Integral Representation of the Solution to the Vector Helmholtz Equation in
the System of Arbitrary Orthogonal Curvilinear Coordinates
ANTIMIROV M.Ya., DZENITE I.A.
Department of Engineering Mathematics
Riga Technical University
Meza street 1/4, Riga LV-1048
LATVIA
Abstract: In the integral representation of the solution to the vector Helmholtz equation known in literature the
electromagnetic field vector potential is expressed in terms of a triple integral of multiplication of current density
vector and fundamental solution of the scalar Helmholtz equation. This representation has the simplest form in the
  
rectangular coordinate system, in which the unit vectors e x , e y , e z do not depend on coordinates. In the present paper
the integral representation of the solution to the vector Helmholtz equation is obtained for the system of arbitrary
  
orthogonal curvilinear coordinates, in which the unit vectors eq1 , eq 2 , eq 3 are prescribed functions of coordinates. As
particular cases of the representation obtained, the integral representations of the solution to the vector Helmholtz
equation are found for the systems of cylindrical and spherical coordinates.
Key-Words: Vector potential, Helmholtz equation, Integral representation, Lame coefficient
1 Introduction
The Helmholtz equation for the vector potential used
in electrodynamics has the form



2
2
2
(1)
 A  k 2 A   0  I e ,   2  2  2 ,
x
y
z


where k 2   0  0   2 , I e  I e (M ) is the external
current vector density;  0 and  0 are the electric and
magnetic constants, respectively;  and  are the
relative
permittivity
and
relative
magnetic
permeability of the medium, respectively.


Vectors A(M ) and I e (M ) in the system of
Cartesian coordinates has the form




(2)
A(M )  Ax (M )e x  Ay (M )e y  Az (M )e z ,
e



e
e
e
(3)
I (M )  I x (M )e x  I y (M )e y  I z (M )e z .
Consider the integral representation of Helmholtz
equation in the vector form for a point M ( x, y, z )
situated in the region, where the external current

vector I e  0 (see [1], page 322):

 
A( M )  0
4

V
 ~ exp(  jkrMM~ ) ~
I e (M )
dV ,
rMM~
(4)
where the integration is performed over the points

~
M (~
x, ~
y, ~
z ) V , in which I e  0 ;   1,   1 since
the wire is situated in free space, rMM~ is the distance
between the points M ( x, y, z ) and M ( ~
x, ~
y, ~
z):
rMM~  ( x  ~
x)2  ( y  ~
y ) 2  (z  ~
z )2 .
(5)
Conditions for all the components, Ax , Ay , Az , of the
vector potential at infinity are so-called Sommerfeld’s
conditions of radiation (see [2], page 509):
 1  A
1
R 2  r 2  z 2   : A  O  ,
 jkA o  ,(6)
 R  R
 R
where symbol O1 / R  denotes that A and 1 R are
infinitesimal of the same order at R   , but symbol
o1 / R  denotes that A / R  jkA is infinitesimal of
higher order than 1 R at R   .
It can be easy verified that if the functions I xe (M ) ,
I ye (M ) , I ze (M ) are continuous in some closed region
V and, consequently, they are bounded in this region,

then the vector function A(M ) in formula (4) satisfy
Sommerfeld’s conditions (5). Consequently, in this
case, formula (4) gives the solution to the problem (1),
(6) under the condition that the vector function

I e (M ) is prescribed.
In the Cartesian coordinate system, the unit vectors
 

e x , e y and e z are constant. Therefore, in this case,
according to formula (4), the each component of the

vector A is expressed in terms of a triple integral of a

corresponding component of the vector I e (i.e. Ax in
terms of I xe , A y in terms of I ye and so on). For
example,
Ax ( M ) 
0
4

V
~ exp( jkrMM~ ) ~
I xe ( M )
dV ,
rMM~
(7)
and so on. However, in the all other orthogonal
curvilinear coordinate systems, the unit vectors are
already the functions of coordinates. For example, in
the system of cylindrical polar coordinates (r ,  , z ) ,
 ~
vector I e (M ) has the form
 ~
~ 
~ 
~ 
(8)
I e (M )  I re (M )er  I e (M )e  I ze (M )e z ,

and the only unit vector e z is constant. In this case, the
equality
~  ~ exp( jkrMM~ ) ~
I re ( M ) er ( M )
dV 
rMM~
V

exp(  jkrMM~ ) ~  ~
dV  er ( M )
(9)
rMM~
V
~  ~
and the similar equality for I e (M )  e (M ) are wrong.

 I
e ~
r (M )
Ar  Ax cos   Ay sin  ,
(12)
A   Ax sin   Ay cos  .
(13)
~
~
The components I xe (M ) , I ye (M ) can be expressed in
~
~
terms of the components I re (M ) and I e (M ) :
~
~
~
I xe (M )  I re (M ) cos ~  I e (M ) sin ~ ,
~
~
~
I ye (M )  I re (M ) sin ~  I e (M ) cos ~ .
(14)
(15)
It follows from (4) that
0
4
 I
 
Ay (M )  0
4
 I
Ax ( M ) 
~ ~
e ~
x ( M )  ( M , M ) dV
,
(16)
~ ~
e ~
y ( M )  ( M , M ) dV
,
(17)
V
V
where
exp( jkrMM~ )
~
,
( M , M ) 
rMM~
(18)
Consequently, in this case, the components Ar (M )
and A (M ) has the form of a triple integral of some
~
linear combination of components I re (M ) and
~
I e (M ) . In authors’ papers [3] and [4] problem (1), (6)
~
and the distance, rMM~ , between the points M and M
is defined by formula (5). It follows from (5) by
substituting
has been solved by using the cylindrical polar
coordinates as follows. At first, triple integral (7) is
transformed into the single integral in the Cartesian
coordinates and only, at second, the passing to the
cylindrical polar coordinates is performed in the
solution obtained.
In this paper, the integral representation of the
solution to the vector Helmholtz equation (4) is
obtained in the system of cylindrical polar, spherical
and also arbitrary orthogonal curvilinear coordinates.

Components of the vector A(M ) are expressed in
terms of triple integrals of the linear combinations of
 ~
the components of the current vector density I e (M ) .
(19)
2 Integral Representation in the System
of Cylindrical Polar Coordinates
In the system of cylindrical polar coordinates ( r ,  , z ),


 ~

r , ~, ~
z)
vectors A( M )  A(r ,  , z ) and I e ( M )  I e (~
have the form




A(M )  Ar (M ) er (M )  A (M ) e (M )  Az (M ) ez ,
(10)
e ~
~
~
e ~ 
e ~ 
e ~ 
I (M )  I r (M ) er (M )  I  (M ) e (M )  I z (M ) e z ,
(11)
the components Ar (M ) , A (M ) can be expressed in
terms of the components Ax (M ) , Ay (M ) as
x  r cos , y  r sin  , z  z;
~
x ~
r cos , ~
y ~
r sin ~, ~
z ~
z
that
rMM~  r 2  ~
r 2  2r ~
r cos(  ~)  ( z  ~
z )2 .
(20)
It follows from (12) by substituting expression (16) for
Ax (M ) and expression (17) for Ay (M ) that
0
4
 I
 
 0
4
 I
Ar ( M ) 
~ ~
e ~
x ( M )  ( M , M ) dV
 cos  
V
~ ~
e ~
y ( M )  ( M , M ) dV  sin
.
(21)
V
~
Substituting the expression (14) for I xe (M ) and (15)
~
for I ye (M ) into (21) yields
Ar ( M ) 
0
4
 [ I
e ~
r ( M ) cos
~
~  I e ( M ) sin ~ ]
V
~ ~
 (M , M ) dV  cos 
 
~
~
 0
[ I re ( M ) sin ~  I e ( M ) cos ~ ]
4 V
~ ~
 ( M , M ) dV  sin  .
(22)

The final expression for the component Ar (M ) can be
easily obtained from (22) by performing some
elementary transforms, and it has the form
 
~
Ar ( M )  0
[ I re ( M ) cos(  ~ ) 
4 V

~
~ ~
 Ie (M ) sin(  ~)] (M , M ) dV ,
where
~
dV  ~
r d~
r d~ d~
z.
(23)
(24)
~
~
I e (M ) and I e (M ) (see [5], page 582):
~
~
~
~
~
I xe (M )  I e (M ) sin  cos~  Ie (M ) cos cos~ 
~
(32)
 I e (M ) sin ~ ,

The expression for the component A (M ) is obtained
by performing the similar transforms and by using
formulae (13) for A (M ) , (16) for Ax (M ) , (17) for
~
~
Ay (M ) , (14) for I xe (M ) and (15) for I ye (M ) . It has
the form
0
~
[ I re ( M ) sin(  ~ ) 

4 V
~
~ ~
 I e (M ) cos(  ~)] (M , M ) dV .
A ( M ) 
(25)
The component Az (M ) has the same form as in the
Cartesian coordinates:
Az ( M ) 
0
4
 I
~ ~
e ~
z ( M )  ( M , M ) dV .
(26)
V
Thus, formulae (10), (23), (25) and (26) give the
integral representation of the solution to the vector
Helmholtz equation (4) in the cylindrical polar
coordinates.
3 Integral Representation in the System
of Spherical Coordinates
In the system of spherical coordinates (  ,  ,  )


 ~

~
vectors A(M )  A(  , ,  ) and I e ( M )  I e ( ~, , ~)
have the form



A(M )  A (M ) e (M )  A (M ) e (M ) 

 A ( M ) e ( M ) ,
(27)
e ~
~
~
e ~ 
e ~ 
I (M )  I  (M ) e (M )  I (M ) e (M ) 
~  ~
(28)
 Ie (M ) e (M ) .
~
~
~
~
~
I ye (M )  I e (M ) sin  sin ~  Ie (M ) cos sin ~ 
~
(33)
 Ie (M ) cos~ ,
~
~
~
~
~
I ze (M )  I e (M ) cos  Ie (M ) sin  .
It follows from formula (4) that
~
 I xe ( M ) 
 Ax ( M ) 


 0
~ 
~ ~
 I ye ( M )  F ( M , M ) dV ,
 Ay (M )  
~ 
 A ( M )  4 V  I e ( M
)
 z

 z

Ax (M ) ,
A (M )  [ Ax (M ) cos  Ay ( M ) sin  ] sin  
 Az (M ) cos ,
(29)
exp( jkrMM~ )
~
,
F (M , M ) 
rMM~
 Az (M ) sin  ,
A (M )   Ax (M ) sin   Ay (M ) cos  ,
(30)
(31)
~
~
~
and the components I xe (M ) , I ye (M ) and I ze (M ) can
~
be expressed in terms of the components I e (M ) ,
(36)
~
and the distance, rMM~ , between the points M and M
is defined by formula (5). It follows from (5) by
substituting
x   sin  cos , y   sin  sin  , z   cos ,
~
~
~
~
x  ~ sin  cos~, ~
y  ~ sin  sin ~, ~
z  ~ cos (37)
that
~
rMM~   2  ~ 2  2  ~[sin  sin  cos(  ~ ) 
~
 cos cos ] .
(38)
It follows from (29) by substituting (35) that
A ( M ) 


0
4
 I
0
4
 I
0
4
 I
~ ~
e ~
x ( M ) F ( M , M ) dV
 cos  sin  
V
~ ~
e ~
y ( M ) F ( M , M ) dV  sin
 sin  
V
~ ~
e ~
z ( M ) F ( M , M ) dV
 cos  .
(39)
V
The final expression for the component A (M ) is
obtained by substituting expressions (32) - (34) of the
 ~
components of I e (M ) into (39) and by performing
some elementary transforms, and it has the form
 
~
~
A ( M )  0
{I e ( M ) [sin  sin  cos(  ~ ) 
4 V
~
 cos cos ] 
~
~
~
 I e (M )[sin  cos cos(  ~)  cos sin  ] 
~
~ ~
 I e (M ) sin  sin(  ~)} F (M , M ) dV , (40)

A (M )  [ Ax (M ) cos  Ay (M ) sin  ] cos 
(35)
where
The components A (M ) , A (M ) and A (M ) can be
expressed in terms of the components
Ay (M ) and Az (M ) :
(34)
where
~
~
~
dV  ~ 2 sin 2  d~ d d~ .
(41)
By performing the similar transforms for the
components A (M ) and A (M ) , the final expression
Aq j ( M ) 
 Az ( M )
for these components has the form
0
~
~
{I e ( M ) [sin  cos cos(  ~ ) 

4 V
~
 cos sin  ] 
~
~
~
 I e (M )[cos cos cos(  ~)  sin  sin  ] 
~
~ ~
 I e (M ) cos sin(  ~)} F (M , M ) dV , (42)
A ( M ) 
0
~
~
{I e ( M ) sin  sin(  ~ ) 

4 V
~
~
 Ie (M ) cos sin(  ~) 
~
~ ~
 I e (M ) cos(  ~)} F (M , M ) dV .
(43)
Let the system of arbitrary orthogonal curvilinear
coordinates ( q1 , q 2 , q 3 ) be given by the functions
3
I
e
qk
k
(49)
(50)
k
z
~ 1 ~
I qek ( M ) ~ ~ ,
H k q k
k 1
3

(51)
~
where H k  H k (q1 , q 2 , q 3 ) , H k  H k (q~1 , q~2 , q~3 ) are
the Lame coefficients of the prescribed coordinate
system (see [5] with the notation that H k  hk1 ).
It follows from formula (4) that
~
 I xe ( M ) 
 Ax ( M ) 


 0
~ 
~ ~
 I ye ( M )  G ( M , M ) dV ,
(52)
 Ay (M )  
~ 
 A ( M )  4 V  I e ( M
)
 z

 z

where
exp( jkrMM~ )
~
,
G(M , M ) 
rMM~
x  x(q1 , q 2 , q 3 ) , y  y (q1 , q 2 , q 3 ) ,
z  z (q1 , q 2 , q3 )
(44)
and, respectively,
~
x  x(q~1 , q~2 , q~3 ) , ~
y  y (q~1 , q~2 , q~3 ) ,
~
z  z (q~ , q~ , q~ ) .
(45)
3
x
~ 1 ~
(M ) ~ ~ ,
H k q k
k 1
3
y
~
~ 1 ~
I ye ( M ) 
I qek ( M ) ~ ~ ,
H q
~
I xe ( M ) 
~
I ze ( M ) 
4 Integral Representation in the System
of Arbitrary Orthogonal Curvilinear
Coordinates
(48)
~
~
~
and the components I xe (M ) , I ye (M ) , I ze (M ) can be
~
expressed in terms of the components I qe1 (M ) ,
~
~
I qe2 (M ) , I qe3 (M ) (see [5], page 561):
k 1
Thus, formulae (27), (40), (42) and (43) give the
integral representation of the solution to the vector
Helmholtz equation (4) in the spherical coordinates.
2
z 
 , j  1, 2, 3,
q j 

A ( M )  
1
1 
x
y
 Ay (M )

 Ax ( M )
H j ( M ) 
q j
q j



Let e q1 , eq2 , eq3
be the units vectors of this

 ~
coordinate system. Then vectors A(M ) and I e (M )
have the form



A(M )  Aq1 (M ) eq1 (M )  Aq2 (M ) eq2 (M ) 

(46)
 Aq3 ( M ) eq3 ( M ) ,
 ~
~  ~
~  ~
I e (M )  I qe1 (M ) eq1 (M )  I qe2 (M ) eq2 (M ) 
~  ~
(47)
 I qe3 (M ) eq3 (M ) .
(53)
~
and the distance, rMM~ , between the points M and M
is defined by formula (5), where x, y, z and ~
x, ~
y, ~
z
~
~
~
q1 , q 2 , q 3 ,
are the functions of q1 , q 2 , q 3 and
respectively, and they are given by formulae (44),
(45).
It follows from (48), at first, by substituting (49)(51) into (52), and, at second, by substituting the new
obtained form of (52) into (48), that
Aq j ( M ) 
0 
1
4 H j ( M ) 
V
3

k 1
1
~
e
~ I qk ( M ) 
H k (M )
 ~
x x
~
y y
~
z z 
 ~
 ~
 ~

 qk q j qk q j qk q j 
~ ~
 G( M , M ) dV , j  1, 2, 3,
The components Aq1 ( M ) , Aq2 ( M ) , Aq3 ( M ) can be
where
expressed in terms of the components
Ay (M ) , Az (M ) :
~
~
~
~
dV  H 1 (M ) H 2 (M ) H 3 (M )dq~1 dq~2 dq~3 .
Ax (M ) ,
(54)
(55)
Formulae (46) and (54) give the integral representation
of the solution to the vector Helmholtz equation (4) in
the system of arbitrary orthogonal curvilinear
coordinates.
The integral representation of the Helmholtz
equation (4) can be obtained for any orthogonal
coordinate system by substituting the Lame
coefficients of these coordinate system into equations
(54) and (55). For example, in the system of
cylindrical polar coordinates ( r ,  , z ), it follows from
(54) by substituting
q1  r , q 2   , q 3  z , q~1  ~
r , q~2  ~ , q~3  ~
z,
~
~
~
~
H 1  1 , H 2  r , H 3  1 , H1  1 , H 2  r , H 3  1 ,
~
~
G ( M , M )  ( M , M ) ,
and by using formula (19) that at j  1
Ar ( M ) 
0
4
 [ I
e ~
r ( M ) (cos
~ cos   sin ~ sin  ) 
V
~
 I (M ) ( sin ~ cos  cos~ sin  )]
~ ~
 (M , M ) dV .
e
(56)
Formula (56) coincides with previously obtained
formula (23). By similar way, formulae (25) of
A (M ) and (26) of Az (M ) can be obtained from (54)
by substituting j  2 and j  3 , respectively.
In the system of spherical coordinates (  , , ),
formulae (40), (42) and (43) can be obtained from (54)
by substituting
~
q1   , q 2   , q 3   , q~1  ~ , q~2   , q~3  ~ ,
~
~
H 1  1 , H 2   , H 3   sin  , H 1  1 , H 2  ~ ,
~
~
~
~
H 3  ~ sin  , G( M , M )  F ( M , M )
and by using formula (37).
5 Conclusion
The integral representation of the solution to the vector
 
Helmholtz equation for the vector potential A  A(M )
has been obtained in the system of cylindrical polar
and spherical coordinates. In the cylindrical polar
coordinate system the radial and axial components of
the vector potential, A (M ) and A (M ) , are
expressed in terms of a triple integral of linear
combination of the radial and axial components of
current vector density. The similar results have been
obtained for the system of spherical coordinates.
The integral representation of the solution to the
vector Helmholtz equation has been also obtained for
the system of arbitrary orthogonal curvilinear
coordinates. The integral representation of the solution
to the vector Helmholtz equation can be easily found
for any orthogonal curvilinear coordinate system by
only substituting the Lame coefficients of this
coordinate system into the representation obtained.
References:
[1] Nikolsky V.V., Nikolskaya T.I., Electrodynamics
and radio waves spreading, Moscow: Science,
1989, 543 pages (Russian).
[2] Tihonov A.N., Samarsky A.A., Equations of
mathematical physics, Moscow: Science, 1972,
735 pages (Russian).
[3] Antimirov M., Dzenite I., Exact solution of the
problem on the vector potential of a rectangular
frame with current, Scientific Proceedings of Riga
Technical
University,
7th
series:
Telecommunications and Electronics, Vol.1, Riga
Technical University, ISSN 1407-8880, Riga,
2001, pp.115-123.
[4] Antimirov M., Dzenite I., Exact solution to the
problem on the vector potential of an arbitrary
form wire with
given
current, Scientific
Proceedings of Riga Technical University, 7th
series: Telecommunications and Electronics,
Vol.1, Riga Technical University, ISSN 14078880, Riga, 2002pp.78-84.
[5] Happel J., Brenner G., Hydrodynamics at small
Reynolds numbers. Moscow: Mir, 1976, 630 pages
(Russian).