Relationship between Laplacian Operator and D’Alembertian Operator

Relationship between Laplacian Operator and D’Alembertian Operator

Relaci´on entre los Operadores Laplaciano y D’Alembertiano Graciela S. Birman

^∗

([email protected]) Graciela M. Desideri ([email protected])

NUCOMPA - Fac. Exact Sc. - UNCPBA Pinto 399 - B7000GHG, Tandil - Argentina.

Abstract

Laplacian and D’Alembertian operators on functions are very impor- tant tools for several branches of Mathematics and Physics. In addition to their relevance, both operators are very used in vector calculus.

In this paper, we show a relationship between the Laplacian and the D’Alembertian operators, not only on functions but also on vector fields defined on hypersurfaces in them-dimensional Lorentzian spaces.

We also define theBm^k1,...,kl-product andBm-congruence.

Key words and phrases:Laplacian, D’Alembertian, Lorentzian space, operator,Bm^k1,...,kl-product.

Resumen

Los operadores Laplaciano y D’Alembertiano aplicados a funciones son herramientas muy importantes en varias ramas de la Matem´atica y de la F´ısica. Sumada a su relevancia, ambos operadores se destacan por ser muy utilizados en el c´alculo vectorial.

En este art´ıculo mostramos la relaci´on entre los operadores La- placiano y D’Alembertiano tanto sobre funciones como sobre campos vectoriales definidos sobre hipersuperficies del espacio Lorentzianom- dimensional. Adem´as, definimos losB^km^1,...,kl- productos y laBm- con- gruencia entre operadores.

Palabras y frases clave: Laplaciano, D’Alembertiano, espacios Lo- rentzianos, operador,B^km^1,...,kl-producto.

Received 2002/09/03. Revised 2003/12/05. Accepted 2004/01/05.

MSC (2000): Primary 53B30.

∗Partially supported by Consejo Nacional de Investigaciones Cient´ıficas y Tecnol´ogicas de la Rep´ublica Argentina.

1 Introduction

In the last three decades the interest in Lorentzian geometry has increased, [1]. We will concentrate on two differential operators of particular interest here: the Laplacian and the D’Alembertian.

Laplacian and D’Alembertian operators on functions are very important tools for several branches of Mathematics and Physics, specificly in investi- gating many geometrical and physical properties. In addition to relevance, both operators are very used in vector calculus.

Moreover, the Laplacian operator on functions is quite different from the Laplacian operator on vector fields and the D’Alembertian on functions is quite different from the D’Alembertian on vector fields.

There are many interesting vector fields in differential geometry, for ex- ample the mean curvature vector field. In [5], Bang-yen Chen developed the Laplacian on vector fields, and he studied its application on mean curvature vector field for submanifolds in Riemannian space. In [3], we studied the Laplacian operator of the mean curvature vector fields on surfaces in the 3- dimensional Lorentzian space, R³₁,and we showed the Laplacian operator of the mean curvature vector fields on the non-lightlike surfacesS₁²,H₀², S₁¹×R, H₀¹×R,R¹₁×S¹ andR²₁.

The purpose of this article is to show the relationship between the Lapla- cian and the D’Alembertian operators, not only on functions but also on vector fields for non null hypersurfaces in the n+ 1-dimensional Lorentzian space.

In order to do that we will first give the definitions of these operators on functions in both Euclidean and Lorentzian spaces.

In the third section, we will generalize the Laplacian and the D’Alembertian on vector fields of Riemannian geometry to Lorentzian geometry, specifically of the hypersurfaces in Riemannian space to non null hypersurfaces in then+ 1- dimensional Lorentzian space, Rⁿ⁺¹₁ . We will introduce the B_n+1^k -product, from which the relationship between Laplacian and D’Alembertian derives.

In the fourth section, we will study theB_n+1^k1,...,kl-product . We will show that the B_n+1^k1,...,kl-product becomes aBn+1-congruence.

In the fifth section we will show many examples of operators on vector fields and B_n+1^k -products.

2 Preliminaries and definitions

LetRⁿbe then-dimensional Euclidean space with natural coordinatesu1,. . . ,un. In classical notation, the metric tensor is

g=gijduⁱ⊗du^j withg=diag(+1, . . . ,+1)

The Laplacian and D’Alembertian operators on functions definided onRⁿ are well known operators, defined as follows.

Definition 1. Letu1, . . . , unbe the natural cordinates inRⁿ.The differential operators

∆ = Xn

i=1

∂²

∂u²_i (1)

¤=−∂²

∂u²₁ + Xn

i=2

∂²

∂u²_i (2)

are called the Laplacian operator and the D’Alembertian operator in Rⁿ, respectively. They are defined on smooth real-valued functions onRⁿ.

Let (Rⁿ₁,g) be ann-dimensional Lorentzian space of zero curvature where the signature of gis (−,+, . . . ,+). We will indicate withh,ithe correspond- ing inner product.

In Lorentzian spaces there are three kinds of vectors: timelike, spacelike and lightlike, according to the inner product of the vector with itself is nega- tive, positive or zero, respectively.

We say that a hypersurface M in Rⁿ₁ is spacelike or timelike if at every point p∈M its tangent space Tp(M) is spacelike or timelike, that is if the normal vector is timelike or spacelike, respectively, (cf. [2] for more details).

We will call these hypersurfacesnon null hypersurfacesfrom now onwards.

ConsideringRⁿ =Rⁿ₀,we denote the set of all smooth real-valued functions onRⁿ_ν withF(Rⁿ_ν), whereν: 0,1.

It is natural then to define Laplacian and D’Alembertian operators on functions in the Lorentzian space Rⁿ₁. Some operators on functions in the Lorentzian space Rⁿ₁ are well known, (cf. [1] and [7]).

Definition 2. Letu1, . . . , unbe the natural coordinates inRⁿ₁. The differen- tial operators ∆ and ¤are given by:

∆ = Xn

i=1

εi ∂²

∂u²_i (3)

and

¤=−ε₁ ∂²

∂u²₁ + Xn

i=2

ε_i ∂²

∂u²_i, (4)

respectively, whereεi=

½ −1 if i= 1, +1 if 2≤i≤n. . Both operators are defined on functionsf ∈ F(Rⁿ₁).

According to Definition 1 and Definition 2, the Laplacian operator is de- fined by using the tensor metric of the respective structure. In some contexts, the Laplacian is defined with opposite sign and others name are used to call it (cf. [7]).

3 Relationships between the Laplacian and D’Alembertian operators

We denote the Laplacian and the D’Alembertian operators on functions inRⁿ₁ with ∆ⁿ₁ and¤ⁿ₁,and on functions inRⁿ with ∆ⁿ₀ and¤ⁿ₀,respectively.

Proposition 3. According to Definitions 1 and 2,

∆ⁿ₁ (f) =¤ⁿ₀ (f)and∆ⁿ₀(f) =¤ⁿ₁ (f).

Proof. By Definition 2,¤ⁿ₀ (f) =−^∂_∂u^2f2 1 +P_n

i=2 ∂²f

∂u²_i. By Definition 1, ∆ⁿ₁ (f) =−^∂_∂u^2f2

1 +P_n

i=2

∂²f

∂u²_i. Thus ∆ⁿ₁ (f) =¤ⁿ₀ (f).

Similarly,¤ⁿ₁ (f) =−³

−^∂_∂u^2f2 1

´ +P_n

i=2∂²f

∂u²_i =P_n

i=1∂²f

∂u²_i = ∆ⁿ₀ (f).

The Laplacian operator on vector fields for submanifolds in Riemannian manifolds is known (cf. [5]). Now, we show the Laplacian and D’Alembertian operators on vector fields for hypersurfaces in an+ 1-dimensional Lorentzian space of zero curvature,Rⁿ⁺¹₁ .

LetM be an n-dimensional non null hypersurface inRⁿ⁺¹₁ with induced connection∇.

Let

Ξ (M) =©

X :M →Rⁿ⁺¹₁ ; X is a vector field andX(p)∈Rⁿ⁺¹₁ ª and

Ξ (M) =

½

X :M → [

p∈M

Tp(M) ;X is vector field and X(p)∈Tp(M)

¾ .

We sayE1, . . . , En is a basis of Ξ (M) andEn+1 is the unit normal vector field on M if at every point p∈M, {E1(p), . . . , En(p)} is a basis ofTp(M) andEn+1(p) is the unit normal vector atp, respectively. Thus,E1, . . . , En+1

is a basis of Ξ (M). If{E1(p), . . . , En(p)} is a orthonormal basis ofTp(M) and En+1(p) is the unit normal vector at p,∀p ∈ M, E1, . . . , En+1 is an orthonormal basis of Ξ (M)

We recall the well known fact that if X ∈Ξ (M) andEi ∈ Ξ (M),then

∇EiX is vector field of Ξ (M). Consequently, ifX ∈Ξ (M) andEij ∈Ξ (M) then ∇Ei1· · · ∇E_imX ∈ Ξ (M). Thus it is possible to define the Laplacian and the D’Alembertian operators on vector fields of Ξ (M).

Definition 4. Let M be an n-dimensional non null hypersurface in Rⁿ⁺¹₁ with induced connection ∇. Let E1, . . . , En be an orthonormal basis of Ξ (M).

a) The Laplacian ∆ on vector fields of Ξ (M) is given by:

∆ = Xn

i=1

εi∇Ei∇Ei, (5)

b) The D’Alembertian¤on vector fields of Ξ (M) is given by:

¤=−ε1∇E1∇E1+ Xn

i=2

εi∇Ei∇Ei, (6) where εi=hEi, Eii, i= 1, . . . , n.

Now we introduce some notation which will be used later. Let∇¹_i1 =∇Ei1,

∇²_i1,i2 =∇Ei1∇Ei2, . . . ,∇^m_i1,...,im =∇Ei1· · · ∇E_im,where 1≤i1, . . . , im≤n and E₁, . . . , E_n+1 is basis of Ξ (M). Let F(M) be the set of all smooth real-valued functions on M. Let

P(M) =©

Q6= 0; Q=P_n

i1=1qi1∇¹_i1+· · ·+P_n

i1,...,im=1qi1,...,im∇^m_i1,...,im, wherem=m(Q)<∞andqi1, . . . , qi1,...,im ∈ F(M)ª

. We define a new application which produces a certain change of sign in some terms of the operators ofP(M). Since this application satisfies prop- erties of inner products, we shall call it “product”. We shall make use of this product when we relate the Laplacian and the D’Alembertian operators.

Definition 5. LetM be ann-dimensional non null hypersurface inRⁿ⁺¹₁ with induced connection ∇. Let E1, . . . , En+1 be an orthonormal basis of Ξ (M).

Fork: 0, . . . , n,theB_n+1^k -product is an application onP(M) toP(M) which is characterized by

¡b^k_i1,...,im¢

jt=

½ −εjt ifk∈ {i1, . . . , im}

εjt ifk /∈ {i1, . . . , im} , (7) where εjt=hEj, Eti, j, t= 1, . . . , n+ 1, and{i1, . . . , im} ⊂ {1, . . . , n}.

We denote theB_n+1^k -product withh,i^k_B

n+1. The equalityQ=P_n+1

t=1 hQ, Eti^k_Bn+1EtmeansQX=P_n+1

t=1 hQX, Eti^k_Bn+1Et

for allX ∈Ξ (M). Hence, theB^k_n+1-product is well defined.

Remark 6. TheB_n+1^k -product isF(M)-bilinear.

Remark 7. From Definition 5, if∇^m_i1,...,imX =P_n+1

j=1X_i^j_1,...,imEj then we have D

∇^m_i1,...,imX, Et

E_k

Bn+1

=P_n+1

j=1X_i^j_1,...,imhEj, Eti^k_B

n+1

=P_n+1

j=1X_i^j_1,...,im¡

b^k_i1,...,im¢

jt=X_i^t_1,...,im¡

b^k_i1,...,im¢

tt

=

½ −εttX_i^t_1,...,im ifk∈ {i1, . . . , im} εttX_i^t_1,...,im ifk /∈ {i1, . . . , im} .

The following theorem relates the Laplacian and the D’Alembertian oper- ators, which are defined in (5) and (6).

Theorem 8. LetM be ann-dimensional non null hypersurface inRⁿ⁺¹₁ with induced connection ∇. Let E1, . . . , En+1 be an orthonormal basis of Ξ (M).

Then, the Laplacian∆ and the D’Alembertian¤operators on vector fields of Ξ (M)are related by:

¤=

n+1X

t=1

h∆, Eti¹_Bn+1Et (8) and

∆ =

n+1X

t=1

h¤, Eti¹_Bn+1Et. (9) Proof. LetX ∈Ξ (M) and let ∇Ei∇EiX =P_n+1

j=1X_ii^jEj. By (5) and (6), P_n+1

t=1 h∆X, E_ti¹_B

n+1E_t=P_n+1

t=1

­P_n

i=1ε_i∇_Ei∇_EiX, E_t®₁

Bn+1E_t

=P_n+1

t=1

nP_n

i=1εi

­∇Ei∇EiX, Et

®1 Bn+1

o Et

=P_n+1

t=1

nP_n

i=1εi

P_n+1

j=1X_ii^j hEj, Eti¹_B

n+1

o Et.

From the orthonormality condition of the basis of Ξ (M),

P_n+1

t=1

nP_n

i=1εi

P_n+1

j=1 X_ii^j hEj, Eti¹_Bn+1 o

Et

=P_n+1

t=1

n

−ε1X₁₁^t hEt, Eti+P_n+1

i=2 εiX_ii^t hEt, Eti o

Et

=−P_n+1

t=1 ε1

­ P_n+1

j=1 X₁₁^j Ej, Et

®Et+P_n+1

t=1

P_n+1

i=2 εi

­ P_n+1

j=1X_ii^jEj, Et

®Et

=−ε₁P_n+1

t=1

­∇_E1∇_E1X, E_t®

E_t+P_n+1

i=2 ε_iP_n+1

t=1

­∇_Ei∇_EiX, E_t® E_t

=−ε1∇E1∇E1X+P_n+1

i=2 εi∇Ei∇EiX =¤X. Therefore,¤=P_n+1

t=1 h∆, Eti¹_Bn+1Et. Analogously, P_n+1

t=1 h¤X, Eti¹_Bn+1Et

=P_n+1

t=1

©−ε1

­∇Ei∇EiX, Et

®1

Bn+1+P_n

i=2εi

­∇Ei∇EiX, Et

®1 Bn+1

ªEt

=P_n+1

t=1

©−ε1

P_n+1

j=1X₁₁^j hEj, Eti¹_B

n+1+P_n

i=2εi

P_n+1

j=1X_ii^jhEj, Eti¹_B

n+1

ªEt

=P_n+1

t=1

© P_n

i=1εi

DP_n+1

j=1X_ii^jEj, Et

E ªEt

=P_n

i=1εi

© P_n+1

t=1

­∇Ei∇EiX, Et

®Et

ª

=P_n

i=1εi∇Ei∇EiX = ∆X.

From now onwards, we will extend Definition 5 and Theorem 8 to general, not necessary orthonormal basis. In order to do that we first define the Lapla- cian and D’Alembertian operators on vector fields whenM is an-dimensional non null hypersurface in Rⁿ⁺¹₁ . In a classical way, we denotegij =hEi, Eji, 1≤i, j≤n+ 1,and¡

g^ij¢

= (gij)⁻¹.

Definition 9. Let M be an n-dimensional non null hypersurface in Rⁿ⁺¹₁ with induced connection ∇. LetE1, . . . , En be a basis of Ξ (M).

a) The Laplacian ∆ on vector fields of Ξ (M) is given by:

∆ = Xn

i,j=1

g^ij∇Ei∇Ej. (10)

b) The D’Alembertian¤on vector fields of Ξ (M) is given by:

¤=−g¹¹∇E1∇E1− Xn

i=2

gⁱ¹¡

∇Ei∇E1+∇E1∇Ei

¢+ Xn

i,j=2

g^ij∇Ei∇Ej. (11)

Naturally, theB_n+1^k -product must also be extended to general basis.

Definition 10. LetM be an n-dimensional non null hypersurface in Rⁿ⁺¹₁ with induced connection ∇. Let E1, . . . , En+1 be an orthonormal basis of

Ξ (M). For k : 0, . . . , n, the B_n+1^k -product is an application on P(M) to P(M) which is characterized by:

¡b^k_i1,...,im¢

jt=

½ −gjt ifk∈ {i1, . . . , im}

gjt ifk /∈ {i1, . . . , im} . (12) We denote theB_n+1^k -product withh,i^k_Bn+1.

Remark 11. Sinceh,iisF(M)-bilinear, theB_n+1^k -product is F(M)-bilinear too.

Remark 12. If∇^m_i1,...,imX=P_n+1

j=1 X_i^j_1,...,imEj then we have D

∇^m_i1,...,imX, Et

E_k

Bn+1

=P_n+1

j=1X_i^j_1,...,imhEj, Eti^k_B

n+1

=P_n+1

j=1X_i^j_1,...,im¡

b^k_i1,...,im¢

jt

=

( −P_n+1

j=1gjtX_i^j_1,...,im ifk∈ {i1, . . . , im} P_n+1

j=1gjtX_i^j_1,...,im ifk /∈ {i1, . . . , im}

=





− D

∇^m_i1,...,imX, Et

E

ifk∈ {i1, . . . , im} D

∇^m_i1,...,imX, E_tE

ifk /∈ {i₁, . . . , i_m} .

Theorem 13. Let M be an n-dimensional non null hypersurface in Rⁿ⁺¹₁ with induced connection ∇. Let E1, . . . , En be a basis ofΞ (M)and letEn+1

be the unit normal vector field. Then,

¤=

n+1X

t=1

h∆, Eti¹_Bn+1Et (13) and

∆ =

n+1X

t=1

h¤, Eti¹_Bn+1Et. (14) Proof. Clearly, the Laplacian ∆ and the D’Alembertian¤are two operators ofP(M).

LetX∈Ξ (M),then∇Ei∇EjX =P_n+1

r=1X_ij^rEr,where X_ij^r =P_n+1

s=1g^sr­

∇Ei∇EjX, Es

®By (10) and (11), P_n+1

t=1 h∆X, Eti¹_Bn+1Et=P_n+1

t=1

DP_n

i,j=1g^ij∇Ei∇EjX, Et

E1 Bn+1

Et

=P_n+1

t=1

nP_n

i,j=1g^ij­

∇Ei∇EjX, Et

®1 Bn+1

o Et

=P_n+1

t=1

nP_n

i,j=1g^ijP_n+1

r=1X_ij^r hEr, Eti¹_B

n+1

o Et

=P_n+1

t=1

nP_n

i,j=1g^ijP_n+1

r=1X_ij^r ¡ b¹_ij¢

rt

o Et

=P_n+1

t,r=1

n

−P_n

j=1g^1jX_1j^rgrt−P_n

i=2gⁱ¹X_i1^rgrt+P_n

i,j=2g^ijX_ij^rgrt

o Et

=−P_n+1

t=1

nP_n

j=1g^1j­

∇E1∇EjX, Et

®+P_n

i=2gⁱ¹­

∇Ei∇E1X, Et

®oEt

+P_n+1

t=1

nP_n

i,j=2g^ij­

∇Ei∇EjX, Et

®oEt

=−P_n

j=1g^1j∇E1∇EjX−P_n

i=2gⁱ¹∇Ei∇E1X+P_n+1

t=1

P_n

i,j=2g^ij∇Ei∇EjX

=¤X.

Therefore,¤=P_n+1

t=1 h∆, Eti¹_Bn+1Et. In similar way,P_n+1

t=1 h¤X, Eji¹_B

n+1Et

=−P_n+1

t,r=1

nP_n

j=1g^1jX_1j^r ¡ b¹_1j¢

rt+P_n

i=2gⁱ¹X_i1^r ¡ b¹_i1¢

rt

o E_t +P_n+1

t,r=1

nP_n

i,j=2g^ijX_ij^r ¡ b¹_ij¢

rt

o Et

=−P_n+1

t,r=1

n

−P_n

j=1g^1jX_1j^rgrt−P_n

i=2gⁱ¹X_i1^rgrt

o Et

+P_n+1

t,r=1

nP_n

i,j=2g^ijX_ij^rgrt

o Et

=P_n+1

t=1

nP_n

i,j=1g^ij­

∇Ei∇EjX, Et

®oEt=P_n

i,j=1g^ij∇Ei∇EjX = ∆X.

Therefore, ∆ =P_n+1

t=1 h¤, Eti¹_Bn+1Et.

4 B

_n+1

-congruence

LetM be ann-dimensional non null hypersurface inRⁿ⁺¹₁ with induced con- nection ∇. From now anwards, we consider E1, . . . , En+1vector fields such that En+1(p) is the unit normal vector at p and {E1(p), . . . , En(p)} is a basis ofTp(M),at allp∈M.

Definition 14. Letk1, . . . , kl be integer numbers such that 0≤k1 <· · · <

kl≤n. TheB^k_n+1^1,...,kl-product is characterized by:

³ b^k_i1¹_,...,i^,...,k_m^l

´

jt= (−1)^cgjt, (15)

withc=|{kt; kt∈ {i1, . . . , im} and 1≤t≤l}|.

In classical way, we consider|∅|= 0. It is obvious that 0≤c≤min{l, m} ≤ n.

We denote theB_n+1^k1,...,kl-product withh,i^k_B¹_n+1^,...,kl.Sinceh,iisF(M)-bilinear, theB^k_n+1^1,...,kl-product is tooF(M)-bilinear.

IfQ∈ P(M),we denote P_n+1

t=1 hQ, Eti^k_B¹_n+1^,...,klEt withB_n+1^k1,...,kl(Q).

From Definition 14, we getB_n+1^k1,...,kl(Q) is a differential operator ofP(M). Remark 15. Let us note that ¡

b⁰_i1,...,im¢

jt =gjt, at all j, t: 1, . . . , n+ 1 and {i1, . . . , im} ⊂ {1, . . . , n}.ThusB_n+1⁰ (Q) =Q.

Definition 16. LetP, Qbe two differential operators ofP(M).We say that P isBn+1-congruent toQifP =B^k_n+1(Q).

Lemma 17. Let ¡

b^k_i1,...,im¢

uv,¡

b^h_j1,...,js¢

rt be as in (15), then

¡b^k_i1,...,im¢

uv

¡b^h_j1,...,js¢

rt=





−guvgrt ifk∈ {i1, . . . , im} ∧h /∈ {j1, . . . , js},

−guvgrt ifk /∈ {i1, . . . , im} ∧h∈ {j1, . . . , js}, guvgrt in other case.

Proof. It follows from the table:

k=i1 ... k=im k /∈ {i1, . . . , im} h=j1 (−guv) (−grt) ... (−guv) (−grt) guv(−grt) h=j2 (−guv) (−grt) ... (−guv) (−grt) guv(−grt)

... ... ... ... ...

h=js (−guv) (−grt) ... (−guv) (−grt) guv(−grt) h /∈ {j1, . . . , js} (−guv)grt ... (−guv)grt guvgrt

We denote the set of all B_n+1^k1,...,kl-products with Bn+1, where 0 ≤ k1 <

· · ·< kl≤n.

Concecutive application of products inBn+1 result in another product in Bn+1.Its proof is more dull than the idea itself. So we have developed it in steps.

Proposition 18. Let P, Q, R ∈ P(M) such that P = B^k_n+1(R) and R = B_n+1^h (Q), then

P =B^k_n+1¡

B^h_n+1(Q)¢

=





B^k,h_n+1(Q) if k < h,

Q if k=h,

B^h,k_n+1(Q) if k > h,

(16)

where 0≤k, h≤n.

Proof. We will explicitly show thatB^k_n+1¡

B^h_n+1(Q)¢ . Let Q=P_n

i1=1qi1∇¹_i1+· · ·+P_n

i1,...,im=1qi1,...,im∇^m_i1,...,im. Since P = B_n+1^k (R) = P_n+1

t=1 hR, Eti^k_B

n+1Et and R = P_n+1

j=1hQ, Eji^h_B

n+1Ej, then

P =B_n+1^k ³P_n+1

j=1hQ, Eji^h_B

n+1Ej

´

=B_n+1^k

µP_n+1

j=1

½P_n

i1=1qi1

D

∇¹_i1, Ej

E_h

Bn+1

¾ Ej

¶ +· · · +B_n+1^k

µP_n+1

j=1

½P_n

i1,...,im=1qi1,...,im

D

∇^m_i1,...,im, Ej

E_h

Bn+1

¾ Ej

¶ . Let us note that

P_n

i1=1qi1

D

∇¹_i1, Ej

E_h

Bn+1

=P_n

i1=1 i16=h

qi1

D

∇¹_i1, Ej

E

−qh

D

∇¹_h, Ej

E , P_n

i1,i2=1q_i1,i2D

∇²_i1,i2, E_jE_h

Bn+1

=P_n

i1,i2=1 i1,i26=h

q_i1,i2D

∇²_i1,i2, E_jE

−P_n

i2=1qh,i2

D

∇²_h,i2, Ej

E

−P_n

i1=1qi1,h

D

∇²_i1,h, Ej

E +qh,h

D

∇²_h,h, Ej

E , and in the same way,

P_n

i1,...,im=1q_i1,...,imD

∇^m_i1,...,im, E_jE_h

Bn+1

=P_n

i1,...,im=1 i1,...,im6=h

qi1,...,im

D

∇^m_i1,...,im, Ej

E

−P_n

i2,...,im=1qh,i2,...,im

D

∇^m_h,i2,...,im, Ej

E

−P_n

i1,i3,...,im=1qi1,h,i3,...,im

D

∇^m_{i1,h,i3,...,im}, Ej

E

− · · · −P_n

i1,...,im−1=1qi1,...,im−1,h

D

∇^m_{i1,...,im−1,h}, Ej

E

+P_n

i3,...,im=1qh,h,i3,...,im

D

∇^m_{h,h,i3,...,im}, Ej

E

+P_n

i2,i4,...,im=1qh,i2,h,i4,...,im

D

∇^m_{h,i2,h,i4,...,im}, Ej

E

+· · ·+P_n

i1,...,im−2=1qi1,...,im−2,h,h

D

∇^m_{i1,...,im−2,h,h}, Ej

E

−P_n

i4,...,im=1qh,h,h,i4,...,im

D

∇^m_{h,h,h,i4,...,im}, Ej

E

− · · · −P_n

i1,...,im−3=1qi1,...,im−3,h,h,h

D

∇^m_{i1,...,im−3,h,h,h}, Ej

E

+· · ·+ (−1)^mqh,...,h

D

∇^m_h,...,h, Ej

E . We distinguish two cases:

a) Ifk6=h,