Download DYNAMIC PROPERTIES OF THE ELECTROMAGNETIC FIELD

Électrotechnique et électroénergétique
DYNAMIC PROPERTIES OF THE ELECTROMAGNETIC FIELD
ANDREI NICOLAIDE1, AURELIU PANAITESCU2
Key words: Energy, Dynamic properties of the electromagnetic field.
In the present paper, there will be determined, in the framework of the macroscopic
Electrodynamics, the dynamic properties of the electromagnetic field in the proper
sense in bodies, a component of the physical system substance-field, as well as the
dynamic properties of the system substance-field. For the proofs, only the general
Maxwell equations and the relationships of inductions, strengths and polarizations will
be used. There is also shown that the Maxwell equations expressed in terms of the state
quantities of the field {E, B}, to which the four state quantities of the electromagnetic
state of the substance have to be added, namely {ρ, J, P, M}, permit the dynamic
characterization of the components of the substance-field system, like in the
microscopic Electrodynamics of Lorentz.
1. INTRODUCTION
One body and the electromagnetic field in which it is placed form a single
physical system. A domain bounded by a closed surface located within the body is
a part of this physical system. In the paper, we shall determine the momentum of
the electromagnetic field and the flux of the momentum of the electromagnetic
field through the surface, which bounds the domain, the electromagnetic field
being a component of the considered physical domain. In the paper, we shall also
determine the momentum and the flux of the momentum for the substance located
within the domain and being in an electromagnetic state, as well as for the whole
physical system substance-field.
The electromagnetic material properties of the body substance may be
whatever ones, the body may be at rest or in motion, and the regime of variation of
the electromagnetic field will be generally variable. No material law will be used,
in order to keep the general character of the equations of the electromagnetic field.
In the proofs, we shall use only the definitions of the state electromagnetic
quantities, the relationships between inductions, strengths and polarizations and the
local forms of the general laws of electromagnetic phenomena (the general
1
2
"Transilvania" University of Braşov, E-mail: [email protected]
"Politehnica" University of Bucharest, E-mail: [email protected]
Rev. Roum. Sci. Techn. – Électrotechn. et Énerg., 53, 3, p. 239–252, Bucarest, 2007
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Andrei Nicolaide, Aureliu Panaitescu
2
Maxwell equations, in the case of domains with continuity and smoothness, adding
the extension of Minkowski for moving media [18]).
The dynamic properties of the system substance-field may be deduced by
using the Maxwell equations, in which occur the field state quantities E, D, B, H,
and the electromagnetic state quantities (material quantities) of the body, ρv, J, P,
M. The electromagnetic field is characterized by two state quantities, E and B.
The primitive electromagnetic state quantities of bodies: q, p, i, m or their
volume densities, ρv, P, J, M have been introduced from the mechanical
(ponderomotive) actions exerted in vacuo by stationary electric or magnetic fields,
upon certain small proof bodies (test bodies, check bodies) being in
electromagnetic state (manifested by electric charge, electric polarization,
electrokinetic condition, magnetization) also stationary [6, 8, 13, 16, 17]. We recall
that these forces, as shown in the previous paper [19], are expressed as follows:
F = qE;
f dV = ρv dV E ;
F = grad( p ⋅ E ) ;
↑
d F = id l × B;
F = grad(m ⋅ B ) ;
(1 a, b)
f d V = grad( P d V ⋅ E ) ;
↑
f d V = J d V × B;
(3 a, b)
f d V = grad( M d V ⋅ B ) ;
↑
(2 a, b)
↑
(4 a, b)
or also:
F = ( p ⋅ ∇) E v ,
(5)
F = (m ⋅ ∇) Bv .
(6)
In formulae above, the arrow indicates the quantity that is to be differentiated, as
mentioned in Appendix.
It is useful to add the following consideration. As the volume element around
any point tends to zero, at that point, the influence of the field produced by the
polarization of that element, on the resultant field may be neglected. The case
differs from that of a molecule of finite dimensions. For this reason, the field state
quantities to which the volume element is submitted will be considered as being the
mean values (average values). For this reason, the field state quantities of the
relations (1b, 2b, 3b, 4b) have not any subscript designing an external field or a
field in vacuo.
The volume densities of the preceding forces are:
f = ρ v E ; f = grad( P ⋅ E ) ; f = grad( M ⋅ B ) ; f = J × B ; (7 a, …, d)
↑
↑
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Dynamic properties of the electromagnetic field
241
or also:
f = ρv E ;
f = (P ⋅ ∇ )E ;
f = (M ⋅ ∇ )B ;
f = J × µ 0 H . (8 a, …, d)
In electrostatic field, the forces of (7 b) and (8 b) are equal. Also, in
magnetostatic field, the forces of (7 c) and (8 c) are equal, but only if the
considered body is in vacuo. An analysis of the expressions of the mentioned type
is not given in literature, but it could be carried out starting from references: [3, p.
192–193, 490–491], [7, p. 258], [17, p. 30, 92, 101, 107], [12], [14].
It is worth noting that Maxwell introduced certain expressions of local forces,
partially similar to expressions above. Also, for more precision, it should be
mentioned that the set of equations established by Maxwell, as referred in [10, vol.
II, p. 180], contains 12 vector equations (called by him equations in quaternion
form) designated in his work by capital letters from (A) to (L). These equations
include 9 general equations, 3 material equations, to which there are also added 2
force equations. The expressions of (7) and (8) have been introduced subsequently.
The density of the polarization electric charges and the densities of the
amperian electric currents are [19]:
ρ pv = − div P ; ρ ps = − div s P ;
J m = curl M ; J ms = curl s M .
(9 a, …, d)
Like in the previous paper [19], we shall consider any closed surface
bounding a domain with continuity within any body, situated in an electromagnetic
field.
2. THE ELECTRIC FORCE ACTING ON A DIELECTRIC BODY
SITUATED IN AN ELECTROSTATIC FIELD
In the macroscopic theory, the mechanical actions on the substance of a
volume element dV are produced either by the field, i.e., by ponderomotive actions
(like the action of the gravitational field), or by direct contact with the
neighbourhood volume elements (e.g., pressure force, elastic stress). If a body
situated in vacuo has on its external surface an electric charge, the electric field
acts on the surface elements dS, with a force that is transmitted by contiguity to all
elements of the body, by action substance-substance. Therefore, a volume element
dV, of the inside of the body, is submitted to a force even if the substance of the
considered volume dV is not in an electromagnetic state. The forces produced by
the electromagnetic causes acting on any volume element can be calculated using
the relationships of the macroscopic Electromagnetic Field Theory, but the
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Andrei Nicolaide, Aureliu Panaitescu
4
resulting distribution of forces have to be determined by using the relationships of
the Theory of Plasticity.
Let us consider any dielectric body with linear electric properties, situated in
vacuo, in an electrostatic field. Let the body be not charged with electric charge.
The external surface of the body will be denoted by Σext. The surface density of the
polarization electric charge on this surface will be ρ ps = P ⋅ n . Let Σ be a closed
surface, situated in the interior of the dielectric body in a domain with continuity.
The polarization electric charge is distributed only in the volume bounded by this
surface, with the volume density ρ pv = − div P . No surface polarization electric
charges exist, the surface being in a domain with continuity.
It is necessary to take into account that in the electrostatic field curl E = 0
and hence ( P ⋅ ∇) E = grad( P ⋅ E ) .
↑
Further on, several vector transformations will occur. There are several
manners to perform them. For instance, it is possible to use the theorems of
Appendix. Also, it is possible to use the basic transformation formula of a closed
surface integral into a volume integral and multiply the relation by any constant
vector that will be renounced, by simplifying, after performing the calculations,
[10, vol. I, p. 151].
The force exerted by the electrostatic field on the substance of the inside of
the surface Σ will be:
Fe = ∫ ρ pv E d V =
VΣ
∫ (− div P )E d V =
VΣ
( )
= − ∫ ( P ⋅ n )E d S + ∫ grad P ⋅ E d V .
Σ
(10)
↑
VΣ
The formula considered as valid for this case, in agreement with the
measured values, is
∫
( )
Fe = grad P ⋅ E d V .
VΣ
↑
(11)
The surface Σ has two faces, Σ+ outside, in contact with the substance of the
outside the domain, with the normal n + = −n , and Σ− inside, in contact with the
substance of the domain, with the normal n − = n . On the two faces of the surface
element dS, there are two surface polarization electric charges, P ⋅ n d S on Σ− and
− P ⋅ n d S on Σ+. The electric field acting on these charges and the resulting force
will be zero. The total force acting on the substance of the domain bounded by the
surface Σ− will be:
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Dynamic properties of the electromagnetic field
243
Fe = ∫ ρ pv E d V + ∫ ρ ps E d S =
Σ−
VΣ
=
∫ (− div P )E d V + ∫ (P ⋅ n)E d S = ∫ grad(P ⋅ E↑ )d V .
Σ−
VΣ
(12)
VΣ
Therefore, the calculated force using the proposed model, with surface
charges on both faces of the surface, is valid. Under certain conditions, formula
(12) can be brought to a form corresponding to known relations [10, vol. I, p. 270].
3. THE MAGNETIC FORCE ACTING ON A BODY SITUATED
IN STATIONARY MAGNETIC FIELD (THE EFFECT OF AMPERIAN
ELECTRIC CURRENTS)
From the study of the magnetization state of a body the following results
have been established: in stationary regime, the magnetization state of a
homogeneous body, with magnetic permeability µ, situated in vacuo, is equivalent
with the magnetic state associated to a distribution of amperian currents within the
volume of the body, with the volume density J m = curl M and to a distribution on
the external surface of the body Σext, with the density J ms = curl s M = M × n . It
could be added that the amperian currents of the calculation model, like the
polarization electric charges, are considered as distributed in vacuo, and the
presence of these currents manifest by producing the permeability µ of that body.
The force by which an exterior and non-uniform magnetic field acts on a
portion of any magnetized body bounded by the surface Σ taken inside the body,
hence having continuity, may be calculated by formulae above (4 a, b):
( )
FVΣ = ∫ grad M ⋅B d V .
VΣ
↑
(13)
The magnetic force can also be calculated using the Laplace formula:
FVΣ =
∫ (curl M ) × B d V ,
VΣ
(14)
where J m = curl M represents the amperian (molecular), macroscopic electric
current. We shall look for the relation that exists between expressions (13) and
(14).
The integrand of formula (13) may be expanded as follows:
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Andrei Nicolaide, Aureliu Panaitescu
(
( )
)
grad M ⋅ B = grad( M ⋅ B ) − grad M ⋅ B =
↑
↑
= grad(M ⋅ B ) − (B ⋅ ∇ )M − B × curl M .
6
(15)
Also, taking into account that div B = 0 , we shall obtain:
∫ (B ⋅ ∇ )M dV = ∫ (B ⋅ n)M d S .
Σ
VΣ
Therefore:
( )
(16)
FVΣ = grad M ⋅B d V = grad(M ⋅ B ) d V −
∫
↑
VΣ
−
∫
VΣ
∫ (B ⋅ n)M d S − ∫ B × curl M d V ,
Σ
(17)
VΣ
where:
n (B ⋅ M ) − M (B ⋅ n ) = B × (n × M ) = (M × n ) × B .
(17 a)
The relation (17) can be rewritten as:
∫
(
)
FVΣ = grad M ⋅ B d V =
VΣ
↑
∫ (M × n) × B d S + ∫ curl M × B dV ,
Σ
VΣ
(18)
which represents the relation we have looked for.
4. FUNDAMENTAL IDENTITIES IN ELECTROMAGNETISM
As already shown in the previous paper [19], starting from the Maxwell
equations, two identities may be derived, the energetic and the dynamic one,
respectively, which are very important in the study of electromagnetic phenomena.
In the previous paper, the energetic identity has been examined, further on, we
shall examine the dynamic identity.
Like in paper [19], the following symbols, in the extended sense, will be used
in the equations of the electromagnetic field (a form of the Maxwell equations):
ρ e = ρ v + ρ pv = div D − div P = div(ε 0 E ) ,
J e = J + J m + J p = J + curl M +
∂P
.
∂t
(19)
(20)
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Dynamic properties of the electromagnetic field
245
Therefore, the general equations of the macroscopic Electrodynamics, denoted by
(a) will have the same form as the equations of the microscopic Electrodynamics,
denoted by (b):
∂B
∂B
; curl E = −
;
∂t
∂t
curl E = −
curl H = J +
∂D
∂E
1
; curl B = J e + ε 0
;
∂t
µ0
∂t
div E =
ρe
ρ
; div E = e ;
ε0
ε0
div B = 0 ; div B = 0 .
(21 a, b)
(22 a, b)
(23 a, b)
(24 a, b)
In the classical microscopic Electrodynamics (the theory of electrons), in
which the microscopic quantities are used, the Lorentz equations, as shown in [19],
with certain slight modifications of symbols, are:
curl e = −
curl
∂b
,
∂t
∂e
b
= j + ε0
,
µ0
∂t
(25)
(26)
ρv
,
ε0
(27)
div b = 0 ,
(28)
dive =
to which the Lorentz law of ponderomotive action has to be added [19]:
f = ρv e + j × b .
(29)
Further on, there will be shown that if beside the general equations of the
electromagnetic field, (21)–(24), two densities of the forces acting on the substance
will be used, namely a volume density and a surface density, then the theorem of
the conservation of the momentum of the macroscopic Electrodynamics will have
the same form as the corresponding theorem of the microscopic theory of
Electrodynamics, i.e., (63) and (65).
As already shown in paper [19], if we consider a closed surface Σ within a
domain with continuity of the physical system substance-field, then on this surface
with two faces Σ− and Σ+, the single densities that could exist are the surface
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Andrei Nicolaide, Aureliu Panaitescu
8
densities of polarization electric charges and the densities of the amperian electric
currents. This occurs because on the surface Σ, situated within a domain with
continuity, the following relations are fulfilled:
divs D = 0 ; curls H = 0 .
(30 a, b)
The mentioned densities are equal in absolute value and have opposite signs,
and are given by relations:
ρ s Σ − = ρ ps = P ⋅ n ; J s Σ − = J ms = M × n .
(31 a, b)
Therefore, like in [19], there follows the two force densities:


∂P
f v = ρ e E + J e × B = [div(ε 0 E )]E +  J +
+ curl M  × B,
∂t


f s = ρ s Σ − E + J sΣ − × B = ( P ⋅ n) E + ( M × n) × B.
(32 a, b)
5. THE FUNDAMENTAL DYNAMIC IDENTITY
5.1. RELATIONS OF VOLUME AND SURFACE FORCES
The fundamental dynamic identity may be derived from the general equations
of the electromagnetic field (Maxwell equations), expressed in terms of vectors E
and B. For obtaining this, the vector product of both sides of equations (21 a) and
(22 a) will be performed by vectors ε0E and B, respectively, and then the two
equations will be added side by side. There follows:

(curl E ) × ε 0 E +  curl

∂P 
1 
× B +
B  × B =  J + curl M +
µ0 
∂ t 


∂ε E
∂B
+ 0 ×B−
× ε0 E .
∂t
∂t
(33)
We shall put the last equation in a convenient form, for emphasizing the
forces, which include gradient operators. We can write the relations:
grad ε 0 E ⋅ E  − (ε 0 E ⋅ ∇ )E = ε 0 E × curl E ,
↑ 

(34 a)
 1
  1

1
grad B ⋅ B  −  B ⋅ ∇  B =
B × curl B .
↑
µ0
 µ0
  µ0

(34 b)
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Dynamic properties of the electromagnetic field
247
Also, from relation (19), we get:
div(ε 0 E + P )E = ρ v E ,
(35 a)
[div(ε0 E )]E = ρv E − E div P .
(35 b)
or:
Replacing in equation (33) the first term by expression (34 a), the second by
(34 b), and summing the obtained equation and the equation (35 a), side by side,
we shall obtain:
 1

B ⋅ ∇  B −
  µ0

(ε0 E ⋅ ∇ )E + (div ε0 E )E − grad ε 0 E ⋅ E↑  + 

 1

− grad B ⋅ B  =
↑
 µ0


∂P 
× B +
= ρv E + (− div P )E +  J + curl M +
∂ t 

∂
+ (ε 0 E × B );
∂t
1
grad ε 0 E ⋅ E  = grad(ε 0 E ⋅ E );
↑ 

2

 1
 1
 1
grad B ⋅ B  = grad B ⋅ B  .
↑

 µ0
 2
 µ0
(36)
(36 a)
In the right-hand side of identity (36), there occurs the force acting on the
substance and produced by the sources of the electromagnetic field, distributed in
volume. Like previously, after integrating over the considered domain, there is to
be added the force acting on the surface Σ− and due to the surface polarization
electric charges and to the surface magnetization current sheets:
FΣ = ∫ [(P ⋅ n )E + (M × n ) × B ]d S .
Σ
(37)
By integrating both sides of relation (36) over the volume VΣ, and
transforming the volume integral of the left-hand side into a surface integral, then
adding expression (37) to both sides, and using relations (A.1)–(A.3), we shall
obtain the following relationship:
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Andrei Nicolaide, Aureliu Panaitescu
10
⌠

1
1
1
2
B 2  d S +
 (ε 0 E ⋅ n )E + (B ⋅ n ) B − n ε 0 E +
2 µ0
µ0

2
⌡ 
Σ
+
∫ (P ⋅ n)E d S + ∫ (M × n)× B d S =
Σ
Σ
⌠

∂P 

=   ρ v + ρ pv E +  J + curl M +
 × B dV +
∂t 

⌡

(
)
(38)
VΣ
+
⌠ ∂
∫ (P ⋅ n)E d S + ∫ ( M × n) × B d S + ⌡ ∂t (ε E × B )d V ;
Σ
Σ
0
VΣ
∫ (B ⋅ ∇ )B d V = ∫ B B ⋅ d S − ∫ B div B d V ;
VΣ
Σ
VΣ
(38 a)
ρ pv E = (− div P )E ; div B = 0 .
In relation (38), we shall replace the volume density of the polarization
electric charge. We can use the following relation:
∫ E (P ⋅ d S ) − ∫ (div P )E d V = ∫ grad(P ⋅ E↑ )d V −
Σ
VΣ
VΣ
−
∫ P × curl E d V .
(39)
VΣ
Also, we shall replace in relation (38) the term containing a curl, by using the
following relations:
B × curl M = grad M ⋅ B − (B ⋅ ∇ )M = grad( M ⋅ B ) −
( )
− grad(M ⋅ B )− (B ⋅ ∇ )M ,
↑
(40 a)
↑
∫ (B ⋅ ∇ )M d V = ∫ M (B ⋅ n) d S − ∫ M div B d V ;
Σ
VΣ
div B = 0 .
VΣ
(40 b)
Therefore, we shall get:
− B × curl M d V = − grad(M ⋅ B ) d V +
∫
VΣ
(
∫
VΣ
)
+ grad M ⋅ B d V + M (B ⋅ n ) d S .
∫
VΣ
↑
∫
Σ
(41)
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Dynamic properties of the electromagnetic field
249
By replacing expressions (39) and (41) into relation (38), and taking also into
account relation (21 b), after reducing the like terms, we shall get:
⌠

1
1
1
B − n  ε 0 E 2 +
B 2  d S +
 (ε 0 E ⋅ n )E + (B ⋅ n )
2 µ0
µ0

2
⌡ 
Σ
+ ∫ ( P ⋅ n) E d S + ∫ ( M × n ) × B d S =
Σ
( )
Σ
(
)
⌠

∂
=  ρ v E + J × B + grad P ⋅ E + grad M ⋅ B + (P × B ) d V +
↑
↑
∂t
⌡

(42)
VΣ
⌠ ∂
+  (ε 0 E × B ) d V .
⌡ ∂t
VΣ
Further, in order to get a certain symmetrical form, we shall write:
grad P ⋅ E  = grad P ⋅ E  ,
↑ 
↑ 


grad P ⋅ E  = grad( P ⋅ E ) − grad P ⋅ E  ,
↑ 
↑ 

(43 a, b)
where both sides of relation (43 a) are identical. By subtracting equations (43 a)
and (43 b) side by side, we shall obtain:
1
1
grad P ⋅ E  = grad P ⋅ E  − grad P ⋅ E  + grad( P ⋅ E ),


↑ 
↑
↑





 2
2
(44 a)
and similarly for magnetic quantities:
1
1
grad M ⋅ B  = grad M ⋅ B  − grad M ⋅ B  + grad(M ⋅ B ),
↑ 
↑ 


 ↑
 2
2 
(44 b)
and if D = εE , B = µH , then:
1
1
grad P ⋅ E  − grad P ⋅ E  = − E 2 grad ε ,
↑ 

 ↑ 
2 
2
1
1
grad M ⋅ B  − grad M ⋅ B  = − B 2 grad µ .


↑ 
 ↑


2
2
(45 a, b)
By replacing expressions (44 a) and (44 b), into (42), and grouping the terms,
we shall obtain the expression:
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Andrei Nicolaide, Aureliu Panaitescu
12
⌠
1
 
1
1
B − n  ε 0 E 2 +
B 2  + 
 (ε 0 E ⋅ n )E + (B ⋅ n )
µ0
2 µ0
2
 



1
1

dS =
 + (P ⋅ n )E − (B ⋅ n )M − n  P ⋅ E + M ⋅ B  +
2

2




 + n (B ⋅ M )





⌡

Σ
[ ( ) ( )]
) ( )]
1
⌠

− grad P ⋅ E + 
 ρ v E + J × B + 2 grad P ⋅ E
↑
↑
 dV +
=
 + 1 grad M ⋅ B − grad M ⋅ B + ∂ ( P × B ) 
 2

↑
↑
∂t
⌡
[ (
(46)
VΣ
⌠ ∂
+  (ε 0 E × B ) d V .
⌡ ∂t
VΣ
By grouping the terms in relation (46), there follows:
⌠
1
1

 (D ⋅ n )E + (B ⋅ n )H − n  2 D ⋅ E + 2 B ⋅ H  d S =


⌡
Σ
⌠
1

1
 
 − grad D⋅ E  + 
 ρ v E + J × B + grad 2 D ⋅ E
↑ 

2 ↑  
dV +
=

1

1
 
+ grad B ⋅ H  − grad B⋅ H  

↑ 
2
2 ↑
 
⌡
(47)
VΣ
⌠ ∂
+  (D × B ) d V .
⌡ ∂t
VΣ
Remark. It is worth noting that the two transformations (44 a, b), which
contain terms similar to those of the Minkowski formulae, are not casual, but have
the aim to obtain the expression of the right-hand side, which represent the single
formula considered as experimentally valid for the volume density of the force
acting on polarized or magnetized isotropic substances at rest [10, vol. I, p. 157].
By using the expression of the right-hand side of relation (42) or (46), for the
same body in vacuo, we shall obtain the same result. However, for calculating the
volume density of the force, without other precaution (e.g., adding some terms), we
13
Dynamic properties of the electromagnetic field
251
shall obtain correct results only by using the integrand of the right-hand side of
(46), because, as mentioned, only in this case the experimental results are satisfied.
It is to be mentioned that relation (47) could be derived directly from the
general equations of the electromagnetic field (Maxwell equations) written in terms
of the vectors of the set {E, B, D, H}, namely by using equations (21 a) and (22 a),
correspondingly multiplied with factors D and B, and added side by side, and then
by performing calculations similar to those of Sub-section 5.1.
Therefore, we have derived two relations concerning the electromagnetic
forces acting upon a domain bounded by a closed surface Σ, namely (46) and (47),
corresponding to both manners of writing the general equations of the
electromagnetic field, i.e., the Maxwell equations, using either the set {E, B} or
{E, B, D, H}, [1, 4, 5, 11], [13–18]. These developments are not given in literature.
APPENDIX
Transformations of Gauss and Ostrogradski type and other relations,
[9, p. 120], [10, p. 474], given below, may be used:
∫ [(a ⋅ ∇ )b + (div a )b]d V = ∫ (a ⋅ n) b d S = ∫ b (a ⋅ d S ),
(A.1)
∫ grad f d V = ∫ f n d S ,
(A.2)
Σ
VΣ
VΣ
Σ
Σ
( )
1
grad a 2 ,
(A.3)
2
where a and b are two field vectors, and f a scalar function, all being continuous
and differentiable. In several papers, parentheses containing terms with arrow
subscript or with the subscript const, having before them a differential operator,
like grad, are used. The letter with arrow subscript will be differentiated, while the
other will be kept as constant as performing that operation. The letters with
subscript const will be kept constant. These parentheses very useful for expressing
certain differential operations have not to be used in integral transformations before
expanding the parentheses, in order to avoid errors.
grad a ⋅ a = grad(a const ⋅ a ) = grad(a c ⋅ a ) =
↑
Received on 18 December, 2007
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