generalization of integral theorem representations

TEKA Kom. Mot. Energ. Roln. – OL PAN, 2008, 8a, 12–16
GENERALIZATION OF INTEGRAL THEOREM
REPRESENTATIONS
*
*
Mariya Branspiz, ** Stanisław Sosnowski
The Volodymyr Dahl East-Ukrainian National University, ** Uniwersytet Rzeszowski
Summary. It is shown that the known theorems about volume integration of some differential-vectorial operations can be written as one analytical expression.
Key words: volume integration, surface integration.
INTRODUCTION
Nowadays the theorems about volume integration of differential-vectorial operations (divergence, rotor, gradient), that allow to bring it to the corresponding surface integration, are known
[1-3]. The Green theorem [1-3] also gives a possibility to bring volume integration of differentialvectorial operations with two scalar functions to the corresponding surface integration. All these
theorems have different forms of representation. However, all these forms have something general
in the structure [3].
In this paper it is shown, that allthe above-mentioned theorems can be generalized in the
following form:
§
w\ (a , b, ] ) ·¸
˜ dV ³ \ (a , b, n ) ˜ dS ,
¸
w]
v © ] x , y, z
S
¹
³ ¨¨ ¦
(1)
where: Ψ – function, defined in V, that represents certain algebraic combination of vectors
ā, b̄ and normal vector n̄ to the surface S.
1. THEOREM ABOUT VOLUME INTEGRATION OF DIVERGENCE OF VECTOR
Let Ψ be the scalar multiplication of vectors ā and n̄:
<
a˜n.
(2)
GENERALIZATION OF INTEGRAL THEOREM REPRESENTATIONS
13
For such type of the function Ψ the following equations can be written:
w\(a , x )
wx
w\(a , y)
wy
w
(a ˜1x )
wx
w
(a ˜ 1y )
wy
wa x
;,
wx
wa y
,;
wy
w\(a , z)
wz
w
(a ˜ 1z )
wz
wa z
,
wz
that allow to write the equation (1) in the form:
wa
§ wa
wa ·
y
x
z
³ (a ˜ n ) ˜ dS ³ ¨¨ wx wy wz ¸¸ ˜ dV ³ diva ˜ dV ,
¹
S
V©
v
which is known as the Ostrogradskiy-Gauss theorem [1, 2].
2. THEOREM ABOUT VOLUME INTEGRATION OF ROTOR OF VECTOR
Let Ψ be vectorial multiplication of vectors ā and n̄:
<
aun .
(3)
For such a type of the function Ψ the following equations can be obtained by factoring of
vectorial multiplication ā × 1̄ζ :
w< (a , x )
wx
w< (a , y)
wy
w
w
a y ˜1z a z ˜1y ;,
([a u 1x ])
wx
wx
w
w
a z ˜1x a x ˜1z ;,
([a u 1y ])
wy
wy
w< (a , z)
wz
w
([a u 1z ])
wz
(4)
w
a x ˜1y a y ˜1x ..
wz
As a result, taking into account (4) and the analytical form of representation of rotor of vector
in cartesian coordinates [1], it is possible to write the next correlation:
w
\a,1] ] x , y, z w]
¦
§ wa z wa y ·
¨
¸
¨ wy wz ¸ ˜1x ©
¹
§ wa y wax ·
waz ·
§ wa
¸ ˜1 rot a ,
¨ x ¸ ˜1 ¨¨
w
w
wy ¸¹
z
x
©
¹
© wx
that in this case allows to present (1) in the following way:
³ a u n ˜ dS ³ rota ˜ dV ,
S
v
which is the notation of the known theorem about volume integration of rotor of vector [1, 2].
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Mariya Branspiz, Stanisław Sosnowski
3. THEOREM ABOUT VOLUME INTEGRATION OF GRADIENT
OF SCALAR FUNCTION
Let the function Ψ be multiplication of vector n̄ and scalar function φ:
<
(5)
n ˜M.
For such kind of the function Ψ it is possible to write the following equations:
w\ a , b, x wx
w\ a , b, y wy
w\ a , b, z wZ
w M ˜ 1x wx
w M ˜ 1y wy
w M ˜ 1z wZ
wM
˜ 1 x ;,
wx
wM
˜ 1 y ;,
wy
wM
˜ 1z ,
wZ
which results in:
§ w
¦ ¨¨
[ x , y, z © w[
M ˜ 1[ ·¸¸
¹
wM
wM
wM
˜ 1x ˜ 1y ˜ 1z
wx
wy
wz
gradM .
(6)
Taking into account (6) the equation (1) in this case can be represented as:
³ M ˜ n ˜ dS ³ gradM ˜ dV ,
S
V
which is the representation of the known theorem about volume integration of gradient of
scalar function [1].
4. GREEN THEOREM
Let the function Ψ be the multiplication of scalar function f and derivative of scalar function
φ in the direction n̄:
<
f˜
wM
.
wn
(7)
For such a type of function Ψ it is possible to write the following equations:
w<
wx
§ wM ·
¨f ˜ ¸
© wx ¹
wf wM
w 2M
˜
f ˜ 2 ,;
wx wx
wx
w<
wy
§ wM ·
¨¨ f ˜ wy ¸¸
¹
©
wf wM
w 2M
˜
f ˜ 2 ;,
wy wy
wy
w<
wz
§ wM ·
¨f ˜ ¸
© wz ¹
wf wM
w 2M
˜
f ˜ 2 ,
wz wz
wz
it results from this that:
§ w 2M w 2M w 2M · wf wM
w § wM ·
¨
¸
¨¨ f ˜ w[ ¸¸ f ˜ ¨ 2 2 2 ¸ wx ˜ wx w
[
wy
wz ¹
©
¹
[ x , y, z
© wx
.
wf wM wf wM
f ˜ 'M gradf ˜ gradM .
˜
˜
wy wy wz wz
¦
(8)
GENERALIZATION OF INTEGRAL THEOREM REPRESENTATIONS
15
Taking into account (8) the equation (1) in this case can be written as:
wM
³ f wn ˜ dS ³ >f ˜ 'M gradf ˜ gradM@˜dV ,
S
V
which is the notation of the Green theorem [1-3].
5. VOLUME INTEGRATION OF EXPRESSION OF THE TYPE
Let the function Ψ be defined by the following equation:
<
b ˜ (a ˜ n ) .
(9)
For such type of the function Ψ it is possible to obtain the following equations:
w
b ˜ a ˜ 1x w a ˜ b b ˜ wa x a x ˜ w b ,;
wx
wx
wx
wx
wa y
wb
w
w
b ˜ a ˜ 1y a ˜ b b ˜ a y ˜ ;,
wy
wy
wy
wy
w
b ˜ a ˜ 1z wz
w
a ˜ b b ˜ wa z a z ˜ w b ,
wz
wz
wz
and taking into account given above correlations it can be obtained that:
wa y wa z ·
§ wa
w
¸
\a , b, 1] b ˜ ¨¨ x x
wy
wz ¸¹
w
]
w
] x , y, z
©
wb
wb
wb
ax
ay
az
b ˜ diva a ˜ grad b.
wx
wy
wz
¦
(9)
It results from (9) that the equation (1) in this case can be represented in the form:
³ bn ˜ a ˜ dS
S
³ [a ˜ grad b b ˜ diva ] ˜ dV ,
V
which coincides with the known correlation [4].
In conclusion of consideration of equation (1) it should be noted that all types of the aboveconsidered functions Ψ (2), (3), (5), (7), (9) satisfy the property of oddness:
< a , b, n < a , b,n ..
It should be expected that the equation (1) allows to obtain new integral equations. But it
requires а general proof of equation (1) which can be found as generalization of the known proofs
of theorems about volume integration of differential-vectorial operations.
CONCLUSIONS
1. Known theorems about volume integration of differential-vectorial operations are generalized in the form of representation.
16
Mariya Branspiz, Stanisław Sosnowski
2. The necessity of general proof of equation (1) for volume integration of differential-vectorial operations is shown in order to obtain new integral equations.
REFERENCES
Korn G., Korn Т., 1968: Mathematical handbook for scientists and engineers. Definitions, theorems
and formulas for reference and review. McGraw – Hill Book Company, New York – San
Francisko –Toronto – London – Sydney.
Bronshtein I.N., Semendjaev K. A., 1986: Reference book on mathematics for engineers and students. Nauka, Мoscow: 544. (in Russian)
Myshkis A. D., 1969: Lectures on higher mathematics. Nauka, Moscow: 640. (in Russian)
Zagirnyak M. V., Branspiz Yu. A., 1996: Calculation of direct magnetic field force on the ferromagnetic body. Proceedings of the XIV-th Symposium “Electromagnetic phenomena in
nonlinear circuits” (EPNC’96), Poznan: 16-164.
UOGÓLNIENIE INTEGRALNYCH REPREZENTACJI TWIERDZEŃ
Streszczenie. Wykazano, że znane twierdzenia o integracji objętościowej pewnych operacji różniczkowo-wektorialnych mogą być zapisane jako jedno działanie analityczne.
Słowa kluczowe: integracja objętościowa, integracja powierzchniowa.