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European Journal of Applied Sciences – Vol. 10, No. 1
Publication Date: February 25, 2022
DOI:10.14738/aivp.101.11865. Ichidayama, K. (2022). Property of Tensor Satisfying Binary Law 4. European Journal of Applied Sciences, 10(1). 556-576.
Services for Science and Education – United Kingdom
Property of Tensor Satisfying Binary Law 4
Koji Ichidayama
Okayama 716-0002, Japan
ABSTRACT
I have already reported "Property of Tensor Satisfying Binary Law 3". This article
improved a proof part in article "Property of Tensor Satisfying Binary Law 3". The
Proposition that improvement was carried out is Proposition4-8, Proposition10-15
in article "Property of Tensor Satisfying Binary Law 3". The Proposition of article
"Property of Tensor Satisfying Binary Law 3" will become stronger by this
improvement.
Keywords: Tensor; Covariant Derivative.
INTRODUCTION
I have already reported "Property of Tensor Satisfying Binary Law 3".[2] This article improved
a proof part in article "Property of Tensor Satisfying Binary Law 3".
The Proposition that improvement was carried out is Proposition4-8, Proposition10-15 in
article "Property of Tensor Satisfying Binary Law 3".
The Proposition of article "Property of Tensor Satisfying Binary Law 3" will become stronger
by this improvement.
In addition, I performed the explanation about this improvement in Discussion chapter.
The improvement technique that I showed here was performed about other Proposition2-5
equally.
DEFINITION
Definition1 �! ≠ �!, �" ≠ �", �! = �", �" = �! is established.[1] I named �! ≠ �!, �" ≠
�", �! = �", �" = �! "������ ���" .[1]
Definition2 If �! ≠ �!, �" ≠ �", �! = �", �" = �! is established, �" = �!is established.[1]
Definition3 If �! ≠ �!, �" ≠ �", �! = �", �" = �! is established, x! = x" is established.[1]
Definition4 If �! ≠ �!, �" ≠ �", �! = �", �" = �! is established, �" = −�! is established.[1]
Definition5 If all coordinate systems �!, �", �$, �%, ⋯ satisfies �! ≠ �!, �" ≠ �", �! = �", �" =
�!,
all coordinate systems �!, �", �$, �%, ⋯ shifts to only two of �!, �".[1]
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Ichidayama, K. (2022). Property of Tensor Satisfying Binary Law 4. European Journal of Applied Sciences, 10(1). 556-576.
URL: http://dx.doi.org/10.14738/aivp.101.11865
Definition6 �!
! = 1, �"
! = 0: (μ ≠ ν) is establishment.[3]
Definition7 &!'"
&'# &'# &'# = � is established for &!'"
&'# &'# &'#.
Definition8 The first-order covariant derivative of the covariant vector satisfied
�!;" = &'"
&'# − �)Γ!"
) = &'"
&'# − �)
*
+ �,) ;
&-"$
&'# + &-#$
&'" − &-"#
&'$ =.[4]
Definition9 The first-order covariant derivative of the contravariant vector satisfied
�;"
! = &'"
&'# + �)Γ)"
! = &'"
&'# + �) *
+ �,! ;
&-%$
&'# + &-#$
&'% − &-%#
&'$ =.[4]
Definition10 The second-order covariant derivative of the covariant vector satisfied
�.;/;0 = 1'&;(
1') − �2;/Γ.0
2 − �.;2Γ/0
2
= 1
1') ;
1'&
1'( − �3Γ./
3 = − ;1'*
1'( − �3Γ2/
3 = Γ.0
2 − ;
1'&
1'* − �3Γ.2
3 = Γ/0
2
= 1+'&
1'(1') − 1
1') ?�3
*
+ �43 ;
1-&,
1'( + 1-(,
1'& − 1-&(
1', =@
− 1'*
1'(
*
+ �42 ;
1-&,
1') + 1-),
1'& − 1-&)
1', = + �3
*
+ �43 ;
1-*,
1'( + 1-(,
1'* − 1-*(
1', = *
+ �42 ;
1-&,
1') + 1-),
1'& − 1-&)
1', =
− &'"
&'-
*
+ �,5 ;
&-#$
&'. + &-.$
&'# − &-#.
&'$ = + �)
*
+ �,) ;
&-"$
&'- + &--$
&'" − &-"-
&'$ = *
+ �,5 ;
&-#$
&'. + &-.$
&'# − &-#.
&'$ =.[4]
Definition11 The second-order covariant derivative of the contravariant vector satisfied
�;/;0
. = 1';(
&
1') + �;/
2 Γ20
. − �;2
.
Γ/0
2
= 1
1') ;
1'&
1'( + �3Γ3/
. = + ;
1'*
1'( + �3Γ3/
2 = Γ20
. − ;
1'&
1'* + �3Γ32
.
= Γ/0
2
= &+'"
&'# &'. + &
&'. ?�) *
+ �,! ;
&-%$
&'# + &-#$
&'% − &-%#
&'$ =@
+ &'-
&'#
*
+ �,! ;
&--$
&'. + &-.$
&'- − &--.
&'$ = + �) *
+ �,5 ;
&-%$
&'# + &-#$
&'% − &-%#
&'$ = *
+ �,! ;
&--$
&'. + &-.$
&'- − &--.
&'$ =
− &'"
&'-
*
+ �,5 ;
&-#$
&'. + &-.$
&'# − &-#.
&'$ = − �) *
+ �,! ;
&-%$
&'- + &--$
&'% − &-%-
&'$ = *
+ �,5 ;
&-#$
&'. + &-.$
&'# − &-#.
&'$ =.[4]
Definition12 The third-order covariant derivative of the contravariant vector satisfied
�
;/;0;6
. = 1';(;)
&
1'/ + �;/;0 7 Γ76
. − �;7;0
. Γ/6
7 − �;/;7
. Γ06
7
= 1
1'/ A 1
1') ;
1'&
1'( + �3Γ3/
. = + ;
1'*
1'( + �3Γ3/
2 = Γ20
. − ;
1'&
1'* + �3Γ32
.= Γ/0
2 B
+ A 1
1') ;
1'0
1'( + �3Γ3/
7 = + ;
1'*
1'( + �3Γ3/
2 = Γ20
7 − ;
1'0
1'* + �3Γ32
7= Γ/0
2 B Γ76
.
− A 1
1') ;
1'&
1'0 + �3Γ37
. = + ;
1'*
1'0 + �3Γ37
2 = Γ20
. − ;
1'&
1'* + �3Γ32
.= Γ70
2 B Γ/6
7
− A 1
1'0 ;
1'&
1'( + �3Γ3/
. = + ;
1'*
1'( + �3Γ3/
2 = Γ27
. − ;
1'&
1'* + �3Γ32
.= Γ/7
2 B Γ06
7
= &!'"
&'# &'. &'1 + &+
&'. &'1 ?�) *
+ �,! ;
&-%$
&'# + &-#$
&'% − &-%#
&'$ =@
+ &
&'1 ?
&'-
&'#
*
+ �,! ;
&--$
&'. + &-.$
&'- − &--.
&'$ =@
+ 1
1'/ ?�3 *
+ �42 ;
1-2,
1'( + 1-(,
1'2 − 1-2(
1', = *
+ �4. ;
1-*,
1') + 1-),
1'* − 1-*)
1', =@
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− &
&'1 ?
&'"
&'-
*
+ �,5 ;
&-#$
&'. + &-.$
&'# − &-#.
&'$ =@
− &
&'1 ?�) *
+ �,! ;
&-%$
&'- + &--$
&'% − &-%-
&'$ = *
+ �,5 ;
&-#$
&'. + &-.$
&'# − &-#.
&'$ =@
+ &+'3
&'# &'.
*
+ �,! ;
&-3$
&'1 + &-1$
&'3 − &-31
&'$ =
+ &
&'. ?�) *
+ �,8 ;
&-%$
&'# + &-#$
&'% − &-%#
&'$ =@ *
+ �,! ;
&-3$
&'1 + &-1$
&'3 − &-31
&'$ =
+ &'-
&'#
*
+ �,8 ;
&--$
&'. + &-.$
&'- − &--.
&'$ = *
+ �,! ;
&-3$
&'1 + &-1$
&'3 − &-31
&'$ =
+�) *
+ �,5 ;
&-%$
&'# + &-#$
&'% − &-%#
&'$ = *
+ �,8 ;
&--$
&'. + &-.$
&'- − &--.
&'$ = *
+ �,! ;
&-3$
&'1 + &-1$
&'3 − &-31
&'$ =
− &'3
&'-
*
+ �,5 ;
&-#$
&'. + &-.$
&'# − &-#.
&'$ = *
+ �,! ;
&-3$
&'1 + &-1$
&'3 − &-31
&'$ =
−�) *
+ �,8 ;
&-%$
&'- + &--$
&'% − &-%-
&'$ = *
+ �,5 ;
&-#$
&'. + &-.$
&'# − &-#.
&'$ = *
+ �,! ;
&-3$
&'1 + &-1$
&'3 − &-31
&'$ =
− &+'"
&'3 &'.
*
+ �,8 ;
&-#$
&'1 + &-1$
&'# − &-#1
&'$ =
− &
&'. ?�) *
+ �,! ;
&-%$
&'3 + &-3$
&'% − &-%3
&'$ =@ *
+ �,8 ;
&-#$
&'1 + &-1$
&'# − &-#1
&'$ =
− &'-
&'3
*
+ �,! ;
&--$
&'. + &-.$
&'- − &--.
&'$ = *
+ �,8 ;
&-#$
&'1 + &-1$
&'# − &-#1
&'$ =
−�) *
+ �,5 ;
&-%$
&'3 + &-3$
&'% − &-%3
&'$ = *
+ �,! ;
&--$
&'. + &-.$
&'- − &--.
&'$ = *
+ �,8 ;
&-#$
&'1 + &-1$
&'# − &-#1
&'$ =
+ &'"
&'-
*
+ �,5 ;
&-3$
&'. + &-.$
&'3 − &-3.
&'$ = *
+ �,8 ;
&-#$
&'1 + &-1$
&'# − &-#1
&'$ =
+�) *
+ �,! ;
&-%$
&'- + &--$
&'% − &-%-
&'$ = *
+ �,5 ;
&-3$
&'. + &-.$
&'3 − &-3.
&'$ = *
+ �,8 ;
&-#$
&'1 + &-1$
&'# − &-#1
&'$ =
− &+'"
&'# &'3
*
+ �,8 ;
&-.$
&'1 + &-1$
&'. − &-.1
&'$ =
− &
&'3 ?�) *
+ �,! ;
&-%$
&'# + &-#$
&'% − &-%#
&'$ =@ *
+ �,8 ;
&-.$
&'1 + &-1$
&'. − &-.1
&'$ =
− &'-
&'#
*
+ �,! ;
&--$
&'3 + &-3$
&'- − &--3
&'$ = *
+ �,8 ;
&-.$
&'1 + &-1$
&'. − &-.1
&'$ =
−�) *
+ �,5 ;
&-%$
&'# + &-#$
&'% − &-%#
&'$ = *
+ �,! ;
&--$
&'3 + &-3$
&'- − &--3
&'$ = *
+ �,8 ;
&-.$
&'1 + &-1$
&'. − &-.1
&'$ =
+ &'"
&'-
*
+ �,5 ;
&-#$
&'3 + &-3$
&'# − &-#3
&'$ = *
+ �,8 ;
&-.$
&'1 + &-1$
&'. − &-.1
&'$ =
+�) *
+ �,! ;
&-%$
&'- + &--$
&'% − &-%-
&'$ = *
+ �,5 ;
&-#$
&'3 + &-3$
&'# − &-#3
&'$ = *
+ �,8 ;
&-.$
&'1 + &-1$
&'. − &-.1
&'$ =.
Definition13 When the next conversion equation is established, �!
! is components of a tensor
of rank zero. �!
! = &'"
&'#
&'#
&'"
�"
"
Definition14 When the next conversion equation is established, �! is contravariant
components of a tensor of the first rank.[4] �! = &'"
&'# �"
Definition15 When the next conversion equation is established, �! is covariant components of
a tensor of the first rank.[4] �! = &'#
&'"
�"
Definition16 When the next conversion equation is established, �!" is contravariant
components of a tensor of the second rank.[4] �!" = &'"
&'.
&'#
&'1 �$%
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Ichidayama, K. (2022). Property of Tensor Satisfying Binary Law 4. European Journal of Applied Sciences, 10(1). 556-576.
URL: http://dx.doi.org/10.14738/aivp.101.11865
Definition17 When the next conversion equation is established, �!" is covariant components
of a tensor of the second rank.[4] �!" = &'.
&'"
&'1
&'# �$%
Definition18 When the next conversion equation is established, �"
! is components of the mixed
tensor of the second rank.[4] �"
! = &'"
&'.
&'1
&'# �%
$
Definition19 When the next conversion equation is established, �!"$ is covariant components
of a tensor of the third rank.[4] �!"$ = &'1
&'"
&'-
&'#
&'$
&'. �%5,
Definition20 When the next conversion equation is established, �"$
! is components of the
mixed tensor of the third rank of the second rank covariant in the first rank contravariant.[4]
�"$
! = &'"
&'1
&'-
&'#
&'$
&'. �5,
%
Definition21 When the next conversion equation is established, �"$%
! is components of the
mixed tensor of the fourth rank of the third rank covariant in the first rank contravariant.[4]
�"$%
! = &'"
&'-
&'$
&'#
&'4
&'.
&'5
&'1 �,9:
5
ABOUT COVARIANT DERIVATIVE FOR THE VECTOR IN TENSOR SATISFYING BINARY
LAW
Proposition1 �.;/ = 1'&
1'( , �.
;. = 1'&
1'&
− �/
*
+
1-&(
1'& is established in tensor satisfying Binary Law.
Proof: If all coordinate systems satisfies Binary Law in &-"$
&'# − &-"#
&'$ ,
&-"$
&'# − &-"#
&'$ = 0 (1)
is established. I get
�!;" = &'"
&'# − �)
*
+ �,) ;
&-#$
&'" = = &'"
&'# − �)
*
+
&-#
%
&'" (2)
from (1),Definision8. I get
�!;" = &'"
&'# − �)
*
+
&-#
%
&'" (3)
as if Binary Law being satisfied for all index except the dummy index of (2). If (3) is a tensor
equation, the dummy index of (3) can't make μ or ν. On the other hand, If all coordinate systems
satisfies Binary Law, dummy index of (3) should be μ or ν in consideration of Definision5. I aim
at the coexistence of these two demands. I rewrite (3) using Definision2 and get
�!
;! = &'"
&'"
− �)
*
+ ;
&-"%
&'" = = &'"
&'"
− �"
*
+ ;
&-"#
&'" =. (4)
(4) satisfies two demands mentioned above together here. I rewrite (3) using Definision4 and
get
−�!;! = − &'"
&'" + �)
*
+ ;
&-"
%
&'"= = − &'"
&'" + �"
*
+ ;
&-"
#
&'"=. (5)
(5) satisfies two demands mentioned above together here. I get
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−�!;! = − &'"
&'" (6)
in consideration of Definision6 for (5). Because the second term of the right side doesn't exist
in (6),
�!;" = &'"
&'# (7)
can rewrite (6) using Definision4. In addition, �!;" can't rewrite �!
;! of (4) using Definision2
because the second term of the right side exists in (4). End Proof
Proposition2 �;"
! = &'"
&'# is established in tensor satisfying Binary Law.
Proof: If all coordinate systems satisfies Binary Law in &-#$
&'% − &-%#
&'$ ,
&-#$
&'% − &-%#
&'$ = 0 (8)
is established. I get
�;"
! = &'"
&'# + �) *
+ �,! ;
&-%$
&'# = = &'"
&'# + �) *
+
&-%
"
&'# (9)
from (8),Definision9. I get
�;"
! = &'"
&'# + �) *
+
&-%
"
&'# (10)
as if Binary Law being satisfied for all index except the dummy index of (9). If (10) is a tensor
equation, the dummy index of (10) can't make μ or ν. On the other hand, If all coordinate
systems satisfies Binary Law, dummy index of (10) should be μ or ν in consideration of
Definision5. I aim at the coexistence of these two demands. I rewrite (10) using Definision2 and
get
�!;! = &'"
&'"
+ �) *
+
&-%
"
&'"
= &'"
&'"
+ �" *
+
&-#
"
&'"
. (11)
(11) satisfies two demands mentioned above together here. I get
�!;! = &'"
&'"
(12)
in consideration of Definision6 for (11). Because the second term of the right side doesn't exist
in (12),
�;"
! = &'"
&'# (13)
can rewrite (12) using Definision2. I rewrite (10) using Definision4 and get
−�;!
! = − &'"
&'" − �) *
+
&-%
"
&'" = − &'"
&'" − �" *
+
&-#
"
&'". (14)
(14) satisfies two demands mentioned above together here. I get
−�;!
! = − &'"
&'" (15)
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Ichidayama, K. (2022). Property of Tensor Satisfying Binary Law 4. European Journal of Applied Sciences, 10(1). 556-576.
URL: http://dx.doi.org/10.14738/aivp.101.11865
in consideration of Definision6 for (14). Because the second term of the right side doesn't exist
in (15), (13) can rewrite (15) using Definision4.
If all coordinate systems satisfies Binary Law in &-%$
&'# − &-%#
&'$ ,
&-%$
&'# − &-%#
&'$ = 0 (16)
is established. I get
�;"
! = &'"
&'# + �) *
+ �,! ;
&-#$
&'% = = &'"
&'# + �) *
+ ;
&-#
"
&'% = (17)
from (8),Definision9. I get
�;"
! = &'"
&'# + �) *
+ ;
&-#
"
&'% = (18)
as if Binary Law being satisfied for all index except the dummy index of (17). If (18) is a tensor
equation, the dummy index of (18) can't make μ or ν. On the other hand, If all coordinate
systems satisfies Binary Law, dummy index of (18) should be μ or ν in consideration of
Definision5. I aim at the coexistence of these two demands. I rewrite (18) using Definision2 and
get
�!;! = &'"
&'"
+ �) *
+ ;
&-""
&'% = = &'"
&'"
+ �" *
+ ;
&-""
&'# =. (19)
(19) satisfies two demands mentioned above together here. I rewrite (18) using Definision4
and get
−�;!
! = − &'"
&'" − �) *
+ ?
&-"
"
&'% @ = − &'"
&'" − �" *
+ ?
&-"
"
&'#@. (20)
(20) satisfies two demands mentioned above together here. I get
−�;!
! = − &'"
&'" (21)
in consideration of Definision6 for (20). Because the second term of the right side doesn't exist
in (21), (13) can rewrite (21) using Definision4. End Proof
Proposition3 �!;":" = &+'"
&'# &'# is established in tensor satisfying Binary Law.
Proof: If all coordinate systems satisfies Binary Law in
A
&-"$
&'# − &-"#
&'$ ,
&-"$
&'. − &-".
&'$ ,
&-#$
&'- − &--#
&'$ ,
&-.$
&'# − &-#.
&'$ ,
&-"$
&'- − &-"-
&'$ B,
1-&,
1'( − 1-&(
1', = 0,
1-&,
1') − 1-&)
1', = 0,
1-(,
1'* − 1-*(
1', = 0,
&-.$
&'# − &-#.
&'$ = 0,
&-"$
&'- − &-"-
&'$ = 0 (22)
is established. I get
�!;";$ = &+'"
&'# &'. − &
&'. ?�)
*
+ �,) ;
&-#$
&'" =@ − &'-
&'#
*
+ �,5 ;
&-.$
&'" =
+�)
*
+ �,) ;
&--$
&'# = *
+ �,5 ;
&-.$
&'" = − &'"
&'-
*
+ �,5 ;
&-#$
&'. = + �)
*
+ �,) ;
&--$
&'"= *
+ �,5 ;
&-#$
&'. =
= &+'"
&'# &'. − &
&'. ;�)
*
+
&-#
%
&'"= − &'-
&'#
*
+
&-.
-
&'" + �)
*
+
&--
%
&'#
*
+
&-.
-
&'" − &'"
&'-
*
+
&-#
-
&'. + �)
*
+
&--
%
&'"
*
+
&-#
-
&'. (23)
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from (22),Definision10. I get
�!;";" = &+'"
&'# &'# − &
&'# ;�)
*
+
&-#
%
&'"= − &'-
&'#
*
+
&-#
-
&'" + �)
*
+
&--
%
&'#
*
+
&-#
-
&'" − &'"
&'-
*
+
&-#
-
&'# + �)
*
+
&--
%
&'"
*
+
&-#
-
&'# (24)
as if Binary Law being satisfied for all index except the dummy index of (23). If (24) is a tensor
equation, the dummy index of (24) can't make μ or ν. On the other hand, If all coordinate
systems satisfies Binary Law, dummy index of (24) should be μ or ν in consideration of
Definision5. I aim at the coexistence of these two demands. I rewrite (24) using Definision2 and
get
�!
;!;! = &+'"
&'" &'"
− &
&'"
;�)
*
+
&-"%
&'" = − &'-
&'"
*
+
&-"-
&'" +�)
*
+
&--
%
&'"
*
+
&-"-
&'" − &'"
&'-
*
+
&-"-
&'"
+ �)
*
+
&--
%
&'"
*
+
&-"-
&'"
= &+'"
&'" &'"
− &
&'"
;�"
*
+
&-"#
&'" = − &'#
&'"
*
+
&-"#
&'" +�"
*
+
&-#
#
&'"
*
+
&-"#
&'" − &'"
&'#
*
+
&-"#
&'"
+ �"
*
+
&-#
#
&'"
*
+
&-"#
&'"
. (25)
(25) satisfies two demands mentioned above together here. I rewrite (24) using Definision4
and get
�!;!;! = &+'"
&'" &'" − &
&'" ;�)
*
+
&-"
%
&'"= − &'-
&'"
*
+
&-"
-
&'" +�)
*
+
&--
%
&'"
*
+
&-"
-
&'" − &'"
&'-
*
+
&-"
-
&'" + �)
*
+
&--
%
&'"
*
+
&-"
-
&'"
= &+'"
&'" &'" − &
&'" ;�"
*
+
&-"
#
&'"= − &'#
&'"
*
+
&-"
#
&'" +�"
*
+
&-#
#
&'"
*
+
&-"
#
&'" − &'"
&'#
*
+
&-"
#
&'" + �"
*
+
&-#
#
&'"
*
+
&-"
#
&'". (26)
(26) satisfies two demands mentioned above together here. I get
�!;!;! = &+'"
&'" &'" (27)
in consideration of Definision6 for (26). Because the second term of the right side doesn't exist
in (27),
�!;";" = &+'"
&'# &'# (28)
can rewrite (27) using Definision4.
If all coordinate systems satisfies Binary Law in
A
&-"$
&'# − &-"#
&'$ ,
&-"$
&'. − &-".
&'$ ,
&--$
&'# − &--#
&'$ ,
&-#$
&'. − &-#.
&'$ ,
&-"$
&'- − &-"-
&'$ B,
1-&,
1'( − 1-&(
1', = 0,
1-&,
1') − 1-&)
1', = 0,
1-*,
1'( − 1-*(
1', = 0,
&-#$
&'. − &-#.
&'$ = 0,
&-"$
&'- − &-"-
&'$ = 0 (29)
is established. I get
�!;";$ = &+'"
&'# &'. − &
&'. ?�)
*
+ �,) ;
&-#$
&'" =@ − &'-
&'#
*
+ �,5 ;
&-.$
&'" =
+�)
*
+ �,) ;
&-#$
&'- = *
+ �,5 ;
&-.$
&'" = − &'"
&'-
*
+ �,5 ;
&-.$
&'# = + �)
*
+ �,) ;
&--$
&'"= *
+ �,5 ;
&-.$
&'# =
= &+'"
&'# &'. − &
&'. ;�)
*
+
&-#
%
&'"= − &'-
&'#
*
+
&-.
-
&'" + �)
*
+
&-#
%
&'-
*
+
&-.
-
&'" − &'"
&'-
*
+
&-.
-
&'# + �)
*
+
&--
%
&'"
*
+
&-.
-
&'# (30)
from (29),Definision10. I get
�!;";" = &+'"
&'# &'# − &
&'# ;�)
*
+
&-#
%
&'"= − &'-
&'#
*
+
&-#
-
&'" + �)
*
+
&-#
%
&'-
*
+
&-#
-
&'" − &'"
&'-
*
+
&-#
-
&'# + �)
*
+
&--
%
&'"
*
+
&-#
-
&'# (31)
as if Binary Law being satisfied for all index except the dummy index of (30). If (31) is a tensor
equation, the dummy index of (31) can't make μ or ν. On the other hand, If all coordinate
Page 8 of 21
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Ichidayama, K. (2022). Property of Tensor Satisfying Binary Law 4. European Journal of Applied Sciences, 10(1). 556-576.
URL: http://dx.doi.org/10.14738/aivp.101.11865
systems satisfies Binary Law, dummy index of (31) should be μ or ν in consideration of
Definision5. I aim at the coexistence of these two demands. I rewrite (31) using Definision2 and
get
�!
;!;! = &+'"
&'" &'"
− &
&'"
;�)
*
+
&-"%
&'" = − &'-
&'"
*
+
&-"-
&'" + �)
*
+
&-"%
&'-
*
+
&-"-
&'" − &'"
&'-
*
+
&-"-
&'"
+ �)
*
+
&--
%
&'"
*
+
&-"-
&'"
= &+'"
&'" &'"
− &
&'"
;�"
*
+
&-"#
&'" = − &'#
&'"
*
+
&-"#
&'" + �"
*
+
&-"#
&'#
*
+
&-"#
&'" − &'"
&'#
*
+
&-"#
&'"
+ �"
*
+
&-#
#
&'"
*
+
&-"#
&'"
. (32)
(32) satisfies two demands mentioned above together here. I rewrite (31) using Definision4
and get
�!;!;! = &+'"
&'" &'" − &
&'" ;�)
*
+
&-"
%
&'"= − &'-
&'"
*
+
&-"
-
&'" + �)
*
+
&-"
%
&'-
*
+
&-"
-
&'" − &'"
&'-
*
+
&-"
-
&'" + �)
*
+
&--
%
&'"
*
+
&-"
-
&'"
= &+'"
&'" &'" − &
&'" ;�"
*
+
&-"
#
&'"= − &'#
&'"
*
+
&-"
#
&'" + �"
*
+
&-"
#
&'#
*
+
&-"
#
&'" − &'"
&'#
*
+
&-"
#
&'" + �"
*
+
&-#
#
&'"
*
+
&-"
#
&'". (33)
(33) satisfies two demands mentioned above together here. I get
�!;!;! = &+'"
&'" &'" (34)
in consideration of Definision6 for (33). Because the second term of the right side doesn't exist
in (34), (28) can rewrite (34) using Definision4. End Proof
Proposition4 �;":"
! = &+'"
&'# &'# is established in tensor satisfying Binary Law.
Proof: If all coordinate systems satisfies Binary Law in
A
&-#$
&'% − &-%#
&'$ ,
&-.$
&'- − &--.
&'$ ,
&-.$
&'# − &-#.
&'$ ,
&--$
&'% − &-%-
&'$ B,
&-#$
&'% − &-%#
&'$ = 0,
&-.$
&'- − &--.
&'$ = 0,
&-.$
&'# − &-#.
&'$ = 0,
&--$
&'% − &-%-
&'$ = 0 (35)
is established. I get
�;";$
! = &+'"
&'# &'. + &
&'. ?�) *
+ �,! ;
&-%$
&'# =@ + &'-
&'#
*
+ �,! ;
&--$
&'. =
+�) *
+ �,5 ;
&-%$
&'# = *
+ �,! ;
&--$
&'.= − &'"
&'-
*
+ �,5 ;
&-#$
&'. = − �) *
+ �,! ;
&-%$
&'- = *
+ �,5 ;
&-#$
&'. =
= &+'"
&'# &'. + &
&'. ;�) *
+
&-%
"
&'#= + &'-
&'#
*
+
&--
"
&'. + �) *
+
&-%
-
&'#
*
+
&--
"
&'. − &'"
&'-
*
+
&-#
-
&'. − �) *
+
&-%
"
&'-
*
+
&-#
-
&'. (36)
from (35),Definision11. I get
�;";"
! = &+'"
&'# &'# + &
&'# ;�) *
+
&-%
"
&'#= + &'-
&'#
*
+
&--
"
&'# + �) *
+
&-%
-
&'#
*
+
&--
"
&'# − &'"
&'-
*
+
&-#
-
&'# − �) *
+
&-%
"
&'-
*
+
&-#
-
&'# (37)
as if Binary Law being satisfied for all index except the dummy index of (36). If (37) is a tensor
equation, the dummy index of (37) can't make μ or ν. On the other hand, If all coordinate
systems satisfies Binary Law, dummy index of (37) should be μ or ν in consideration of
Definision5. I aim at the coexistence of these two demands. I rewrite (37) using Definision2 and
get
�!;!;! = &+'"
&'" &'"
+ &
&'"
?�) *
+
&-%
"
&'"
@ + &'-
&'"
*
+
&--
"
&'"
+ �) *
+
&-%
-
&'"
*
+
&--
"
&'"
− &'"
&'-
*
+
&-"-
&'"
− �) *
+
&-%
"
&'-
*
+
&-"-
&'"
= &+'"
&'" &'"
+ &
&'"
?�" *
+
&-#
"
&'"
@ + &'#
&'"
*
+
&-#
"
&'"
+ �" *
+
&-#
#
&'"
*
+
&-#
"
&'"
− &'"
&'#
*
+
&-"#
&'"
− �" *
+
&-#
"
&'#
*
+
&-"#
&'"
. (38)
Page 9 of 21
564
European Journal of Applied Sciences (EJAS) Vol. 10, Issue 1, February-2022
Services for Science and Education – United Kingdom
(38) satisfies two demands mentioned above together here. I get
�!;!;! = &+'"
&'" &'"
− &'"
&'#
*
+
&-"#
&'"
(39)
in consideration of Definision6 for (38). I rewrite (37) using Definision4 and get
�;!;!
! = &+'"
&'" &'" + &
&'" ;�) *
+
&-%
"
&'"= + &'-
&'"
*
+
&--
"
&'" + �) *
+
&-%
-
&'"
*
+
&--
"
&'" − &'"
&'-
*
+
&-"
-
&'" − �) *
+
&-%
"
&'-
*
+
&-"
-
&'"
= &+'"
&'" &'" + &
&'" ;�" *
+
&-#
"
&'"= + &'#
&'"
*
+
&-#
"
&'" + �" *
+
&-#
#
&'"
*
+
&-#
"
&'" − &'"
&'#
*
+
&-"
#
&'" − �" *
+
&-#
"
&'#
*
+
&-"
#
&'". (40)
(40) satisfies two demands mentioned above together here. I get
�;!;!
! = &+'"
&'" &'" (41)
in consideration of Definision6 for (40). Because the second term of the right side doesn't exist
in (41),
�;";"
! = &+'"
&'# &'# (42)
can rewrite (41) using Definision4.
If all coordinate systems satisfies Binary Law in
A
&-%$
&'# − &-%#
&'$ ,
&--$
&'. − &--.
&'$ ,
&-#$
&'. − &-#.
&'$ ,
&-%$
&'- − &-%-
&'$ B,
&-%$
&'# − &-%#
&'$ = 0,
&--$
&'. − &--.
&'$ = 0,
&-#$
&'. − &-#.
&'$ = 0,
&-%$
&'- − &-%-
&'$ = 0 (43)
is established. I get
�;";$
! = &+'"
&'# &'. + &
&'. ?�) *
+ �,! ;
&-#$
&'% =@ + &'-
&'#
*
+ �,! ;
&-.$
&'- =
+�) *
+ �,5 ;
&-#$
&'% = *
+ �,! ;
&-.$
&'- = − &'"
&'-
*
+ �,5 ;
&-.$
&'# = − �) *
+ �,! ;
&--$
&'% = *
+ �,5 ;
&-.$
&'# =
= &+'"
&'# &'. + &
&'. ;�) *
+
&-#
"
&'% = + &'-
&'#
*
+
&-.
"
&'- + �) *
+
&-#
-
&'%
*
+
&-.
"
&'- − &'"
&'-
*
+
&-.
-
&'# − �) *
+
&--
"
&'%
*
+
&-.
-
&'# (44)
from (43),Definision11. I get
�;";"
! = &+'"
&'# &'# + &
&'# ;�) *
+
&-#
"
&'% = + &'-
&'#
*
+
&-#
"
&'- + �) *
+
&-#
-
&'%
*
+
&-#
"
&'- − &'"
&'-
*
+
&-#
-
&'# − �) *
+
&--
"
&'%
*
+
&-#
-
&'# (45)
as if Binary Law being satisfied for all index except the dummy index of (44). If (45) is a tensor
equation, the dummy index of (45) can't make μ or ν. On the other hand, If all coordinate
systems satisfies Binary Law, dummy index of (45) should be μ or ν in consideration of
Definision5. I aim at the coexistence of these two demands. I rewrite (45) using Definision2 and
get
�!;!;! = &+'"
&'" &'"
+ &
&'"
;�) *
+
&-""
&'% = + &'-
&'"
*
+
&-""
&'- + �) *
+
&-"-
&'%
*
+
&-""
&'- − &'"
&'-
*
+
&-"-
&'"
− �) *
+
&--
"
&'%
*
+
&-"-
&'"
= &+'"
&'" &'"
+ &
&'"
;�" *
+
&-""
&'# = + &'#
&'"
*
+
&-""
&'# + �" *
+
&-"#
&'#
*
+
&-""
&'# − &'"
&'#
*
+
&-"#
&'"
− �" *
+
&-#
"
&'#
*
+
&-"#
&'"
. (46)
(46) satisfies two demands mentioned above together here. I rewrite (45) using Definision4
and get
Page 10 of 21
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Ichidayama, K. (2022). Property of Tensor Satisfying Binary Law 4. European Journal of Applied Sciences, 10(1). 556-576.
URL: http://dx.doi.org/10.14738/aivp.101.11865
�;!;!
! = &+'"
&'" &'" + &
&'" ?�) *
+
&-"
"
&'% @ + &'-
&'"
*
+
&-"
"
&'- + �) *
+
&-"
-
&'%
*
+
&-"
"
&'- − &'"
&'-
*
+
&-"
-
&'" − �) *
+
&--
"
&'%
*
+
&-"
-
&'"
= &+'"
&'" &'" + &
&'" ?�" *
+
&-"
"
&'#@ + &'#
&'"
*
+
&-"
"
&'# + �" *
+
&-"
#
&'#
*
+
&-"
"
&'# − &'"
&'#
*
+
&-"
#
&'" − �" *
+
&-#
"
&'#
*
+
&-"
#
&'". (47)
(47) satisfies two demands mentioned above together here. I get
�;!;!
! = &+'"
&'" &'" (48)
in consideration of Definision6 for (47). Because the second term of the right side doesn't exist
in (48), (42) can rewrite (48) using Definision4. End Proof
Proposition5 �;";";"
! = &!'"
&'# &'# &'# is established in tensor satisfying Binary Law.
Proof: If all coordinate systems satisfies Binary Law in
A
&-#$
&'% − &-%#
&'$ ,
&-.$
&'- − &--.
&'$ ,
&-.$
&'# − &-#.
&'$ ,
&--$
&'% − &-%-
&'$ ,
&-1$
&'3 − &-31
&'$ ,
&-1$
&'# − &-#1
&'$ ,
&-3$
&'% − &-%3
&'$ B,
A
&-.$
&'3 − &-3.
&'$ ,
&-1$
&'. − &-.1
&'$ ,
&-3$
&'- − &--3
&'$ ,
&-3$
&'# − &-#3
&'$ B
&-#$
&'% − &-%#
&'$ = 0,
&-.$
&'- − &--.
&'$ = 0,
&-.$
&'# − &-#.
&'$ = 0,
&--$
&'% − &-%-
&'$ = 0,
&-1$
&'3 − &-31
&'$ = 0,
&-1$
&'# − &-#1
&'$ = 0,
&-3$
&'% − &-%3
&'$ = 0,
&-.$
&'3 − &-3.
&'$ = 0,
&-1$
&'. − &-.1
&'$ = 0,
&-3$
&'- − &--3
&'$ = 0,
&-3$
&'# − &-#3
&'$ = 0 (49)
is established. I get
�
;/;0;6
. = 1!'&
1'(1')1'/ + 1+
1')1'/ ?�3 *
+ �4. ;
1-2,
1'( =@ + 1
1'/ ?
1'*
1'(
*
+ �4. ;
1-*,
1') =@
+ &
&'1 ?�) *
+ �,5 ;
&-%$
&'# = *
+ �,! ;
&--$
&'. =@ − &
&'1 ?
&'"
&'-
*
+ �,5 ;
&-#$
&'. =@ − &
&'1 ?�) *
+ �,! ;
&-%$
&'- = *
+ �,5 ;
&-#$
&'. =@
+ &+'3
&'# &'.
*
+ �,! ;
&-3$
&'1 = + &
&'. ?�) *
+ �,8 ;
&-%$
&'# =@ *
+ �,! ;
&-3$
&'1 = + &'-
&'#
*
+ �,8 ;
&--$
&'.= *
+ �,! ;
&-3$
&'1 =
+�) *
+ �,5 ;
&-%$
&'# = *
+ �,8 ;
&--$
&'. = *
+ �,! ;
&-3$
&'1 = − &'3
&'-
*
+ �,5 ;
&-#$
&'. = *
+ �,! ;
&-3$
&'1 =
−�) *
+ �,8 ;
&-%$
&'- = *
+ �,5 ;
&-#$
&'. = *
+ �,! ;
&-3$
&'1 = − &+'"
&'3 &'.
*
+ �,8 ;
&-#$
&'1 =
− &
&'. ?�) *
+ �,! ;
&-%$
&'3 =@ *
+ �,8 ;
&-#$
&'1 = − &'-
&'3
*
+ �,! ;
&--$
&'.= *
+ �,8 ;
&-#$
&'1 =
−�) *
+ �,5 ;
&-%$
&'3 = *
+ �,! ;
&--$
&'.= *
+ �,8 ;
&-#$
&'1 = + &'"
&'-
*
+ �,5 ;
&-3$
&'. = *
+ �,8 ;
&-#$
&'1 =
+�) *
+ �,! ;
&-%$
&'- = *
+ �,5 ;
&-3$
&'. = *
+ �,8 ;
&-#$
&'1 = − &+'"
&'# &'3
*
+ �,8 ;
&-.$
&'1 =
− &
&'3 ?�) *
+ �,! ;
&-%$
&'# =@ *
+ �,8 ;
&-.$
&'1 = − &'-
&'#
*
+ �,! ;
&--$
&'3= *
+ �,8 ;
&-.$
&'1 =
−�) *
+ �,5 ;
&-%$
&'# = *
+ �,! ;
&--$
&'3= *
+ �,8 ;
&-.$
&'1 = + &'"
&'-
*
+ �,5 ;
&-#$
&'3 = *
+ �,8 ;
&-.$
&'1 =
+�) *
+ �,! ;
&-%$
&'- = *
+ �,5 ;
&-#$
&'3 = *
+ �,8 ;
&-.$
&'1 =
= &!'"
&'# &'. &'1 + &+
&'. &'1 ;�) *
+
&-%
"
&'#= + &
&'1 ;
&'-
&'#
*
+
&--
"
&'.= + &
&'1 ;�) *
+
&-%
-
&'#
*
+
&--
"
&'.= − &
&'1 ;
&'"
&'-
*
+
&-#
-
&'.=
− &
&'1 ;�) *
+
&-%
"
&'-
*
+
&-#
-
&'.= + &+'3
&'# &'.
*
+
&-3
"
&'1 + &
&'. ;�) *
+
&-%
3
&'#= *
+
&-3
"
&'1 + &'-
&'#
*
+
&--
3
&'.
*
+
&-3
"
&'1
+�) *
+
&-%
-
&'#
*
+
&--
3
&'.
*
+
&-3
"
&'1 − &'3
&'-
*
+
&-#
-
&'.
*
+
&-3
"
&'1 − �) *
+
&-%
3
&'-
*
+
&-#
-
&'.
*
+
&-3
"
&'1 − &+'"
&'3 &'.
*
+
&-#
3
&'1
− &
&'. ;�) *
+
&-%
"
&'3= *
+
&-#
3
&'1 − &'-
&'3
*
+
&--
"
&'.
*
+
&-#
3
&'1 − �) *
+
&-%
-
&'3
*
+
&--
"
&'.
*
+
&-#
3
&'1 + &'"
&'-
*
+
&-3
-
&'.
*
+
&-#
3
&'1
Page 11 of 21
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European Journal of Applied Sciences (EJAS) Vol. 10, Issue 1, February-2022
Services for Science and Education – United Kingdom
+�) *
+
&-%
"
&'-
*
+
&-3
-
&'.
*
+
&-#
3
&'1 − &+'"
&'# &'3
*
+
&-.
3
&'1 − &
&'3 ;�) *
+
&-%
"
&'#= *
+
&-.
3
&'1 − &'-
&'#
*
+
&--
"
&'3
*
+
&-.
3
&'1
−�) *
+
&-%
-
&'#
*
+
&--
"
&'3
*
+
&-.
3
&'1 + &'"
&'-
*
+
&-#
-
&'3
*
+
&-.
3
&'1 + �) *
+
&-%
"
&'-
*
+
&-#
-
&'3
*
+
&-.
3
&'1 (50)
from (49),Definision12. I get
�;";";"
! = &!'"
&'# &'# &'# + &+
&'# &'# ;�) *
+
&-%
"
&'#= + &
&'# ;
&'-
&'#
*
+
&--
"
&'#= + &
&'# ;�) *
+
&-%
-
&'#
*
+
&--
"
&'#=
− &
&'# ;
&'"
&'-
*
+
&-#
-
&'#= − &
&'# ;�) *
+
&-%
"
&'-
*
+
&-#
-
&'#= + &+'3
&'# &'#
*
+
&-3
"
&'# + &
&'# ;�) *
+
&-%
3
&'#= *
+
&-3
"
&'#
+ &'-
&'#
*
+
&--
3
&'#
*
+
&-3
"
&'# + �) *
+
&-%
-
&'#
*
+
&--
3
&'#
*
+
&-3
"
&'# − &'3
&'-
*
+
&-#
-
&'#
*
+
&-3
"
&'# − �) *
+
&-%
3
&'-
*
+
&-#
-
&'#
*
+
&-3
"
&'#
− &+'"
&'3 &'#
*
+
&-#
3
&'# − &
&'# ;�) *
+
&-%
"
&'3= *
+
&-#
3
&'# − &'-
&'3
*
+
&--
"
&'#
*
+
&-#
3
&'# − �) *
+
&-%
-
&'3
*
+
&--
"
&'#
*
+
&-#
3
&'#
+ &'"
&'-
*
+
&-3
-
&'#
*
+
&-#
3
&'# + �) *
+
&-%
"
&'-
*
+
&-3
-
&'#
*
+
&-#
3
&'# − &+'"
&'# &'3
*
+
&-#
3
&'# − &
&'3 ;�) *
+
&-%
"
&'#= *
+
&-#
3
&'#
− &'-
&'#
*
+
&--
"
&'3
*
+
&-#
3
&'# − �) *
+
&-%
-
&'#
*
+
&--
"
&'3
*
+
&-#
3
&'# + &'"
&'-
*
+
&-#
-
&'3
*
+
&-#
3
&'# + �) *
+
&-%
"
&'-
*
+
&-#
-
&'3
*
+
&-#
3
&'# (51)
as if Binary Law being satisfied for all index except the dummy index of (50). If (51) is a tensor
equation, the dummy index of (51) can't make μ or ν. On the other hand, If all coordinate
systems satisfies Binary Law, dummy index of (51) should be μ or ν in consideration of
Definision5. I aim at the coexistence of these two demands. I rewrite (51) using Definision2 and
get
�!;!;!;! = &!'"
&'" &'" &'"
+ &+
&'" &'"
?�) *
+
&-%
"
&'"
@ + &
&'"
?
&'-
&'"
*
+
&--
"
&'"
@ + &
&'"
?�) *
+
&-%
-
&'"
*
+
&--
"
&'"
@
− &
&'"
?
&'"
&'-
*
+
&-"-
&'"
@ − &
&'"
?�) *
+
&-%
"
&'-
*
+
&-"-
&'"
@ + &+'3
&'" &'"
*
+
&-3
"
&'"
+ &
&'"
?�) *
+
&-%
3
&'"
@ *
+
&-3
"
&'"
+ &'-
&'"
*
+
&--
3
&'"
*
+
&-3
"
&'"
+ �) *
+
&-%
-
&'"
*
+
&--
3
&'"
*
+
&-3
"
&'"
− &'3
&'-
*
+
&-"-
&'"
*
+
&-3
"
&'"
− �) *
+
&-%
3
&'-
*
+
&-"-
&'"
*
+
&-3
"
&'"
− &+'"
&'3 &'"
*
+
&-"3
&'"
− &
&'"
;�) *
+
&-%
"
&'3= *
+
&-"3
&'"
− &'-
&'3
*
+
&--
"
&'"
*
+
&-"3
&'"
− �) *
+
&-%
-
&'3
*
+
&--
"
&'"
*
+
&-"3
&'"
+ &'"
&'-
*
+
&-3
-
&'"
*
+
&-"3
&'"
+ �) *
+
&-%
"
&'-
*
+
&-3
-
&'"
*
+
&-"3
&'"
− &+'"
&'" &'3
*
+
&-"3
&'"
− &
&'3 ?�) *
+
&-%
"
&'"
@ *
+
&-"3
&'"
− &'-
&'"
*
+
&--
"
&'3
*
+
&-"3
&'"
− �) *
+
&-%
-
&'"
*
+
&--
"
&'3
*
+
&-"3
&'"
+ &'"
&'-
*
+
&-"-
&'3
*
+
&-"3
&'"
+ �) *
+
&-%
"
&'-
*
+
&-"-
&'3
*
+
&-"3
&'"
= &!'"
&'" &'" &'"
+ &+
&'" &'"
?�" *
+
&-#
"
&'"
@ + &
&'"
?
&'#
&'"
*
+
&-#
"
&'"
@ + &
&'"
?�" *
+
&-#
#
&'"
*
+
&-#
"
&'"
@
− &
&'"
?
&'"
&'#
*
+
&-"#
&'"
@ − &
&'"
?�" *
+
&-#
"
&'#
*
+
&-"#
&'"
@ + &+'#
&'" &'"
*
+
&-#
"
&'"
+ &
&'"
?�" *
+
&-#
#
&'"
@ *
+
&-#
"
&'"
+ &'#
&'"
*
+
&-#
#
&'"
*
+
&-#
"
&'"
+ �" *
+
&-#
#
&'"
*
+
&-#
#
&'"
*
+
&-#
"
&'"
− &'#
&'#
*
+
&-"#
&'"
*
+
&-#
"
&'"
− �" *
+
&-#
#
&'#
*
+
&-"#
&'"
*
+
&-#
"
&'"
− &+'"
&'# &'"
*
+
&-"#
&'"
− &
&'"
;�" *
+
&-#
"
&'#= *
+
&-"#
&'"
− &'#
&'#
*
+
&-#
"
&'"
*
+
&-"#
&'"
− �" *
+
&-#
#
&'#
*
+
&-#
"
&'"
*
+
&-"#
&'"
+ &'"
&'#
*
+
&-#
#
&'"
*
+
&-"#
&'"
+ �" *
+
&-#
"
&'#
*
+
&-#
#
&'"
*
+
&-"#
&'"
− &+'"
&'" &'#
*
+
&-"#
&'"
− &
&'# ?�" *
+
&-#
"
&'"
@ *
+
&-"#
&'"
− &'#
&'"
*
+
&-#
"
&'#
*
+
&-"#
&'"
− �" *
+
&-#
#
&'"
*
+
&-#
"
&'#
*
+
&-"#
&'"
+ &'"
&'#
*
+
&-"#
&'#
*
+
&-"#
&'"
+ �" *
+
&-#
"
&'#
*
+
&-"#
&'#
*
+
&-"#
&'"
. (52)
(52) satisfies two demands mentioned above together here. I get
Page 12 of 21
567
Ichidayama, K. (2022). Property of Tensor Satisfying Binary Law 4. European Journal of Applied Sciences, 10(1). 556-576.
URL: http://dx.doi.org/10.14738/aivp.101.11865
�!;!;!;! = &!'"
&'" &'" &'"
− &
&'"
?
&'"
&'#
*
+
&-"#
&'"
@ − &+'"
&'# &'"
*
+
&-"#
&'"
− &+'"
&'" &'#
*
+
&-"#
&'"
+ &'"
&'#
*
+
&-"#
&'#
*
+
&-"#
&'"
(53)
in consideration of Definision6 for (52). I rewrite (51) using Definision4 and get
−�;!;!;!
! = − &!'"
&'" &'" &'" − &+
&'" &'" ;�) *
+
&-%
"
&'"= − &
&'" ;
&'-
&'"
*
+
&--
"
&'"= − &
&'" ;�) *
+
&-%
-
&'"
*
+
&--
"
&'"=
+ &
&'" ;
&'"
&'-
*
+
&-"
-
&'"= + &
&'" ;�) *
+
&-%
"
&'-
*
+
&-"
-
&'"= − &+'3
&'" &'"
*
+
&-3
"
&'" − &
&'" ;�) *
+
&-%
3
&'"= *
+
&-3
"
&'"
− &'-
&'"
*
+
&--
3
&'"
*
+
&-3
"
&'" − �) *
+
&-%
-
&'"
*
+
&--
3
&'"
*
+
&-3
"
&'" + &'3
&'-
*
+
&-"
-
&'"
*
+
&-3
"
&'" + �) *
+
&-%
3
&'-
*
+
&-"
-
&'"
*
+
&-3
"
&'"
+ &+'"
&'3 &'"
*
+
&-"
3
&'" + &
&'" ;�) *
+
&-%
"
&'3= *
+
&-"
3
&'" + &'-
&'3
*
+
&--
"
&'"
*
+
&-"
3
&'" + �) *
+
&-%
-
&'3
*
+
&--
"
&'"
*
+
&-"
3
&'"
− &'"
&'-
*
+
&-3
-
&'"
*
+
&-"
3
&'" − �) *
+
&-%
"
&'-
*
+
&-3
-
&'"
*
+
&-"
3
&'" + &+'"
&'" &'3
*
+
&-"
3
&'" + &
&'3 ;�) *
+
&-%
"
&'"= *
+
&-"
3
&'"
+ &'-
&'"
*
+
&--
"
&'3
*
+
&-"
3
&'" + �) *
+
&-%
-
&'"
*
+
&--
"
&'3
*
+
&-"
3
&'" − &'"
&'-
*
+
&-"
-
&'3
*
+
&-"
3
&'" − �) *
+
&-%
"
&'-
*
+
&-"
-
&'3
*
+
&-"
3
&'"
= − &!'"
&'" &'" &'" − &+
&'" &'" ;�" *
+
&-#
"
&'"= − &
&'" ;
&'#
&'"
*
+
&-#
"
&'"= − &
&'" ;�" *
+
&-#
#
&'"
*
+
&-#
"
&'"=
+ &
&'" ;
&'"
&'#
*
+
&-"
#
&'"= + &
&'" ;�" *
+
&-#
"
&'#
*
+
&-"
#
&'"= − &+'#
&'" &'"
*
+
&-#
"
&'" − &
&'" ;�" *
+
&-#
#
&'"= *
+
&-#
"
&'"
− &'#
&'"
*
+
&-#
#
&'"
*
+
&-#
"
&'" − �" *
+
&-#
#
&'"
*
+
&-#
#
&'"
*
+
&-#
"
&'" + &'#
&'#
*
+
&-"
#
&'"
*
+
&-#
"
&'" + �" *
+
&-#
#
&'#
*
+
&-"
#
&'"
*
+
&-#
"
&'"
+ &+'"
&'# &'"
*
+
&-"
#
&'" + &
&'" ;�" *
+
&-#
"
&'#= *
+
&-"
#
&'" + &'#
&'#
*
+
&-#
"
&'"
*
+
&-"
#
&'" + �" *
+
&-#
#
&'#
*
+
&-#
"
&'"
*
+
&-"
#
&'"
− &'"
&'#
*
+
&-#
#
&'"
*
+
&-"
#
&'" − �" *
+
&-#
"
&'#
*
+
&-#
#
&'"
*
+
&-"
#
&'" + &+'"
&'" &'#
*
+
&-"
#
&'" + &
&'# ;�" *
+
&-#
"
&'"= *
+
&-"
#
&'"
+ &'#
&'"
*
+
&-#
"
&'#
*
+
&-"
#
&'" + �" *
+
&-#
#
&'"
*
+
&-#
"
&'#
*
+
&-"
#
&'" − &'"
&'#
*
+
&-"
#
&'#
*
+
&-"
#
&'" − �" *
+
&-#
"
&'#
*
+
&-"
#
&'#
*
+
&-"
#
&'". (54)
(54) satisfies two demands mentioned above together here. I get
−�;!;!;!
! = − &!'"
&'" &'" &'" (55)
in consideration of Definision6 for (54). Because the second term of the right side doesn't exist
in (55),
�;";";"
! = &!'"
&'# &'# &'# (56)
can rewrite (55) using Definision4.
If all coordinate systems satisfies Binary Law in
A
&-%$
&'# − &-%#
&'$ ,
&--$
&'. − &--.
&'$ ,
&-#$
&'. − &-#.
&'$ ,
&-%$
&'- − &-%-
&'$ ,
&-3$
&'1 − &-31
&'$ ,
&-#$
&'1 − &-#1
&'$ ,
&-%$
&'3 − &-%3
&'$ B,
A
&-3$
&'. − &-3.
&'$ ,
&-.$
&'1 − &-.1
&'$ ,
&--$
&'3 − &--3
&'$ ,
&-#$
&'3 − &-#3
&'$ B,
&-%$
&'# − &-%#
&'$ = 0,
&--$
&'. − &--.
&'$ = 0,
&-#$
&'. − &-#.
&'$ = 0,
&-%$
&'- − &-%-
&'$ = 0,
&-3$
&'1 − &-31
&'$ = 0,
&-#$
&'1 − &-#1
&'$ = 0,
&-%$
&'3 − &-%3
&'$ = 0,
&-3$
&'. − &-3.
&'$ = 0,
&-.$
&'1 − &-.1
&'$ = 0,
&--$
&'3 − &--3
&'$ = 0,
&-#$
&'3 − &-#3
&'$ = 0 (57)
is established. I get
�
;";$;%
! = &!'"
&'# &'. &'1 + &+
&'. &'1 ?�) *
+ �,! ;
&-#$
&'% =@ + &
&'1 ?
&'-
&'#
*
+ �,! ;
&-.$
&'- =@
+ &
&'1 ?�) *
+ �,5 ;
&-#$
&'% = *
+ �,! ;
&-.$
&'- =@ − &
&'1 ?
&'"
&'-
*
+ �,5 ;
&-.$
&'# =@ − &
&'1 ?�) *
+ �,! ;
&--$
&'% = *
+ �,5 ;
&-.$
&'# =@
Page 13 of 21
568
European Journal of Applied Sciences (EJAS) Vol. 10, Issue 1, February-2022
Services for Science and Education – United Kingdom
+ &+'3
&'# &'.
*
+ �,! ;
&-1$
&'3 = + &
&'. ?�) *
+ �,8 ;
&-#$
&'% =@ *
+ �,! ;
&-1$
&'3 = + &'-
&'#
*
+ �,8 ;
&-.$
&'- = *
+ �,! ;
&-1$
&'3 =
+�) *
+ �,5 ;
&-#$
&'% = *
+ �,8 ;
&-.$
&'- = *
+ �,! ;
&-1$
&'3 = − &'3
&'-
*
+ �,5 ;
&-.$
&'# = *
+ �,! ;
&-1$
&'3 =
−�) *
+ �,8 ;
&--$
&'% = *
+ �,5 ;
&-.$
&'# = *
+ �,! ;
&-1$
&'3 = − &+'"
&'3 &'.
*
+ �,8 ;
&-1$
&'# =
− &
&'. ?�) *
+ �,! ;
&-3$
&'% =@ *
+ �,8 ;
&-1$
&'# = − &'-
&'3
*
+ �,! ;
&-.$
&'- = *
+ �,8 ;
&-1$
&'# =
−�) *
+ �,5 ;
&-3$
&'% = *
+ �,! ;
&-.$
&'- = *
+ �,8 ;
&-1$
&'# = + &'"
&'-
*
+ �,5 ;
&-.$
&'3 = *
+ �,8 ;
&-1$
&'# =
+�) *
+ �,! ;
&--$
&'% = *
+ �,5 ;
&-.$
&'3 = *
+ �,8 ;
&-1$
&'# = − &+'"
&'# &'3
*
+ �,8 ;
&-1$
&'. =
− &
&'3 ?�) *
+ �,! ;
&-#$
&'% =@ *
+ �,8 ;
&-1$
&'. = − &'-
&'#
*
+ �,! ;
&-3$
&'- = *
+ �,8 ;
&-1$
&'. =
−�) *
+ �,5 ;
&-#$
&'% = *
+ �,! ;
&-3$
&'- = *
+ �,8 ;
&-1$
&'. = + &'"
&'-
*
+ �,5 ;
&-3$
&'# = *
+ �,8 ;
&-1$
&'. =
+�) *
+ �,! ;
&--$
&'% = *
+ �,5 ;
&-3$
&'# = *
+ �,8 ;
&-1$
&'. =
= 1!'&
1'(1')1'/ + 1+
1')1'/ ;�3 *
+
1-(
&
1'2 = + 1
1'/ ;
1'*
1'(
*
+
1-)
&
1'* = + 1
1'/ ;�3 *
+
1-(
*
1'2
*
+
1-)
&
1'* = − &
&'1 ;
&'"
&'-
*
+
&-.
-
&'#=
− &
&'1 ;�) *
+
&--
"
&'%
*
+
&-.
-
&'#= + &+'3
&'# &'.
*
+
&-1
"
&'3 + &
&'. ;�) *
+
&-#
3
&'%= *
+
&-1
"
&'3 + &'-
&'#
*
+
&-.
3
&'-
*
+
&-1
"
&'3
+�) *
+ �,5 ;
&-#$
&'% = *
+
&-.
3
&'-
*
+
&-1
"
&'3 − &'3
&'-
*
+
&-.
-
&'#
*
+
&-1
"
&'3 − �) *
+
&--
3
&'%
*
+
&-.
-
&'#
*
+
&-1
"
&'3 − &+'"
&'3 &'.
*
+
&-1
3
&'#
− &
&'. ;�) *
+
&-3
"
&'% = *
+
&-1
3
&'# − &'-
&'3
*
+
&-.
"
&'-
*
+
&-1
3
&'# − �) *
+
&-3
-
&'%
*
+
&-.
"
&'-
*
+
&-1
3
&'# + &'"
&'-
*
+
&-.
-
&'3
*
+
&-1
3
&'#
+�) *
+
&--
"
&'%
*
+
&-.
-
&'3
*
+
&-1
3
&'# − &+'"
&'# &'3
*
+
&-1
3
&'. − &
&'3 ;�) *
+
&-#
"
&'% = *
+
&-1
3
&'. − &'-
&'#
*
+
&-3
"
&'-
*
+
&-1
3
&'.
−�) *
+
&-#
-
&'%
*
+
&-3
"
&'-
*
+
&-1
3
&'. + &'"
&'-
*
+
&-3
-
&'#
*
+
&-1
3
&'. + �) *
+
&--
"
&'%
*
+
&-3
-
&'#
*
+
&-1
3
&'. (58)
from (57),Definision12. I get
�;";";"
! = &!'"
&'# &'# &'# + &+
&'# &'# ;�) *
+
&-#
"
&'% = + &
&'# ;
&'-
&'#
*
+
&-#
"
&'- = + &
&'# ;�) *
+
&-#
-
&'%
*
+
&-#
"
&'-=
− &
&'# ;
&'"
&'-
*
+
&-#
-
&'#= − &
&'# ;�) *
+
&--
"
&'%
*
+
&-#
-
&'#= + &+'3
&'# &'#
*
+
&-#
"
&'3 + &
&'# ;�) *
+
&-#
3
&'%= *
+
&-#
"
&'3
+ &'-
&'#
*
+
&-#
3
&'-
*
+
&-#
"
&'3 + �) *
+
&-#
-
&'%
*
+
&-#
3
&'-
*
+
&-#
"
&'3 − &'3
&'-
*
+
&-#
-
&'#
*
+
&-#
"
&'3 − �) *
+
&--
3
&'%
*
+
&-#
-
&'#
*
+
&-#
"
&'3
− &+'"
&'3 &'#
*
+
&-#
3
&'# − &
&'# ;�) *
+
&-3
"
&'%= *
+
&-#
3
&'# − &'-
&'3
*
+
&-#
"
&'-
*
+
&-#
3
&'# − �) *
+
&-3
-
&'%
*
+
&-#
"
&'-
*
+
&-#
3
&'#
+ &'"
&'-
*
+
&-#
-
&'3
*
+
&-#
3
&'# + �) *
+
&--
"
&'%
*
+
&-#
-
&'3
*
+
&-#
3
&'# − &+'"
&'# &'3
*
+
&-#
3
&'# − &
&'3 ;�) *
+
&-#
"
&'% = *
+
&-#
3
&'#
− &'-
&'#
*
+
&-3
"
&'-
*
+
&-#
3
&'# − �) *
+
&-#
-
&'%
*
+
&-3
"
&'-
*
+
&-#
3
&'# + &'"
&'-
*
+
&-3
-
&'#
*
+
&-#
3
&'# + �) *
+
&--
"
&'%
*
+
&-3
-
&'#
*
+
&-#
3
&'# (59)
as if Binary Law being satisfied for all index except the dummy index of (58). If (59) is a tensor
equation, the dummy index of (59) can't make μ or ν. On the other hand, If all coordinate
systems satisfies Binary Law, dummy index of (59) should be μ or ν in consideration of
Definision5. I aim at the coexistence of these two demands. I rewrite (59) using Definision2 and
get
�!;!;!;! = &!'"
&'" &'" &'"
+ &+
&'" &'"
;�) *
+
&-""
&'% = + &
&'"
?
&'-
&'"
*
+
&-""
&'- @ + &
&'"
;�) *
+
&-"-
&'%
*
+
&-""
&'- =
− &
&'"
?
&'"
&'-
*
+
&-"-
&'"
@ − &
&'"
?�) *
+
&--
"
&'%
*
+
&-"-
&'"
@ + &+'3
&'" &'"
*
+
&-""
&'3 + &
&'"
;�) *
+
&-"3
&'% = *
+
&-""
&'3
Page 14 of 21
569
Ichidayama, K. (2022). Property of Tensor Satisfying Binary Law 4. European Journal of Applied Sciences, 10(1). 556-576.
URL: http://dx.doi.org/10.14738/aivp.101.11865
+ &'-
&'"
*
+
&-"3
&'-
*
+
&-""
&'3 + �) *
+
&-"-
&'%
*
+
&-"3
&'-
*
+
&-""
&'3 − &'3
&'-
*
+
&-"-
&'"
*
+
&-""
&'3 − �) *
+
&--
3
&'%
*
+
&-"-
&'"
*
+
&-""
&'3
− &+'"
&'3 &'"
*
+
&-"3
&'"
− &
&'"
;�) *
+
&-3
"
&'%= *
+
&-"3
&'"
− &'-
&'3
*
+
&-""
&'-
*
+
&-"3
&'"
− �) *
+
&-3
-
&'%
*
+
&-""
&'-
*
+
&-"3
&'"
+ &'"
&'-
*
+
&-"-
&'3
*
+
&-"3
&'"
+ �) *
+
&--
"
&'%
*
+
&-"-
&'3
*
+
&-"3
&'"
− &+'"
&'" &'3
*
+
&-"3
&'"
− &
&'3 ;�) *
+
&-""
&'% = *
+
&-"3
&'"
− &'-
&'"
*
+
&-3
"
&'-
*
+
&-"3
&'"
− �) *
+
&-"-
&'%
*
+
&-3
"
&'-
*
+
&-"3
&'"
+ &'"
&'-
*
+
&-3
-
&'"
*
+
&-"3
&'"
+ �) *
+
&--
"
&'%
*
+
&-3
-
&'"
*
+
&-"3
&'"
= &!'"
&'" &'" &'"
+ &+
&'" &'"
;�" *
+
&-""
&'# = + &
&'"
?
&'#
&'"
*
+
&-""
&'# @ + &
&'"
;�" *
+
&-"#
&'#
*
+
&-""
&'# =
− &
&'"
?
&'"
&'#
*
+
&-"#
&'"
@ − &
&'"
?�" *
+
&-#
"
&'#
*
+
&-"#
&'"
@ + &+'#
&'" &'"
*
+
&-""
&'# + &
&'"
;�" *
+
&-"#
&'# = *
+
&-""
&'#
+ &'#
&'"
*
+
&-"#
&'#
*
+
&-""
&'# + �" *
+
&-"#
&'#
*
+
&-"#
&'#
*
+
&-""
&'# − &'#
&'#
*
+
&-"#
&'"
*
+
&-""
&'# − �" *
+
&-#
#
&'#
*
+
&-"#
&'"
*
+
&-""
&'#
− &+'"
&'# &'"
*
+
&-"#
&'"
− &
&'"
;�" *
+
&-#
"
&'#= *
+
&-"#
&'"
− &'#
&'#
*
+
&-""
&'#
*
+
&-"#
&'"
− �" *
+
&-#
#
&'#
*
+
&-""
&'#
*
+
&-"#
&'"
+ &'"
&'#
*
+
&-"#
&'#
*
+
&-"#
&'"
+ �" *
+
&-#
"
&'#
*
+
&-"#
&'#
*
+
&-"#
&'"
− &+'"
&'" &'#
*
+
&-"#
&'"
− &
&'# ;�" *
+
&-""
&'# = *
+
&-"#
&'"
− &'#
&'"
*
+
&-#
"
&'#
*
+
&-"#
&'"
− �" *
+
&-"#
&'#
*
+
&-#
"
&'#
*
+
&-"#
&'"
+ &'"
&'#
*
+
&-#
#
&'"
*
+
&-"#
&'"
+ �" *
+
&-#
"
&'#
*
+
&-#
#
&'"
*
+
&-"#
&'"
. (60)
(60) satisfies two demands mentioned above together here. I get
�!;!;!;! = &!'"
&'" &'" &'"
+ &+
&'" &'"
;�" *
+
&-""
&'# = + &
&'"
?
&'#
&'"
*
+
&-""
&'# @ + &
&'"
;�" *
+
&-"#
&'#
*
+
&-""
&'# =
− &
&'"
?
&'"
&'#
*
+
&-"#
&'"
@ + &+'#
&'" &'"
*
+
&-""
&'# + &
&'"
;�" *
+
&-"#
&'# = *
+
&-""
&'# + &'#
&'"
*
+
&-"#
&'#
*
+
&-""
&'#
+�" *
+
&-"#
&'#
*
+
&-"#
&'#
*
+
&-""
&'# − &'#
&'#
*
+
&-"#
&'"
*
+
&-""
&'# − &+'"
&'# &'"
*
+
&-"#
&'"
− &'#
&'#
*
+
&-""
&'#
*
+
&-"#
&'"
+ &'"
&'#
*
+
&-"#
&'#
*
+
&-"#
&'"
− &+'"
&'" &'#
*
+
&-"#
&'"
− &
&'# ;�" *
+
&-""
&'# = *
+
&-"#
&'"
(61)
in consideration of Definision6 for (60). I rewrite (59) using Definision4 and get
−�;!;!;!
! = − &!'"
&'" &'" &'" − &+
&'" &'" ?�) *
+
&-"
"
&'%@ − &
&'" ?&'-
&'"
*
+
&-"
"
&'- @ − &
&'" ?�) *
+
&-"
-
&'%
*
+
&-"
"
&'- @
+ &
&'" ;
&'"
&'-
*
+
&-"
-
&'"= + &
&'" ;�) *
+
&--
"
&'%
*
+
&-"
-
&'"= − &+'3
&'" &'"
*
+
&-"
"
&'3 − &
&'" ;�) *
+
&-"
3
&'% = *
+
&-"
"
&'3
− &'-
&'"
*
+
&-"
3
&'-
*
+
&-"
"
&'3 − �) *
+
&-"
-
&'%
*
+
&-"
3
&'-
*
+
&-"
"
&'3 + &'3
&'-
*
+
&-"
-
&'"
*
+
&-"
"
&'3 + �) *
+
&--
3
&'%
*
+
&-"
-
&'"
*
+
&-"
"
&'3
+ &+'"
&'3 &'"
*
+
&-"
3
&'" + &
&'" ;�) *
+
&-3
"
&'% = *
+
&-"
3
&'" + &'-
&'3
*
+
&-"
"
&'-
*
+
&-"
3
&'" + �) *
+
&-3
-
&'%
*
+
&-"
"
&'-
*
+
&-"
3
&'"
− &'"
&'-
*
+
&-"
-
&'3
*
+
&-"
3
&'" − �) *
+
&--
"
&'%
*
+
&-"
-
&'3
*
+
&-"
3
&'" + &+'"
&'" &'3
*
+
&-"
3
&'" + &
&'3 ?�) *
+
&-"
"
&'%@ *
+
&-"
3
&'"
+ &'-
&'"
*
+
&-3
"
&'-
*
+
&-"
3
&'" + �) *
+
&-"
-
&'%
*
+
&-3
"
&'-
*
+
&-"
3
&'" − &'"
&'-
*
+
&-3
-
&'"
*
+
&-"
3
&'" − �) *
+
&--
"
&'%
*
+
&-3
-
&'"
*
+
&-"
3
&'"
= − &!'"
&'" &'" &'" − &+
&'" &'" ?�" *
+
&-"
"
&'#@ − &
&'" ?
&'#
&'"
*
+
&-"
"
&'#@ − &
&'" ?�" *
+
&-"
#
&'#
*
+
&-"
"
&'#@
+ &
&'" ;
&'"
&'#
*
+
&-"
#
&'"= + &
&'" ;�" *
+
&-#
"
&'#
*
+
&-"
#
&'"= − &+'#
&'" &'"
*
+
&-"
"
&'# − &
&'" ;�" *
+
&-"
#
&'#= *
+
&-"
"
&'#
− &'#
&'"
*
+
&-"
#
&'#
*
+
&-"
"
&'# − �" *
+
&-"
#
&'#
*
+
&-"
#
&'#
*
+
&-"
"
&'# + &'#
&'#
*
+
&-"
#
&'"
*
+
&-"
"
&'# + �" *
+
&-#
#
&'#
*
+
&-"
#
&'"
*
+
&-"
"
&'#
Page 15 of 21
570
European Journal of Applied Sciences (EJAS) Vol. 10, Issue 1, February-2022
Services for Science and Education – United Kingdom
+ &+'"
&'# &'"
*
+
&-"
#
&'" + &
&'" ;�" *
+
&-#
"
&'#= *
+
&-"
#
&'" + &'#
&'#
*
+
&-"
"
&'#
*
+
&-"
#
&'" + �" *
+
&-#
#
&'#
*
+
&-"
"
&'#
*
+
&-"
#
&'"
− &'"
&'#
*
+
&-"
#
&'#
*
+
&-"
#
&'" − �" *
+
&-#
"
&'#
*
+
&-"
#
&'#
*
+
&-"
#
&'" + &+'"
&'" &'#
*
+
&-"
#
&'" + &
&'# ?�" *
+
&-"
"
&'#@ *
+
&-"
#
&'"
+ &'#
&'"
*
+
&-#
"
&'#
*
+
&-"
#
&'" + �" *
+
&-"
#
&'#
*
+
&-#
"
&'#
*
+
&-"
#
&'" − &'"
&'#
*
+
&-#
#
&'"
*
+
&-"
#
&'" − �" *
+
&-#
"
&'#
*
+
&-#
#
&'"
*
+
&-"
#
&'". (62)
(62) satisfies two demands mentioned above together here. I get
−�;!;!;!
! = − &!'"
&'" &'" &'" (63)
in consideration of Definision6 for (62). Because the second term of the right side doesn't exist
in (63), (56) can rewrite (63) using Definision4. End Proof
ABOUT A COORDINATE TRANSFORMATIONS EQUATION IN TENSOR SATISFYING
BINARY LAW
Proposition6 When all coordinate systems satisfies Binary Law, �!
! = �"
" is established for �!
!
components of a tensor satisfying Binary law of rank zero.
Proof: If all coordinate systems satisfies Binary Law, I get
�!
! = &'"
&'#
&'#
&'"
�"
" (64)
from Definision13. Because (64) accords in Definision13, the components of a tensor of rank
zero is equivalent with components of a tensor satisfying Binary law of rank zero. If all
coordinate systems satisfies Binary Law, I get
�! = &'"
&'# �" (65)
from Definision14. Because (65) accords in Definision14, the contravariant components of a
tensor of the first rank is equivalent with contravariant components of a tensor satisfying
Binary law of the first rank. If all coordinate systems satisfies Binary Law, I get
�! = &'#
&'"
�" (66)
from Definision15. Because (66) accords in Definision15, the covariant components of a tensor
of the first rank is equivalent with covariant components of a tensor satisfying Binary law of
the first rank. End Proof
Proposition7 When all coordinate systems satisfies Binary Law, �!" = �"! is established for
�!" contravariant components of a tensor satisfying Binary law of the second rank.
Proof: I get
�!" = &'"
&'.
&'#
&'1 �$% (67)
as if Binary Law being satisfied for all index except the dummy index of Definision16. If all
coordinate systems satisfies Binary Law, dummy index of (67) should be μ or ν in consideration
of Definision5. Thus, I rewrite dummy index of (67) and get
�!" = &'"
&'"
&'#
&'# �!" = �!", (68)
Page 16 of 21
571
Ichidayama, K. (2022). Property of Tensor Satisfying Binary Law 4. European Journal of Applied Sciences, 10(1). 556-576.
URL: http://dx.doi.org/10.14738/aivp.101.11865
�!" = &'"
&'#
&'#
&'# �"" = &'"
&'# �"", (69)
�!" = &'"
&'#
&'#
&'"
�"!, (70)
�!" = &'"
&'"
&'#
&'"
�!! = &'#
&'"
�!!. (71)
I decide not to handle (68),(69),(71) because &'"
&'" ,
&'#
&'# exists in (68),(69),(71). I rewrite (70)
using Definision2,Definision3 and get
�!
! = &'"
&'#
&'"
&'#
�"
". (72)
I rewrite �!" = �!", �"! = �"! using Definision2,Definision3 and get
�!" = �!
!
, �"! = �"
". (73)
I get
&'"
&'#
&'#
&'"
�"! = &'"
&'#
&'"
&'#
�"
"
= &'"
&'#
&'"
&'#
�"! (74)
from (70),(72),(73). I get
�!" = �!
! = �"
" = �"! (75)
from (72),(73). End Proof
Proposition8 When all coordinate systems satisfies Binary Law, �!" = �"! is established for
�!" covariant components of a tensor satisfying Binary law of the second rank.
Proof: I get
�!" = &'.
&'"
&'1
&'# �$% (76)
as if Binary Law being satisfied for all index except the dummy index of Definision17. If all
coordinate systems satisfies Binary Law, dummy index of (76) should be μ or ν in consideration
of Definision5. Thus, I rewrite dummy index of (76) and get
�!" = &'"
&'"
&'#
&'# �!" = �!", (77)
�!" = &'#
&'"
&'#
&'# �"" = &'#
&'"
�"", (78)
�!" = &'#
&'"
&'"
&'# �"!, (79)
�!" = &'"
&'"
&'"
&'# �!! = &'"
&'# �!!. (80)
I decide not to handle (77),(78),(80) because &'"
&'" ,
&'#
&'# exists in (77),(78),(80). I rewrite (79)
using Definision2,Definision3 and get
�"
" = &'"
&'#
&'"
&'# �!
!
. (81)
I rewrite �!" = �!", �"! = �"! using Definision2,Definision3 and get
�!" = �"
", �"! = �!
!
. (82)
I get
&'#
&'"
&'"
&'# �"! = &'"
&'#
&'"
&'# �!
!
Page 17 of 21
572
European Journal of Applied Sciences (EJAS) Vol. 10, Issue 1, February-2022
Services for Science and Education – United Kingdom
= &'"
&'#
&'"
&'# �"! (83)
from (79),(81),(82). I get
�!" = �"
" = �!
! = �"! (84)
from (81),(82). End Proof
Proposition9 When all coordinate systems satisfies Binary Law, �"
! = &'"
&'#
&'"
&'# �!
" is established
for �"
! components of the mixed tensor satisfying Binary law of the second rank.
Proof: I get
�"
! = &'"
&'.
&'1
&'# �%
$ (85)
as if Binary Law being satisfied for all index except the dummy index of Definision18. If all
coordinate systems satisfies Binary Law, dummy index of (85) should be μ or ν in consideration
of Definision5. Thus, I rewrite dummy index of (85) and get
�"
! = &'"
&'"
&'#
&'# �"
! = �"
!
, (86)
�"
! = &'"
&'#
&'#
&'# �"
" = &'"
&'# �"
", (87)
�"
! = &'"
&'#
&'"
&'# �!
", (88)
�"
! = &'"
&'"
&'"
&'# �!
! = &'"
&'# �!
!
. (89)
I decide not to handle (86),(87),(89) because &'"
&'" ,
&'#
&'# exists in (86),(87),(89). End Proof
Proposition10 When all coordinate systems satisfies Binary Law, �!"" = &'#
&'"
&'"
&'#
&'"
&'# �"!! is
established for �!"" covariant components of a tensor satisfying Binary law of the third rank.
Proof: I get
�!"" = &'1
&'"
&'-
&'#
&'$
&'# �%5, (90)
as if Binary Law being satisfied for all index except the dummy index of Definision19. If all
coordinate systems satisfies Binary Law, dummy index of (90) should be μ or ν in consideration
of Definision5. Thus, I rewrite dummy index of (90) and get
�!"" = &'"
&'"
&'#
&'#
&'#
&'# �!"" = �!"", (91)
�!"" = &'#
&'"
&'#
&'#
&'#
&'# �""" = &'#
&'"
�""", (92)
�!"" = &'#
&'"
&'"
&'#
&'#
&'# �"!" = &'#
&'"
&'"
&'# �"!", (93)
�!"" = &'#
&'"
&'"
&'#
&'"
&'# �"!!, (94)
�!"" = &'"
&'"
&'"
&'#
&'"
&'# �!!! = &'"
&'#
&'"
&'# �!!!, (95)
�!"" = &'"
&'"
&'#
&'#
&'"
&'# �!"! = &'"
&'# �!"!. (96)
I decide not to handle (91),(92),(93),(95),(96) because &'"
&'" ,
&'#
&'# exists in
(91),(92),(93),(95),(96). Because two same index is existing in �!"", �"!! of (94), it is a problem.
I rewrite (94) using Definision2,Definision3 and get
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URL: http://dx.doi.org/10.14738/aivp.101.11865
�""
" = &'"
&'#
&'"
&'#
&'"
&'# �!!
! . (97)
Two same index is existing in �""
" , �!!
! of (97), but the problem doesn't occur because one is
dummy index. Because (97) could rewrite (94), the problem of (94) was solved. I rewrite �!"" =
�!"", �"!! = �"!! using Definision2,Definision3 and get
�!"" = �""
" , �"!! = �!!
! . (98)
I get
&'#
&'"
&'"
&'#
&'"
&'# �"!! = &'"
&'#
&'"
&'#
&'"
&'# �!!
!
= &'"
&'#
&'"
&'#
&'"
&'# �"!! (99)
from (94),(97),(98). End Proof
Proposition11 When all coordinate systems satisfies Binary Law, �""
! = &'#
&'"
�!!
" is established
for �""
! components of the mixed tensor satisfying Binary law of the third rank of the second
rank covariant in the first rank contravariant.
Proof: I get
�""
! = &'"
&'1
&'-
&'#
&'$
&'# �5,
% (100)
as if Binary Law being satisfied for all index except the dummy index of Definision20. If all
coordinate systems satisfies Binary Law, dummy index of (100) should be μ or ν in
consideration of Definision5. Thus, I rewrite dummy index of (100) and get
�""
! = &'"
&'"
&'#
&'#
&'#
&'# �""
! = �""
! , (101)
�""
! = &'"
&'#
&'#
&'#
&'#
&'# �""
" = &'"
&'# �""
" , (102)
�""
! = &'"
&'#
&'"
&'#
&'#
&'# �!"
" = &'"
&'#
&'"
&'# �!"
" , (103)
�""
! = &'"
&'#
&'"
&'#
&'"
&'# �!!
" , (104)
�""
! = &'"
&'"
&'"
&'#
&'"
&'# �!!
! = &'"
&'#
&'"
&'# �!!
! , (105)
�""
! = &'"
&'"
&'#
&'#
&'"
&'# �"!
! = &'"
&'# �"!
! . (106)
I decide not to handle (101),(102),(103),(105),(106) because &'"
&'" ,
&'#
&'# exists in
(101),(102),(103),(105),(106). Because two same index is existing in �""
! , �!!
" of (104), it is a
problem. I rewrite (104) using Definision4 and get
�!!
! = &'"
&'#
&'#
&'"
&'#
&'"
�""
" . (107)
Two same index is existing in �!!
! , �""
" of (107), but the problem doesn't occur because one is
dummy index. Because (107) could rewrite (104), the problem of (104) was solved. I rewrite
�""
! = �""
! , �!!
" = �!!
" using Definision4 and get
�""
! = �!!
! , �!!
" = �""
" . (108)
I get
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European Journal of Applied Sciences (EJAS) Vol. 10, Issue 1, February-2022
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&'"
&'#
&'"
&'#
&'"
&'# �!!
" = &'"
&'#
&'#
&'"
&'#
&'"
�""
"
= &'"
&'#
&'#
&'"
&'#
&'# �!!
" (109)
from (104),(107),(108). I get
&'"
&'#
&'"
&'# = &'#
&'"
&'#
&'" (110)
from (109). End Proof
Proposition12 When all coordinate systems satisfies Binary Law, �"""
! = �!!!
" is established
for �"""
! components of the mixed tensor satisfying Binary law of the fourth rank of the third
rank covariant in the first rank contravariant.
Proof: I get
�"""
! = &'"
&'-
&'$
&'#
&'4
&'#
&'5
&'# �,9:
5 (111)
as if Binary Law being satisfied for all index except the dummy index of Definision21. If all
coordinate systems satisfies Binary Law, dummy index of (111) should be μ or ν in
consideration of Definision5. Thus, I rewrite dummy index of (111) and get
�"""
! = &'"
&'"
&'#
&'#
&'#
&'#
&'#
&'# �"""
! = �"""
! , (112)
�"""
! = &'"
&'#
&'#
&'#
&'#
&'#
&'#
&'# �"""
" = &'"
&'# �"""
" , (113)
�"""
! = &'"
&'#
&'"
&'#
&'#
&'#
&'#
&'# �!""
" = &'"
&'#
&'"
&'# �!""
" , (114)
�"""
! = &'"
&'#
&'"
&'#
&'"
&'#
&'#
&'# �!!"
" = &'"
&'#
&'"
&'#
&'"
&'# �!!"
" , (115)
�"""
! = &'"
&'#
&'"
&'#
&'"
&'#
&'"
&'# �!!!
" , (116)
�"""
! = &'"
&'"
&'"
&'#
&'"
&'#
&'"
&'# �!!!
! = &'"
&'#
&'"
&'#
&'"
&'# �!!!
! , (117)
�"""
! = &'"
&'"
&'#
&'#
&'"
&'#
&'"
&'# �"!!
! = &'"
&'#
&'"
&'# �"!!
! , (118)
�"""
! = &'"
&'"
&'#
&'#
&'#
&'#
&'"
&'# �""!
! = &'"
&'# �""!
! . (119)
I decide not to handle (112),(113),(114),(115),(117),(118),(119) because &'"
&'" ,
&'#
&'# exists in
(112),(113),(114),(115),(117),(118),(119). Because three same index is existing in �"""
! , �!!!
" of
(116), it is a problem. I rewrite (116) using Definision4 and get
�!!"
! = &'"
&'#
&'#
&'"
&'#
&'"
&'"
&'# �""!
" . (120)
Two same index is existing in �!!"
! , �""!
" of (120), but the problem doesn't occur because one is
dummy index. Because (120) could rewrite (116), the problem of (116) was solved. I rewrite
�"""
! = �"""
! , �!!!
" = �!!!
" using Definision4 and get
�"""
! = �!!"
! , �!!!
" = �""!
" . (121)
I get
&'"
&'#
&'"
&'#
&'"
&'#
&'"
&'# �!!!
" = &'"
&'#
&'#
&'"
&'#
&'"
&'"
&'# �""!
"
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Ichidayama, K. (2022). Property of Tensor Satisfying Binary Law 4. European Journal of Applied Sciences, 10(1). 556-576.
URL: http://dx.doi.org/10.14738/aivp.101.11865
= &'"
&'#
&'#
&'"
&'#
&'"
&'"
&'# �!!!
" (122)
from (116),(120),(121). I get
&'"
&'#
&'"
&'# = &'#
&'"
&'#
&'" (123)
from (122). I get
�"""
! = �!!"
! = �""!
" = �!!!
" ,
�"""
! = �!" = �"! = �!!!
" (124)
from (120),(121). I get
�"""
! = �!" = �"
" = �!
! = �"! = �!!!
" (125)
from (84),(124). I rewrite (125) by consideration of �///
. → �;/;/;/
. , �!!!
" → �;!;!;! " , �!" → �!;",
�"! → �";!, �"
" → �"
;"
, �!
! → �!
;!
,(4),(7),(56), μ, ν-inversion form of (4),(7),(56) and get
&!'"
&'# &'# &'# = &'"
&'# = &'#
&'#
− �!
*
+ ;
&-#"
&'# =
= &'"
&'"
− �"
*
+ ;
&-"#
&'" = = &'#
&'" = &!'#
&'" &'" &'". (126)
I get
&!'"
&'# &'# &'# = &'"
&'# = &'#
&'#
− �!
*
+ ;
&-#"
&'# =
= &'"
&'"
− �"
*
+ ;
&-"#
&'" = = &'#
&'" = &!'#
&'" &'" &'" = � (127)
from (126),Definision7. End Proof
DISCUSSION
About Proposition1
If all coordinate systems satisfies Binary Law, I get
�!;" = &'"
&'# − �"
*
+ �"" ;
&-"#
&'# + &-##
&'" − &-"#
&'# =
= &'"
&'# − �"
*
+ �"" ;
&-##
&'" = = &'"
&'# − �"
*
+
&-#
#
&'" (128)
from Definision8. At first I applied Binary Law only for a part to show in
�!;" = &'"
&'# − �)
*
+ �,) ?
&-"$
&'#
E + &-#$
&'" − &-"#
&'$
E @ (129)
for Definision8 and got (2) in this article. Then, I applied Binary Law only for a part to show in
�!<;"= = &'"6
&'#6 − �)
*
+
&-#6
%
&'"6 (130)
for (2) and got (3). I get (128) as if Binary Law being satisfied for all index of (3).
(128) isn't tensor satisfying Binary Law here. On the other hand, I rewrite (3) using Definision2
and get (4). (4) is tensor satisfying Binary Law here.
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European Journal of Applied Sciences (EJAS) Vol. 10, Issue 1, February-2022
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References
Ichidayama, K., Introduction of the Tensor Which Satisfied Binary Law, Jounal of Modern Physics, 2017. 8: p. 126-
132.
Ichidayama, K., Property of Tensor Satisfying Binary Law 3, Advanced Studies in Theoretical Physics, 2021.
15(4): p. 201-234.
Dirac, P.A.M, General Theory of Relativity, Tokyo Tosho, 1988.
Fleisch, D., A Student’s Guide to Vector and Tensors, Iwanami Shoten, 2014.