Journal of Electromagnetic Analysis and Applications, 2012, 4, 400-409
http://dx.doi.org/10.4236/jemaa.2012.410055 Published Online October 2012 (http://www.SciRP.org/journal/jemaa)
Generalization of the Second Order Vector Potential
Formulation for Arbitrary Non-Orthogonal Curvilinear
Coordinates Systems from the Covariant Form of
Maxwell’s Equations
Denis Prémel
CEA, LIST, F-91191 Gif sur Yvette CEDEX, France.
Received July 18th, 2012; revised August 17th, 2012; accepted August 29th, 2012
ABSTRACT
A great number of semi-analytical models, notably the representation of electromagnetic fields by integral equations are
based on the second order vector potential (SOVP) formalism which introduces two scalar potentials in order to obtain
analytical expressions of the electromagnetic fields from the two potentials. However, the scalar decomposition is often
known for canonical coordinate systems. This paper aims in introducing a specific SOVP formulation dedicated to arbi-
trary non-orthogonal curvilinear coordinates systems. The electromagnetic field representation which is derived in this
paper constitutes the key stone for the development of semi-analytical models for solving some eddy currents modelling
problems and electromagnetic radiation problems considering at least two homogeneous media separated by a rough
interface. This SOVP formulation is derived from the tensor formalism and Maxwell’s equations written in a non-or-
thogonal coordinates system adapted to a surface characterized by a 2D arbitrary aperiodic profile.
Keywords: Second Order Vector Potential (SOVP); Curvilinear Coordinate System; Eddy Current Non-Destructive
Testing (ECNDT)
1. Introduction
A great number of semi-analytical models for simulating
Eddy Current Non-Destructive Testing (ECNDT) of con-
ductive test pieces have been developed since the re-
search works of several pioneers [1-7]. Most of canonical
ECNDT configurations can be simulated today by nu-
merically implementing some closed-form expression of
the solution of the forward problem to be solved [8,9].
Most of these semi-analytical models lead to fast nu-
merical models and are thus very useful for running ana-
lysis or parametric studies. Most of these semi-analyti-
cal models are based on the scalar decomposition of the
electromagnetic field in some specific curvilinear ortho-
gonal coordinate systems. The numerical models coming
from these analytical models are therefore limited to ca-
nonical geometries. The purpose of this paper is to pre-
sent a more general scalar potential representation of the
electromagnetic field which can be applied for a non or-
thogonal coordinate system in order to prepare the de-
velopment of some semi-analytical model which has the
capability to compute the quasi-static electromagnetic
fields due to an eddy current probe scanning a conduct-
ing half-space. The shape of the boundary surface is
complex since two regions are separated by a rough sur-
face. The framework of this project aims in generalizing
the previous work [10] which has been firstly introduced
for 2D eddy current problems.
The scalarization of the electromagnetic field is well
known to researchers for a long time [11] and it has been
extensively used for different applications in electro-
magnetism such as the radiation and scattering theory
[12,13] and in the analysis of eddy currents [14]. A great
number of authors have made use of the scalar potential
formulation of the electromagnetic field since it is the
starting point which allows to derive some analytical
expressions of the field arising from specific canonical
geometries implying an arbitrary time harmonic current
above a conducting half space [3,7], or above a slab of
finite thickness [4]. This scalar formulation, also called
the second order vector potential formulation has been
also used in the cylindrical coordinate system for an ar-
bitrary current source inside a borehole [15,16] or in the
spherical coordinate system [17,18]. This paper concerns
more particularly the calculation of the electromagnetic

Generalization of the Second Order Vector Potential Formulation for Arbitrary Non-Orthogonal
Curvilinear Coordinates Systems from the Covariant Form of Maxwell’s Equations
401
field in the quasi-static limit when the geometry of the
separating surface between two homogeneous media is
described by an arbitrary non-orthogonal coordinate sys-
tem.
This paper is organized as follows. In Section 2, the
second-order vector potential formulation is briefly in-
troduced. Since a Laplacian operator is only applied on a
scalar potential, this formulation can minimize the num-
ber of unknowns and consequently the computer storage
when calculating the electromagnetic fields in 3D scat-
tering electromagnetic problems and 3D eddy currents
problems. However, this formulation has been investi-
gated for a finite number of orthogonal coordinates sys-
tems. The third section describes an extended formula-
tion based on the covariant form of Maxwell’s equations.
The curvilinear coordinate method is summarized for
obtaining the relationship between the components of the
electromagnetic field and two longitudinal components.
In Section 4, the second-order vector formulation is finally
derived and some examples of different coordinate sys-
tem are given in Section 5 in order to give some illustra-
tion of the formulas. Finally, Section 6 gives the conclu-
sion and future works.
2. The Second Order Vector Potential
Formulation for Eddy Current Problems
In curvilinear coordinate systems, the components of the
magnetic vector potential cannot be separated due to the
coupling between them [12]. Thus, the problem of the
computation of the vector field leads to a great number of
coupled unknowns since it is not possible to obtain sepa-
rable Helmholtz equations. In order to overcome this
drawback, it is usual to split the vector field into a longi-
tudinal part and a transversal part. The longitudinal part
is obtained from the calculation of the gradient of a scalar
potential, this part is irrotational (rotational free). The
transversal part is derived from another vector potential;
this part is called solenoidal (divergence free).
L
T
AA A
with 0
LL

 AA
2
(1)
12
with 0
TTT T
 AAAA (2)
This vector potential can also derived from two other
scalar potentials and a fixed unit vector judiciously chosen
[11] according to the coordinate system used. The second
order vector potential results in the separation of the
Helmholtz equation in several coordinate systems [19].

1TWuWuW A (3)
According to this decomposition, the magnetic vector
potential depends on three scalar quantities 1 ,W

and
2. Since the magnetic vector potential W
A
is derived
from the curl of W, this implies the coulomb gauge.
This vector potential is also called the Second Order
Vector Potential (SOVP). The longitudinal part of the
electromagnetic field is not necessary for representing
the magnetic flux density since the definition of


B
A and


0 

 . Only two scalar potential
are finally necessary to represent the magnetic flux den-
sity


B
W and the current density can be
derived by 1



J
B.
In such a way, this formulation has been extensively
used to derive some analytical solutions for eddy current
modeling problems. The goal of this paper is to genera-
lize this formulation for a non-orthogonal curvilinear
coordinates system which could describe the arbitrary
shape of a separating rough surface between two regions
of the space.
3. Problem Formulation
Let us consider an isotropic, homogeneous conducting
half-space characterized by its conductivity

and its
magnetic permeability

and the permittivity

. The
global planar surface delimiting this half-space is locally
corrugated according to a cylindrical surface profile. In a
Cartesian coordinate system

,,

x
yz
z

, the surface is as-
sumed to be invariant along the axis and is described
by a parametric function . Eddy currents are in-
duced in the conducting region due to 3D arbitrary cur-
rent sources in air above the half-space. The goal of this
paper is to introduce a generalized scalar potentials for-
malism to build up a fast semi-analytical model which is
able to compute the electromagnetic fields considering
the quasi-static regime. This formalism is based on the
introduction of a curvilinear coordinate system which
conforms the rough surface. Writing the boundary condi-
tions is thus easier since one of the new system of curvi-
linear coordinates is set to zero for each point on the
boundary surface. Moreover, the tangential components
and the normal components of the electromagnetic field
are easily written by using the covariant and contravari-
ant basis respectively. In the next section, the covariant
form of Maxwell’s equations governing the electromag-
netic field are summarized and discussed.

ax
3.1. The Covariant Form of Maxwell’s Equations
In curvilinear coordinates systems, the Maxwell’s equa-
tion are based on the tensorial formalism. By assuming a
standard time dependence


it 
e, the covariant form is
given by:

Generalization of the Second Order Vector Potential Formulation for Arbitrary Non-Orthogonal
Curvilinear Coordinates Systems from the Covariant Form of Maxwell’s Equations
402


0
0no charge sources
with,,1, 2,3
j
j
j
j
i
ijk i
jk
i
ijkii ii
jk s
B
D
B
EiBijk
t
D
HJ JJiD
t





 


 


(4)
where ijk

stands for the Lévi-Cività indicator [20] and
are the indices associated to the compo-
nents of the fields on the the three coordinate axes. The
notation
, ,3ijk1, 2,
j
 means
j
x
 . This formalism is invariant
to a change of referential. The components and
k
Ek
H
are the covariant components of the vectors
E
and
H
.
The components ,
i
Bi
J
and are the contravariant
components of the vectors ,
i
D
B
J
and respectively.
They are themselves related to the covariant components
of the vectors
D
E
and
H
by the constitutive relations:
i iijij
j
j
i iijij
j
j
i iijij
jj
BHggHH
J
EggEE
DEggE E
 
 
 
 
 
 
(5)
where ij
g
are the contravariant components of the met-
ric tensor of the coordinate system. The covariant com-
ponents of the metric tensor verify the condition
ik kj
gg ij

 and

det ij
g
g. The pseudo-tensors
ij

, ij

and ij

depend on the choice of the metric
system and they contain the physical and geometrical
information of the problem. In the quasi-static regime,
the permittivity of the conducting material is neglected
and the wave number becomes 2=
c
ki

 . To exhibit a
symmetry in the Maxwell’s equation, let us introduce
some notations. The complex impedance of the material
is defined so that c
kZ

  and c
Z
ik

 . By de-
noting
j
j
ZHGi, Maxwell’s equations may be simply
written by:
0
0
ij
c
ij ij
sc
kgg
iZ ggkgg
 
 
 
 
E
B
EG
GJ E
(6)
The dot product and the cross product are used exactly
as in the Cartesian case.
3.2. Boundary Conditions
The main interest of this approach based on the coordi-
nate transformation is that boundary conditions may be
written in an analytical form since the boundary surface
conforms exactly with the curvilinear coordinate system.
The new system of coordinates is chosen so that one
variable, at least, is constant for each point conforming
the interface between the two media. Let us consider the
coordinates


123
,,
x
xx
u
and let u a field vector. The
boundary surface is defined by for instance.
Then, the continuity of the tangential components of the
field vector may be expressed by the continuity of
the covariant components while the continuity of
the normal component of may be translated in the
continuity of the contravariant component . These
two conditions will be explicitly described in the follow-
ing of this paper. The main goal of this paper is to define
two scalar potentials, usually the transverse electric and
the transverse magnetic potentials for solving 3D eddy
currents problems by using the curvilinear coordinate
method.
30x
21
,uu
u
3
u
3.3. The Curvilinear Coordinate Method
The boundary surface separating two isotropic homoge-
neous regions may be described by some parametric
equations. Starting from the Cartesian coordinate system


,,
x
yz , we are looking for a new coordinate system
such that the boundary surface conforms with one sur-
face of coordinates. The Cartesian coordinates are la-
beled using index notation so that



123
,,, ,
x
yzxxx.
Now, let us define a set of curvilinear coordinates


312
,,


:


23


11
212
312
,,
,,
,,
xf
xg
xh

3
3









(7)
Two natural sets of base vectors are associated with a
curvilinear system: the covariant vectors i that are
tangent to the coordinate lines and the contravariant vec-
tors that are normal to the coordinate surfaces.
u
i
u
The Jacobian matrix
J
defines the transformation
from the Cartesian coordinates

123
,,

x
xx to the curvi-
linear coordinates


12 3
,,

 :
123
123
123
.
f
ff
x
ggg
J
hhh











 


















(8)
In the following of the paper, let us denote ii
F
F




.
The matrix representation of the covariant components of
the metric tensor is given by:
T
ij
x
x
g



 





 




 
(9)

Generalization of the Second Order Vector Potential Formulation for Arbitrary Non-Orthogonal
Curvilinear Coordinates Systems from the Covariant Form of Maxwell’s Equations
403
and we denote by
g
the determinant ij
g
g. Simi-
larly, the matrix representation of the contravariant
components of the metric tensor are given by:
Let us consider a compact form:
nij
nc t
nij t
cn
nijn
tc n
nijn n
ct
kgg
kgg
kgg E
G
kgg













  


Ix E
G
xI
xx
xx
(17)
T
ij
g
x
x






 

(10)
Assume that this matrix may be expressed by:
where means an identity dyadic operator. The left-
hand side of (17) may be multiplied by the matrix:
I
11 12 13
21 22 23
31 32 33
ij
g
gg
g
ggg
g
gg








(11)
ni
nc
nij
cn
kgg
kgg
j





 




Ix
xI
(18)
Due to the relationship between these two representa-
tions of the metric tensors , the deter-
1
ij
ij
gg



 such as:
minant of the matrix ij
g


is equal to 1
g
and (see the
formula below):
11 12
12 11
nij
nc
nij
cn
nij
nc t
nij t
cn
ct
ct
kgg
kgg
kgg
kgg
CkC
kC C



 







 
 




Ix
xI
Ix
E
G
xI
E
G
(19)
To fit boundary conditions, it is convenient to split the
electromagnetic field into two components:
tu
tu
E
G


E
Eu
GG u
(13)
Any vector may be chosen for this decomposition
though there are suitable choices [11]. In what follows in
the paper, is a vector in the directions of the coordi-
nates .
u
,1,
u
n2,
n
x
with


2
11
12
nijnij
nn c
nijnij
nn
Ck gggg
Cgggg


 



 

Ix x
xx
(20)
3
3.4. Decomposition Transversal/Longitudinal
Components So, in the general case, the transverse fields t
E
and
t are coupled. Similarly, the right side of (17) is mul-
tiplied by the same matrix (18):
G
Let us consider the longitudinal component u and u
and the transversal fields
E G
t
E
and . The nabla opera-
tor can be decomposed:
t
G
11 12
12 11
nij
nc
nij
cn
nijn
tc n
nijnn
ct
n
n
kgg
kgg
kgg E
G
kgg
EMM
GMM



  








  



 


 
Ix
xI
xx
xx
(21)



nn
tntn
nn
ttt nn
E
E
 
 
ExEx
t
E
xx E
(14)
By applying the operator to this equation and
taking into account Equation (6), one obtains:
nx

n
tn nt
nij nij
ctc
E
kggkggG
  
 
xE E
xGxn
n
x
(15)
with
The same equation may be applied to the field vector
by substituting G
E
B and . These cou-
pled equations may be expressed:
n
EG



2
11
12
nijnij
nt c
nij nijn
ctn
Mk gggg
M kgggg
 
 
xx
xx
n
x
x
(22)
n
nijn nij
tn cnntct
nijn nij
tn cnntct
E kggGkgg
GkggEk gg
  
 
xxEx
xxGx
G
E
(16) In particular, the second term of the right side of the
matrix 12
M
can be transformed:
11121322 3323321332123312 2313 22
121222331 23213311 3313 31132111 23
31323321 3231 223112113211 221221
ij
ij
g
gggggggggg gg gg
gggggg gggggggggggg
g
gggggggg gggggg


  


 






  

 
(12)

Generalization of the Second Order Vector Potential Formulation for Arbitrary Non-Orthogonal
Curvilinear Coordinates Systems from the Covariant Form of Maxwell’s Equations
404


12
nij nij
ctn
nij n
ctn
nij
c
Mk gggg
kgg
kgg
 

 

 
xx
xx
x
n
x
(23)
The first term on the left side of the matrix can be
written:




2
2
2
nijnijn
nt c
nnijnij
nnn c
nijnij
nnnc
kgggg
kgggg
kgggg
 
 

  

xxx
xx x
Ix xx
n
n
x (24)
The first term on the right side of the Equation (24)
may be translated on the left side so that an extended
formulation can be obtained for all the components of the
electromagnetic field in terms of the two longitudinal
components:
11 12
12 11
=
c
c
nij
nc n
nij n
cn
CkC
kCC
kgg E
G
kgg
 
 
 

 


 


E
G
x
x

(25)
Finally, the term on the left side of (25) may be
substituted by:
11
C


nijnij nn
ij nn
gggg gg

 

xxFxxFF
(26)
and the expressions of the electromagnetic fields may be
obtained from the longitudinal components by the sim-
plified equation:


2
12
2
12
2
2
nncnnc
cnncnn
nnij
ncij cn
nij n
n
cncij
kg kC
kCk g
kgkgg E
G
kgg kg

 



 


 
 
 



E
G
xx
xx
(27)
In the general case, the matrix operator cannot be
translated on the right side of the equation but this form
can help us to find a scalar decomposition. This is the
main goal of the next section.
4. The Scalar Decomposition
In the following of the paper, all developments are per-
formed according to the vector but any vector may
be chosen. Let us introduce two auxiliary scalar functions
and such that:
3
x
1
W2 W
2
1
22
n
n
nnij
cijc
nij n
ccij
kgkgg W
W
kgg kg

 
 

 
 
 

xx
E
Gx
When there is no source term, Maxwell’s equation are
written in a compact form:
0
0
ij
c
ij
c
kgg
kgg


 
 
 
    

IE
G
I
(29)
so, by introducing (28) in (29),
2
1
22
0
0
n
n
ij
c
ij
c
nnij
cijc
nij n
ccij
kgg
kgg
kgkgg W
W
kgg kg



 



 








 

 

I
I
xx
xx
(30)
which becomes:
2
1
2
2
cc
cc
W
kk
W
kk








ab
ba
0
0 (31)
with:


2ijnnij
ncij
g
gkggg

 axx (32)


ij nijn
ij
gg gggbx x (33)
Since the product of the covariant metric tensor by the
contravariant metric tensor is equal to identity,
ij
ij d
g
gI
 

  and due to the property



 
,  ababbabaab   (34)
the operator may be transformed:
a



2
2
2
2
ijnn ij
nc
ijn nijij
nc n
ijn nijij
nc n
ij n
c
ggk ggg
g
gkggg gg
ggk ggggg
kg gg
 

 

 

axx
xx
xx
x
(35)
Moreover, after some tedious calculations, if 3n

xx,
it is possible to show that:


x
(28)


33
21 2223
11 1213
32 13
3123
21 33
000
0
0
0
ij ij
ij
ij
gg ggg
ggg
gg ggg
g
g
g





















 

 


 


bx x
(36)
Since the tensor matric is symmetric, one obtains:

Generalization of the Second Order Vector Potential Formulation for Arbitrary Non-Orthogonal
Curvilinear Coordinates Systems from the Covariant Form of Maxwell’s Equations
405
33233 23233
33133 131 33
23132 131 23
233 323323 233
1 333 133 13133
1 232 132131 23
0
0
0
0
g
gg
ggg
g
g
g
gg
gg gg
gg gg
gg gg








 


 


 

b
g








(37)
The same result may be also obtained for 1n

xx
W and . Finally, the two scalar potentials 1 and
are governed by the same propagation equation:
2nxx
2
W

210 ij
c
kgg
g





 (38)
According to the tensor analysis, the Laplacian opera-
tor of a scalar may be written: 
210 ij
ij
gg
g



 




(39)
So, Equation (38) is the diffusion Helmholtz equation
expressed in the curvilinear coordinate system. This
equation will be then described for some examples in
Section 5. Finally, we can verify if the two first equations
of Maxwell’s Equation in (4) are satisfied:


ij
gg
iZ

  

B
HG (40)
By using (36) and (38), after some calculations:


1
12
ijnn ij
cij
nn ij
cij
2
g
gkgW gg
kgWgg W
 

Gxx
xx
W
(41)
Due to the vector identity

0

 a, one obtains
0 
B
. Likewise, the condition 0
D
is also
verified. Finally, let us define two scalar potentials so
that:
1
2
, Wi
WiZ

  

(42)
we can write the scalar decomposition:


2
22
n
n
nn ij
cij
nij n
ccij
ikg gg
kgg kg


 

 
Exx
Hx x

(43)
Since b = 0 in Equation (31), a occurs and by using
(32), we can write two substitutions:


2
1
n
n
cij
nij
ij
kg
gg
g


x
x


1
nij
n
ij ij
gg
gg
g

 
x
x
(45)
So, the covariant components of the electromagnetic
fields are expressed in the scalar potential decomposi-
tion:
1nij n
ij ij
iggg g
g







Ex x (46)
2
1nn ij
ijc ij
gkg gg
g



 


Hxx
(47)
In the following of the paper, some particular curvi-
linear coordinate systems are introduced in order to ver-
ify the validity of this decomposition with respect to the
works in the literature.
5. Examples of Different Curvilinear
Coordinate Systems
In this section, several coordinate systems are introduced
in order to compare the formulation to other calculations
exciting in the literature on the second order potential.
Finally, the last Section 5.4 provides a new writing con-
cerning the translation coordinate system.
5.1. Application to the Planar Coordinate System
In the Cartesian coordinate system, the tensor metric is
equal to Identity and the scalar decomposition becomes:
33
i 



 


Exx
(48)
3
i 
3



 


xx (49)


23 3
c
k

 Hxx (50)
This formula may be compared to the equation de-
scribed in [3,11,21].

is often called the Transverse
Electric (TE) potential while is the Transverse Mag-
netic (TM) potential.

5.2. Application to the Cylindrical Coordinate
System
In a cylindrical coordinate system, let us consider




123
,, ,,rz   .
cosxr


(51)
sinyr


(52)
zz

(53)
g
(44) The Jacobian matrix
J
defining the transformation
from the Cartesian coordinates system to the cylindrical

Generalization of the Second Order Vector Potential Formulation for Arbitrary Non-Orthogonal
Curvilinear Coordinates Systems from the Covariant Form of Maxwell’s Equations
406
coordinates system is given by:
12
12
12
0
cossin 0
0sincos
00
1
ff
xgg
Jr0
1
hh
 






 


 

 
 


 







0


r
(54)
The matrix representation of the covariant components
of the metric tensor is given by:
2
100
0
001
ij
gr







(55)
and we denote by g the determinant 2
ij
g
gr. The
matrix representation of the contravariant components of
the metric tensor are given by:
22
2
100100
1
000
001
001
ij
gg
r

0


















(56)
The transverse fields t
E
and t are not coupled
since the components of the metric tensor does not de-
pend on the variable . From the scalar decomposition,
the covariant components of the electrical field are given
by:
G
3
x

33
1ij
ij ij
ig ggg
g
 Ex

x (57)

233
1ij
ijc ij
gkg gg
g
 Hxx (58)
2
1
1
r
EEi rz  






r

(59)
2
2
1
Ei rrz 


 






(60)
2
32
11
Eir
rrr r



 


 
 


(61)
The covariant vectors are related to the unit vectors in
the Cartesian coordinate system:
1
2
3
x
y
z
tt
tJt
tt














 
(62)
The unit vectors of the cylindrical coordinates are re-
lated to the covariant vectors:
1
1
2
3
cossin 0
sincos 0
001
100
cossin 01
sincos 000
001001
rx
y
zz
x
y
z
et
et
et
tt
J
tt
r
tt






 
 

 
 
 





 



 
 



 






 

(63)
and finally
2
1
2
2
2
32
1
11
11
r
z
EEi rzr
EEi
rrrz
EEi r
rrr r

 
 




 




 
 





 

 



 



(64)
These last equations are similar to those existing in the
literature [15]. The propagation Equation (38) is given
by:

2
11 23
2
11 0
c
r
rrk

  (65)
This equation is quite similar to the scalar Helmholtz
equation expressed in the cylindrical coordinate system:
22
0
c
k

 (66)
5.3. Application to the Spherical Coordinate
System
In a spherical coordinate system, let us consider




123
,, ,,r

 

.
12
sin cossincosxr 3

   (67)
12
sin sinsinsinyr 3

   (68)
1
cos coszr 3

  (69)
The Jacobian matrix
J
defining the transformation from
the Cartesian coordinates system to the spherical coordi-
nates system is given by:
12
12
12
0
0
1
sin cossin sincos
cos coscos sinsin
sin sinsin cos0
ff
xgg
J
hh
rr r
rr







































(70)

Generalization of the Second Order Vector Potential Formulation for Arbitrary Non-Orthogonal
Curvilinear Coordinates Systems from the Covariant Form of Maxwell’s Equations
407
The matrix representation of the covariant components
of the metric tensor is given by:
2
22
100
00
00 sin
ij
gr
r








(71)
and 2singr

. The matrix representation of the
contravariant components of the metric tensor are given
by:
3
2
0
sin
sin
ij r
g
r






 





(72)
in this case, we choose a longitudinal orientation along
. The scalar decomposition is given by:
r
rru

1
1ij
ij ij
ig ggg
g
 Ex
1
x (73)

21 1
1ij
ijcij
gkggg
g
 Hxx (74)
The covariant vectors are related to the unit vectors in
the Cartesian coordinate system:
1
2
3
x
y
z
tt
tJt
tt
 
 

 
 
 
(75)
The unit vectors of the spherical coordinates are re-
lated to the covariant vectors:
1
1
2
3
1
2
3
sin cossinsincos
cos coscos sinsin
sin cos0
10 0
1
00
1
00 sin
r
et
eJt
et
t
t
rt
r


 
 





 


 

























(76)
and finally
22
1222
2
2
2
3
cos1 1
sin sin
111
sin
111
sin sin
r
EEi rr
r
EEi
rrr
EEi
rrr



   
 





 




 


 


 
 


 



These last equations are rigorously similar to those ex-
isting in the literature [17]. The propagation Equation (38)
is given by:


2
11 2
22
2
33
2
11
sin sin
sin sin
11 0
sin sin c
r
rr
k
r


 
2

 




(78)
This equation is quite similar to the scalar Helmholtz
equation expressed in the spherical coordinate system:
22
0
c
k

 (79)
5.4. Application to the Translation Coordinate
System
Starting from the Cartesian coordinate system


,,
x
yz ,
a 2D boundary surface may be described by a parametric
form


,axy and let us consider the translation coor-
dinate system so that:
1
x
x

(80)
2
y
x

(81)

31
,zx axx

2

(82)
The height of each point conforming
the surface is translated in a simplified condition
z

,,Pxyz
30x

.
The jacobian matrix is given by:
100
010
1
xy
A
aa












(83)
with


1,
x
aax
y

y and . The metric
tensor is given by:

2,
y
aax

2
2
1
1
1
x
xy x
ijxyyy
xy
aaaa
g
aaaa
aa








 







(84)
and
13
23
13 23 33
10
01
ij
g
g
g
g
gg













(85)
with:
13
x
g
a



(77)
23
y
g
a


332 2
1
x
y
g
aa



The components of the metric tensor does not depend

Generalization of the Second Order Vector Potential Formulation for Arbitrary Non-Orthogonal
Curvilinear Coordinates Systems from the Covariant Form of Maxwell’s Equations
408
on the variable . The field vector t
3
x
E
and t G are
not coupled since the operator t in Equation (20) is
null. The propagation Equation (38) is given by:
G



22233 21313
12331 1
23 23
322 0
c
kgg g
gg
 
 
(86)
This last equation may be compared to the equations
described by [22,23].
6. Conclusion and Future Work
In this paper, a generalized second order potential for-
mulation (SOVP) is proposed for solving scattering or
radiation problems described in an arbitrary non-or-
thogonal curvilinear coordinate system. This formulation
takes advantages from the tensor analysis but no exper-
tise is finally required for developing the expressions of
the electromagnetic field in terms of two scalar potentials,
usually the transverse electric potential and the transverse
magnetic potential. For writing the components of the
electrical field and the magnetic field for any curvilinear
coordinate system, it is necessary to write the metric
tensor which is easily defined in the paper and to use the
vector cross product as usual in a Cartesian coordinate
system. This SOVP formulation represents the key stone
for implementing new numerical models dedicated to
eddy current calculations based on the covariant form of
the Maxwell’ equations. By using a specific curvilinear
coordinate system matching the geometry of the bound-
ary surface, it is possible to write easily and analytically
the boundary conditions implying the covariant and con-
travariant components of the electromagnetic field. In
future work, a numerical method will be developed for
calculating eddy currents induced in a conducting work-
piece due to a 3D eddy current probe scanning the boun-
dary surface described by an arbitrary and irregular ge-
ometry.
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