example of a closed electric field is presented in Fig. 1, . Lines of force

of the induced (solenoidal) vector field have no boundaries, and this field has no boundary. In

accordance with Section 1, we mark a closed field by a circle.

ijB

03 ][ =∂=∂ ijkij BB 0=∂ ∧i

i Eo

iE ∧=oo

E

The double derivation gives zero, 0=∂∂ . For example, if φ∂= iiEo

, then . We say

that the boundary of a boundary is zero, or that a boundary is closed. A boundary has no boundary, but a boundary has a filling, according to the Poincare theorem. In the case,

02][][ =φ∂=∂ kiik E

o

φ∂= iiEo

, φ is a filling of the

boundary . This case is depicted in Fig. 11. Another example: if , then , i.e.

is a boundary, and the vector electric potential is the filling. iE

o

ijj

iE ∧∧ Π∂=o

0=∂ ∧i

i Eo

iE ∧o

ij∧Π

The raising and lowering of tensor indices is usually performed by a metric tensor or . But, in the electrodynamics, this process is accompanied by the transition between differential forms and contravariant densities, for example, between the covector and the vector density . So this process

uses the root of the metric tensor determinant

ikg ikg

iE iE∧

∧g , which is a scalar density of weight . So, the

tensor densities

1+

∧∧ = ggg ikik or ∧∧ = ggg ikik / is used instead of or ,. If Cartesian coordinates

are in use, the absolute value of the determinant equals one, but the root has a specific geometrical properties. The process of the raising or lowering of tensor indices changes the geometrical sense of a field; it is referred to as the conjugation here, and we designate the process by the Courier star * (in contrast to the Hodge star operation *), for example,

ikg ikg

mnij

jnimijij

mnnjmi

mni

kik

kki

iki BBggBBBggBEEgEEEgE ∧∧∧

∧∧∧

∧∧∧∧ ======== **** ,,, . (2.6)

The conjugation is obviously involutory: 1** = . We say that a field and the conjugate field make up a tandem. For example ( ) and ( & ) are tandems (see Fig. 5 and Fig. 6). i

ii EEE *=∧& ijB∧

ijij BB ∧= *

The conjugation * differ from the Hodge star operation * [10, 11, 15]. Hodge operator performs our

conjugation of a field and then renumbers components of the field by the antisymmetric tensor pseudo density (Levi Civita density). For example, . Here Hodge operator transforms a

1-form into the pseudo 2-form . However, the renumbering has no physical and geometrical meaning because has the same geometrical meaning as the vector density (Fig. 5). Therefore, the addition of the renumbering to the conjugation has no sense. We do not use the renumbering and so we reduced Hodge operation to the conjugation. Note, the Hodge operator cannot be applied to a tensor density.

~ijkε ∗

∧ =ε=∗ mniij

mnji EEgE ~

iE ∗mnE

∗mnE i

j EE *=∧

It is important that when the Hodge operator is applied two times in the structure , the result differs from by a sign only [15, p. 315]:

∗∂∗**∂

ppnnp

pω∂−=ω∂ +++ **1)1(** , (2.7)

where n is the dimension of the space and p is the degree of the form . Because a so-called codifferential is defined as

pω

pnnp

pω∂−=ωδ ++ **)1( 1 , (2.8)

we have for the codifferential p

pp

ω∂−=ωδ **)1( . (2.9) It is remarkable that the conjugation often transforms a closed field into a not closed field. For

example, 0=∂ ijB , but . (2.10) mmn

nijjnim

nij jBBggB ∧∧∧ µ=∂=∂=∂ 0)(*

Such fields, closed before or after conjugation, is named conjugate-closed fields, or, simply, coclosed fields: is closed, but is coclosed because . ijB mnB∧ 0=∂=∂ ∧ ij

mn BB*

Now recall Helmholtz’s theorem (1.1). It must be written down in terms of vector densities, iii EEE ∧×∧∧ +=

o. (2.11)

The solenoidal vector field , which satisfies iE ∧=oo

E 0div =⋅∇≡ooEE , is closed,

. The irrotational Coulomb vector field , which satisfies

, is coclosed:

BEE &oo

−=×∇≡curl

0=∂ ∧i

i Eo

iE ∧××=E ,0curl =×∇≡

××EE

0/div ερ=⋅∇≡××EE

0)(2 ][ =∂=∂=×∇×∧××EE *i

ijk Eg . (2.12)

Thus, the mark × , which was used already in (1.1), marks the coclosed fields. As a result, we see that Helmholtz’s decomposition (1.1) is a decomposition into closed and coclosed

components. It is important that Helmholtz’s decomposition (1.1) can be rewritten in the conjugate form in term of

covectors

×+= EEE ***

o ⇔ iii EEE ∧×∧∧ += ***

o⇔ . (2.13) iii EEE

o+=

×

Here , the first component of decomposition (2.13), which corresponds to the induced solenoidal

closed vector field, is not closed now; it has a boundary: (1.2). This boundary is the time-

dependent magnetic field, and the bisurfaces, which depict the covector field , are bounded by tubes of

force of the magnetic field, the bisurfaces start from tubes of force of the magnetic field in Fig. 7. These bisurfaces is orthogonal to lines of force in Fig. 1.The field , is obviously coclosed: .

iE×

][2 ijji EB×

∂=−o

&

iE×

iE×

0=∂×∧ i

kik Eg

The second component in (2.13), , which corresponds to the irrotational vector field , is closed

now because Eq. (1.1), means . Accordingly, -field is

depicted by spherical bisurfaces in Fig. 8. These bisurfaces is orthogonal to lines of force in Fig. 2.

iEo ×

E

,0curl =×∇≡××EE 022 ][][ =∂=∂ ∧×

∧ji

kkji EEg

oiE

o

3. Chains of fields

The property of the conjugation to transform closed fields into not closed fields leads to an existence of infinite or finite chains of fields. We present here, as an example, the infinite chain of the electrostatic.

⋅⋅⋅∂∂φφ∂∂ρρ∂∂δδ∂⋅⋅⋅ ∧×∧∧×∧∧×∧ )()()()()()()()()()())(())(()( iii

iiii

iiii

ii FFEEGGxxooo

****** (3.1)

The sections of the chain, i.e. , etc., are joined by the symbols and (*). It means, for

example, . In (3.1),

ρδ ∧×∧ ,),( iGx )( i∂

φ∂==∂=ρ ∧×∧×∧ iiiii

i EEEEoo

,, * )(xδ is the Dirac delta function if the electric

charge density equals ∧ρ rπ−=ρ∧ 4/1 . The explicit form of the chain in this case is

⋅⋅∂π

−π

−∂

π−

π−

∂π−

π−

∂π−

π−

∂ππ

∂δδ∂⋅⋅ ∧ )(32

)(32

)(8

)(8

)(8

)(8

)(4

1)(4

1)(4

)(4

))(())(()( 33 ii

i

iii

i

iii

i

iirrrrrr

rr

rr

rrrr

rrxx ******

(3.2) Really, a coclosed electric intensity, corresponding to the density rπ−=ρ∧ 4/1 , is

because

rxE ii π−=∧×8/

rrr ii π−=π−∂ 4/1)8/( ; the corresponding potential is π−=φ 8/r ; the density π−=φ=φ∧ 8/r* is the

boundary of a hypothetical coclosed field ; etc. On the other hand, the boundary of

is ; the boundary of is the

π−=∧×32/ii rrF

rπ−=ρ=ρ ∧ 4/1* 34/)4/1( rrrG iii π=π−∂=o

34/ rrGG iii π== ∧×o

* δ -

function: ; etc.: )4/()( 3rrx ii π∂=δ∧

We can present a complementary electrostatic chain: ⋅⋅⋅∂∂ΠΠ∂∂∂∂⋅⋅⋅ ∧×∧×∧×∧×∧×

)()()()()()()()()()()( ki

ikkiki

ik

kiikik

ki

ik FFEEJJGGooooo

***** . (3.3)

Here is a closed vector density, , i.e. kE ∧o

kii

kE ∧×∧ Π∂=o

Π= curlE , and Π is so-called electric vector

potential. Magnetic current density is the boundary of the coclosed covector electric intensity

: , i.e. .

ikJo

kk EE ∧×=

o* ][2 kiik EJ

×∂=

oEJ curl=

We can present also a chain that is complementary to chain (3.3): ⋅⋅⋅∂θθ∂ΠΠ∂ηη∂∂ςς∂⋅⋅⋅ ∧×∧×∧

×∧×∧

×)()()()()()()()()()()( kkij

kijj

kikijjki

jkii

jkjkiijk

ijkk JJ

ooooo

***** . (3.4)

Here the closed electric bivector potential is the boundary of a hypothetic -field: .

This can be expressed as where

ki∧Π

o

kij∧×

θ kijj

ki∧×∧ θ∂=Π

o

** gradθ=Π *Π is a pseudo covector and is a pseudo scalar because , .

*θ2/~* ki

jkij ∧Πε=Πo

6/~* kijkij ∧×θε=θ

Two complementary magnetostatic chains can be obtained by renaming of sections of chains (3.3) and (3.4),

mAFBjE ρ⇒η⇒⇒Π⇒ ,,, :

⋅⋅⋅∂∂∂∂⋅⋅⋅ ∧×∧×∧×

)()()()()()()( ki

ikkiki

ik

ki AABBjjooo

*** , (3.5)

⋅⋅⋅∂θθ∂∂ρρ∂⋅⋅⋅ ∧×∧×∧ )()()()()()()( kkijkij

jki

kijjkim

jki

mi BB

oo*** . (3.6)

Here is the magnetic charge, ][3 kijjkim

B×

∂=ρ Bdiv=ρm

, and is the magnetic pseudo scalar

potential: , i.e. .

kij∧×

θ

kijj

kiB ∧×∧ θ∂=o

θ= gradB

4. The Laplace operator

The derivation of a sum of a closed and a coclosed field equals the derivation of the coclosed term only, because the derivation eliminates closed term, like (1.10),

ii

iii EEE ∧×∧×∧ ∂=+∂ )(

o. (4.1)

However, Laplacian, the second order operator, , treats both terms of such a sum. As is known [15, p. 316],

jiijg ∂∂=∇ 2

δ∂−∂δ−=∇ 2 (4.2) (the signature of ). Here is the codifferential (2.8), (2.9). Because of (2.9), +++=ijg δ

pp

pp

ppω∂∂−−=α∂−=αδ=ω∂δ

++

+

**** )1()1(1

11

, . (4.3) p

pp

ω∂∂−=ωδ∂ **)1(

Therefore, for a p -form , pω

pp

pω∂∂∂∂−=ω∇ )()1(2 **-** , (4.4)

see also [16]. It is easy to show that for a contravariant density of valence p , ∧βp

∧+

∧ β∂∂∂∂−=β∇p

pp

)()1( 12 **-** . (4.5) Thus, Laplacian realizes a transition to four sections of a chain to the left and, maybe, changes the sign. For example, according to (3.2) and (4.5) for 0=p ,

rrx

π−

∇=π−

∂∂=δ∧ 41

41)( 2** , (4.6)

according to (3.5) and (4.4) for , 1=p

iii AAj×××

−∇=∂∂= 2** . (4.7)

or, according to (3.5) and (4.5) for [1, (5.31)] 1=piii AAj ∧∧∧ −∇=∂∂=

ooo

2** . (4.8)

If the vector potential is not satisfied (3-dimension) Lorentz gauge, , then, according to (4.5) and [1, (5.30)],

0≠∂ ∧i

i A

jA −=∂∂∂∂=∇ ∧∧∧ divgrad2 iii AAA **-** . (4.9) 5. The generations

Given a closed differential form or contravariant density, we can find their fillings by an integral generative operator

∫ ′π=

∧

),(4)',(† 3

'

xxrxxrdV i

i , (5.1)

instead of iK from (1.7). For example, the filling of )(x∧δ (see (3.2)) is

)(4)(

),(4)',(

)'( 33

'

' xrxr

xxrdVxxr

xii

π=

′πδ ∧

∧∧

∧∫ . (5.2)

We say that is a source which generates )(x∧δ 34)(

rxr i

π∧ and write 34

)()(†

rxr

xi

i

π=δ ∧

∧ , or 34)(

)(rxr

xi

π→δ ∧

∧ .

Generally, we say that a source generates the generation, i.e. †(source) = (generation).

Next example: 34 rri

π generates

rπ−4

1 , i.e. rr

rii

π−

=π 4

14

† 3 , i.e. rr

ri

π−

→π 4

14 3 , i.e.

∫∫∫ ′π′′

∂′π

=′π′′

′π∂−=

′π′′

′π′

′′′

),(4),(

)(41

),(4),(

)(41

),(4),(

)(4)(

3333 xxrVdxxr

xrxxrVdxxr

xrxxrVdxxr

xrxr i

i

i

i

ii

∫ π−

=′′δ′π

−=)(4

1),()(4

1xr

Vdxxxr

, (5.3)

because

),(4),(

3 xxrVdxxr i

i ′π′′

∂ ′ ),(4),(

3 xxrVdxxr i

i ′π′′

−∂=

Thus, we can rewrite the chain (3.2), for example, in terms of the generation instead of the bourdary operator . ∂

⋅⋅→π

−π

−→

π−

π−

→π−

π−

→π−

π−

→ππ

→δδ→⋅⋅ ∧ 32)(

328)(

88)(

841)(

41

4)(

4)())(( 33

ii

ii

ii rrrrrr

rr

rr

rrrr

rrxx ******

(5.4) We can say, in particular, that ρ is a source of the vector field E , or ρ generates E , or E is the generation from : ρ

ii

i Er

dVxxrx ∧×

∧∧

∧∧ =π

ρ=ρ ∫ 3

'

' 4)',()'(† . (5.5)

This generating is depicted in Fig. 2. Electric charge emits lines of force of . It is a source of E. This example shows visually that generations are coclosed, i.e.

E

0†=∂* . (5.6) Really, bisurfaces, which are orthogonal to the emitted lines, are closed (Fig. 8). This assertion is proved

by a simple identity 03][ =∂rrg

i

kji . Accordingly, we mark generations by the cross × .

The generating (magnetic field generates the magnetic covector potential), according to (3.5),

AB →

k

i

kiiki A

rdVxxrBB

×

∧∧ =

π= ∫ 3

'''

' 4)',(† ( generates ), (5.7) B A

is depicted in Fig. 9. determines the potential uniquely. This potential stands out against a

background of all gauge equivalent vector potentials [16]. ikB kA

×

However, the magnetic vector potential usually is depicted by lines of force of the vector as in

Fig. 10 from [17], but Fig. 9 shows visually how B-tubes emit bisurfaces of .

iAo

iA×

Formula (5.7) is analogous to the Biot-Savarat law. ik

kiik B

rdVxxrjj ∧×

∧∧′

∧′∧ =π

= ∫ 3

']'[

4)',(2† ( generates B ), (5.8) j

The only distinction between (5.8) and (5.7) is that -tubes and -bisurfaces have an outer orientation, but -tubes and -bisurfaces have an inner orientation.

ikB kAij∧

ikB∧

A scalar field is depicted by hatching or darkening of the space. Fig. 11 shows how the closed electric field bounds , i.e. how

)(xφφ∂= iiE

o)(xφ φ -field fills the closed -field, or how generates iE

oiE

oφ ,

)(4

)',(† 3

'''

' xr

dVxxrEEi

iii φ=

π= ∫

∧∧ ( E generates ); (5.9) φ

iE is a source of and, at the same time, φ φ is a filling of . Note, Eq. (5.9) determines the potential

uniquely as well as Eq. (5.7) determines . iE

o

)(xφ kA×

As an example, we apply Eq. (5.9) for a solution of the problem: “What potential is generated by a

thin two-dimensional ‘spherical’ capacitor?” So, in a two-dimensional (for simplicity) space there are two concentric circles between which a given radial electric field E exists. It is necessary to find the potential

in this space by the formula )(xφ

∫ π⋅

=φ ,2

)()( 2rdax rE (5.10)

where da is an element of the space (plane). We have (see Fig 12): )cos(sinsin)cos(cos RxERERxErErE y

yx

x −α=α⋅α−α−⋅α=+=⋅rE , (5.11)

r R x R R x xR2 2 2 2 2 2 2= + − = + −sin ( cos ) cos .α α α . (5.12) If we write δ for the small gap between the circles, then da R d= δ α. , and

φδ

πα

αα

π

=+

+∫ER

xv

u vd

22

0

2 cos /cos

, u R x= +2 2 1/ , v R x= −2 / . (5.13)

Integrating yields:

φδ( ) ( )

( ).x ER

xxR

R x xR R x

RxR x

= − ++−

−−

⎡

⎣⎢⎢

⎤

⎦⎥⎥2 2

2 2

2 2 2 2 (5.14)

I.e. φ = on the outside of the circles, that is at R < x, and 0 δ−=φ E inside the circles, that is at R > x, just as expected.

The generations have an important property: the generation generates zero, 0†† = . (5.15)

In other words, the generations are sterile as well as the Poincare generation (1.7). We prove this assertion here for from (5.5), i.e. we prove that the integral iE ∧×

( )∫ π

≡∧

∧∧×∧× )',(4

)',('† 3

']'

[

][

xxr

dVxxrxEE

ki

ki (5.16)

equals zero. Indeed, inserting from (5.5) into (5.16) gives iE ∧×

∫∫ =π⋅π

ρ≡

∧∧∧∧∧

∧×0

)',(4)",'(4)',()",'()"(† 33

1

'"]'

[1"][

xxrxxrdVdVxxrxxrxE

kiki . (5.17)

To prove the last equality, fix the points x" and x. Then, because of the symmetry of the space, for each x' exists such '~x that the vector product in x' and ][

1ki rr '~x differ in the sign only. So, integrating over

gives zero.

'∧dV

It can be shown that coclosed fields are sterile, i.e.

0† =∂* . (5.18) As a simple example, consider the constant coclosed density 1=ψ ∧

×. We have:

∫ ==ψ ∧×

0')',(† 3i

rdVxxr i

, (5.19)

because of the symmetry of the space. Thus, generating eliminates the sterile part of a source as well as the derivation eliminates the

closed part of a filling (4.1). Only closed part of a source generates. For example,

××φ==+ i

iii

i EEEoo

†)(† . (5.20)

Therefore, we can calculate the potential of a non-potential field by the generative operator. The solenoidal part of the vector field (1.1), which satisfies , and which correspond

to sterile in Eq. (2.13), may be present in the integrand of Eq. (5.9), but its contribution is zero. In

contrast to Eq. (5.8), the standard formula gives an ambiguous result for a non-potential field

E.

oE BEE &

oo−== curl,0div

iE×

∫=ϕ0

xdlE

According to Eq. (5.20),

×φ∂= iiE

o & entails . (5.21)

×φ=i

i Eo

†××φ∂=φ i

i† & jj

ii EEoo

†∂=

In that sense, and ∂ are the mutually inverse operators (compare with (1.7)). For example, we have †

33 44†

rr

rr ij

ji

π=

π∂ ∧∧ and

rrii

π−

=π−

∂4

14

1† .

If relations between E and φ are φ∂=E , as in (3.2) – (3.6), and , as in (5.4), we will write and so on. For example ,

φ→EφaE

⋅⋅⋅φφρρ ∧×∧∧×∧∧×aaaaaaL

oook

ki

ik

k FFEEGG ***** (5.22)

⋅⋅⋅ΠΠ⋅⋅⋅ ∧×∧×∧×∧×∧×aaaaaa

ooooo

kkik

ikiiki

kikk FFEEJJGG ***** . (5.23)

instead of (3.1), (3.3) 6. The generative operator squared

We define the generative operator squared, ‡, by the equations (compare with (4.4), (4.5)) † p

pp

ω−=ω + )††††()1(‡ 1 **-** , (6.1)

∧∧ β−=βp

pp

)††††()1(‡ **-** . (6.2) The result follows from the definition:

1‡‡ 22 =∇=∇ . (6.3) Really, for examlpe,

ppppppp

pω=ω∂∂+ω∂∂=ω+ω∂∂∂∂−−=ω∇

××

+ **********-****-** ††††))(()1)(††††()1(‡ 12

oo, (6.4)

because of (4.1), (5.18). Eqs. (6.3) yields

∫ ′π−=

∧′

),(4‡

xxrdV . (6.5)

Thus, the generative operator squared makes a transition to four sections of a chain to the right and, maybe, changes the sign. For example, according to (3.2) and (6.2) for 0=p ,

rxx

π−

=δ=δ ∧∧ 41)(††)(‡

oo** , i.e.

rxxrdVx

π=

′π′δ∫

∧′

∧′ 41

),(4)( . (6.6)

According to (3.5) and (6.1) for , 1=p

iii Ajj×××

−=−= **††‡ , i.e. ∫ =′π

∧′′

ii A

xxrdVj

),(4 [1, (5.32)]. (6.7)

7. Various variants of Helmholtz’s decomposition

There are many different ways of the Helmholtz’s decomposition. Recall formula (1.1) ×

+= EEEo

.

The simplest decomposition of the vector (density) field is ,)††( EEEE ∂+∂=+=

×o i.e. EEEE ∂=∂=

׆,†

o. (7.1)

This decomposition does not use the conjugation. An explicit form of the decomposition is

∫ ∫ π×′

×∇=π

∂=∂==∧

∧′

∧∧∧ 33

']['

4'

42†

rdV

rdVrEEE

ki

kii rEE

oo, (7.2)

∫ ∫ π⋅∇

=π

∂=∂==

∧∧∧

∧∧×× 33

''''

4')''(

4†

rdV

rdVrE

EEik

kii rEE . (7.3)

It is depicted in Figures 13 and 14.

There is another decompisition of a vector field, which uses operators †,∂ only:

,)††( EEEE **** ∂+∂=+=×o

i.e. EEEE **** †,† ∂=∂=×o

. (7.4)

An explicit form of the decomposition is

∫ ∫ π××∇

=π

∂=∂=

∧∧

′∧′

∧∧ 33

''']'[

4')''(

42†

rdV

rdVrEg

gEEkl

ljkijli rE**

o, (7.5)

∫ ∫ π⋅′

∇=π

∂=∂=∧

∧′

∧′′∧∧× 33

'''

4')(

4†

rdV

rdVrEg

gEEkj

jkl

ilii rE** . (7.6)

Eq. (7.5) is depicted in Fig.15.

Eqs. (1.3), (1.4) use the generative operator squared, ‡, and two the boundary operators, ∂ , e.g., for

(1.4), . It is depicted in Fig. 16. (Note, according to (6.2), ). ii EE ∧∧ ∂−∂= ‡**o

ii EE ∧∧ −=oo

††‡ **

Eq. (1.5) uses the operators ‡ and ∂ in another order then Eq. (1.4): . ii EE ∧∧ ∂−∂= **‡o

Also we can offer the operator and two operators † for obtaining , for example, 2∇ iE ∧o

∫ ∫ π′

×⎟⎟⎠

⎞⎜⎜⎝

⎛′π

′′′×′′∇=∇= ∧∧ 33

22

4)(4)(††

rVd

rVdEE ii rrE

**o

. (7.7)

8. An example of the use of the simplest decomposition (7.1)

Consider a semi-infinite straight thin wire carrying an electric current I along the positive z-axis. Let the current density j is singular in the wire territory:

)0,(RIjz δ= if , and if , (8.1) 0>z 0=zj 0<z

where 22 yxR += , and satisfies )0,(Rδ 12)0,( =πδ∫ RdRR . So,

)0,(xIt δ=ρ−∂=∇j , (8.2)

where satisfies . )0,(xδ ∫ =δ 1)0,( dVxOur aim is to decompose the density j into solenoidal and irrotational parts by applying Eq. (7.1) to j:

EBrjrjjjj txxrdVxx

xxrdVxx

∂−×∇=π

∇+

π×

×∇=+= ∫∫× )',(4

')',()''()',(4

')',(''33

o

. (8.3)

We have step by step. The Biot-Savarat law yields:

∫∫ π×

=π

×=

)',(4)',(''

)',(4')',(''

33 xxrxxI

xxrdVxx rdlrjB , (8.4)

)(,14

,14])'([

'4

22222

0'2/3222 yxR

rz

RxIB

rz

RyI

zzyxdzyIB y

zx +=⎟

⎠⎞

⎜⎝⎛ +

π=⎟

⎠⎞

⎜⎝⎛ +

π−=

−++π= ∫

∞

=

. (8.5)

Bj ×∇=o

for all points except the semi axis (z 0,0 ==> yxz ), where the magnetic field is singular, is

3333 4i.e.,

4,

4,

4 rI

rzIBBj

ryIBj

rxIBj xyyxzxzyyzx π

−=π

−=∂−∂=π

−=∂=π

−=−∂=rj

oooo

. (8.6)

B×∇ for can be determined from (8.5) by the Stokes theorem for a circle of radius 0,0 ==> yxz R when : 0→R

B×∇ IdIxdyydxRI

=ϕπ

=+−π

==×∇∫ ∫ ∫∫π2

02 22

)( dlBdaB . (8.7)

Thus, according to (8.7), )0,()( RIz δ=×∇ B . In other words, the solenoidal part of the current , o

j j

,4 3 z

Ir

I zrj +π

−=o

if , 0>z 34 rIπ

−=rj

o

, if , (8.8) 0<z

consists of radial converged field tubes (8.6) and the semi axis . 0>zThe irrotational part is

∫ π=

πδ

=× )(4

)()',(4

')',()0,'()( 33 xrxI

xxrdVxxxIx rrj . (8.9)

The decomposition is depicted at Fig. 17. Field tubes are used instead of common field lines because the current density j is a vector density. One can see that the components of the decomposition extend over all space, despite j is localized. Many authors pointed out this fact [1]. 9. The Minkowski space. Tandem-closed fields

Chains (3.1) – (3.4) use the Euclidean metric tensor for conjugating, 1}1,1,diag{ +++=ijg , but chains of electromagnetic fields use }11,1,1,diag{ −−−+=µνg :

KKooooo

)()()()()()()()()()( ∂∂∂∂∂ α∧ν×µν

αβ∧

×

α∧ν×µν

αβ∧×

α∧α

×CCQQBBjj ***** AA (9.1)

We denote the electromagnetic tensor µνµν −= FB instead of in order to instead of

common , and our magnetic vector 4-potential satisfies ] . So, e.g.,

, and ;

µνF αβ∧β

α∧ ∂= Bj

αβ∧β

α∧ −∂= Fj νA Aνµµν ∂= [2B

31132

31 AA ∂−∂== BB iiii AA −== AA , 1001110 AA ∂−∂== EB , and . φ== 0

0 AAIn (9.1), Q and C are hypothetic fields. Besides closed and coclosed fields, there are tandem-closed fields, i.e.fields, which are closed and

coclosed simultaneously. We mark these fields by the pair of signs o× . For example, an electromagnetic plane wave makes up an tandem-closed field , . Indeed, if αβ

∧×oB µν×o

Bitizitizitizitiz eBBeBeBBeEB −

××

−

×

−∧×

−∧×

=−=−===== 133101231101 ,,,

ooooo, (9.2)

then 0,0 310103][

133

100

1 =∂+∂=∂=∂+∂=∂××µν×λ∧×∧×

β∧×β

ooooooBBBBBB . (9.3)

In this case, both the fields of a tandem are closed. We name such a tandem an end-tandem because a chain ends at the tandem. Obviously, an end-tandem is the end of two (complementary) chains. For example

Kooooo

)()()()()()()()()(0 ∂∂∂∂∂ α∧ν×µν

αβ∧

×

α∧ν×µν×

αβ∧×

CCQQBB **** AA (9.4)

Kooooo

)()()()()()()()()(0 ∂∂∂ΠΠ∂∂ λµναβγ∧×

αβ∧µν

×λµν

αβγ∧×

αβ∧×µν×

RRQQBB **** (9.5)

Electromagnetic field in chain (9.4) is the boundary of the magnetic vector potential :

, , however the same electromagnetic field in chain (9.5) is the

boundary of the electric three-vector potential : , . Some next

sections of the chains are

µν×oB ν×

A

][2 ν×µµν×∂= A

oB itizie −

∧×−=−= 1

1oAA αβ

∧×oB

αβγ∧×

Π αβγ∧×γ

αβ∧×

Π∂=o

B itizie −∧×

=Π=Π 103103

o

itizitizitiz eztCCeztiQQeztiQQ −∧×

−∧

×

−∧

×

+−=−=

++==

++−=−=

4)(,

4)(1,

4)(1 1

11313

1010

ooo

, (9.6)

itizitizitiz eztRReztiQQeztiQQ −∧×

−∧

×

−∧

×

+−==

+−==

++=−=

4)(,

4)(1,

4)(1

10310313

1310

10ooo

, (9.7)

Because Laplacian realizes a transition to four sections of a chain to the left and, maybe, changes the sign, we have, according to (9.4),

αβ∧

×

αβ∧

×

αβ∧×

−∇=∂∂= QQB 2**o

. (9.8)

But, the same field ,according to (9.5), equals αβ∧×o

B

αβ∧

αβ∧

αβ∧×

∇=∂∂=ooo

QQB 2** . (9.9)

Now we have arrived at an interesting conclusion:

(9.10) Really,

0)(22 =+−=+∇=∇ αβ∧×

αβ∧×

αβ∧

αβ∧

×

αβ∧

ooo

BBQQQ . (9.11)

By the way, we can consider the harmonic field as a plane polarized electromagnetic plane wave because it satisfies the wave equation (9.11). By analogy with (9.2) we have from (9.6), (9.7)

αβ∧Q

itizitiz eBQQQeEQQQ −∧∧

×∧

−∧∧

×∧ −==+===+=

21,

21

23131311010101

oo

. (9.12)

But it is a very strange wave. The Poynting vector, 2/)( BES ×= , has the -component directed opposite to the direction of wave propagation:

z8/1−=zS , and the wave is accompanied by electric

and magnetic currents:

α∧j

λµνJα∧

αβ∧

×β

α∧ =∂=

oAj Q , λµνµν

×λλµν Π=∂=

o][3 QJ . (9.13)

We present here another example of the end-tandem which is very simple. Let . Then and are the tandem-closed fields if . Really:

, and . So, we have two complementary chains:

2,1,,1}1,diag{ =++= jigijia∧ ia xaya == ∧∧

21 ,

xaayaa ==== ∧∧2

21

1 , ** 0,0 ][ =∂=∂ ∧ jii

i aa

Looo

)()()()()()()(0 ∂∂∂∂ ∧×∧×∧× ii

ii ccbbaa *** , (9.14)

Looo

)()()()()()()(0 ∂∂∂∂ ∧×∧∧××

iiik

ikii ccddaa *** , (9.15)

where

6,

6,

22,

124,

124,

32

2

31

1

22

1212

32

22

32

11 xccyccyxddxxyccyyxccxyb −==−==+−==+==+=== ∧×∧×∧∧×∧× oooo

.

(9.16) According to (9.10), must be harmonic. Really, iii ccc

×+=

o

124,

124

32

222

32

111xxycccyyxccc −=+=−=+=

×× oo, (9.17)

and 02211 =∂+∂=∂+∂ cccc yyxxyyxx . This is OK. 10. Conclusion

Usefulness of concepts of differential forms and tensor densities in the electromagnetism is shown. Concepts of boundary and its filling, source and its generation are introduced. These concepts extend an understanding of electrodynamics because they explain mutual relations between the electromagnetic fields. References 1. J. D. Jackson Classical Electrodynamics. (John Wiley & Sons, Inc., 1999) 2. G. A. Korn, T. M. Korn, Mathematical Handbook (McGraw-Hill, 1968) 3. H. C. Ohanian Physics. (N.Y.: W.W.Norton, 1985) 4. Д. В. Сивухин Общий курс физики. Том 3, часть 1. (М.: Наука, 1996) 5. J. D. Jackson “From Lorenz to Coulomb and other explicit gauge transformations” American J.

Physics. 70 917-928 (2002) 6. A. M. Stewart “Vector potential of the Coulomb gauge” European J. Physics 24 519 (2003) 7. V. Hnzido “Comment on `Vector potential of the Coulomb gauge’” European J. Physics 25 L21 (2004)

8. B. P. Miller “Interpretations from Helmholtz’ theorem in classical electromagnetism” American J. Physics 52 948-950 (1984)

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http://www.mai.ru/projects/mai_works/articles/num4/article7/auther.htm (2005) 14. J. Napolitano and R. Lichtenstein “Answer to Question #55 Are the pictorial examples that

distinguish covariant and contravariant vectors?” American J. Physics 65 1037 (1997) 15. C. Von Westenholz, Differential Forms in Mathematical Physics (North Holland, 1978) 16. R. I. Khrapko. “Violation of the gauge equivalence,” physics/0105031 17. R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics (Addison–Wesley,

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