1. Introduction
Definition 1. For every coordinate system, there is no immediate reason for pre- ferring certain systems of co-ordinates to others.
Definition 2. I named
“Binary Law”.
Definition 3.
is established.
Definition 4.
is established.
Definition 5.
is established.
Definition 6. Convariant and contravariant tensor of the first rank
satisfied
[1] .
Definition 7. Tensor of rank zero
satisfied
[1] .
Definition 8. If tensor
satisfied
, this tensor
was named sym- metric tensor [1] .
Definition 9. Convariant differentiation for Convariant Bector
satisfied
[1] .
Definition 10.
and
are establishment [2] .
Definition 11. Convariant differentiation for contravariant bector
satisfied
[2] .
Definition 12. Convariant differentiation for Scalar
satisfied
[2] .
2. About Reason to Take Binary Law into Consideration
We will have to receive existence of the absolute coordinate system if Definition 1 is not established. Therefore, we must accept establishment of Definition 1.
Proposition 1. Definition 1 is not established for all coordinate systems ![]()
Proof: All coordinate systems
thinks about
in a standard and can divide it into two next groups.
![]()
(1)
I think that I change the coordinate systems of the standard
of (1) for all coordi- nate systems
sequentially now. By the way, the difference cannot occur between each conclusion to be provided here if Definition 1 is established. This reason is that all coordinate systems
has a privilege of the equality each other if Definition 1 is established. At first (1) gets an invariable conclusion for
exchange. Therefore, at least (1) must get an invariable conclusion for the next
exchange if Definition 1 is established. Here, I get
![]()
(2)
by
exchange from (1). Therefore, (2) must be equal with (1) if Definition 1 is established. By the way,
of (1) is equal with
of (2), but
of (1) is not equal with
of (2). In other words, (2) is not equal with (1). Therefore, Definition 1 is not established for all coor- dinate systems
.
-End Proof
Establishment of Proposition 1 is a problem in thinking that Definition 1 must be established. Therefore, I aim at getting establishment of Definition 1 for all coordinate systems
.
Proposition 2. If all coordinate systems
satisfies
, Definition 1 is established for all coordinate systems
.
Proof: I get
![]()
(3)
![]()
(4)
from (1), (2) if all coordinate systems
satisfies
(5)
(3) is equal with (4) here. In other words, (2) is equal with (1) if all coordinate sys- tems
satisfies (5). Therefore, Definition 1 is established for all coordi- nate systems
if all coordinate systems
satisfies (5).
-End Proof
Proposition 3. If all coordinate systems
satisfies
, all coordinate systems
shifts to only two of ![]()
Proof: If all coordinate systems
satisfies (5), I get
than all coordinate systems
.
-End Proof
Proposition 4. If
is established,
is esta- blished.
Proof: I get
(6)
from (5), (7) if I assume establishment of
(7)
when (5) is established. Because (6) includes contradiction,
(8)
is established when (5) is established.
-End Proof
Proposition 5. If
is established,
are established.
Proof: When (5) is established, (8) is established from Proposition 4. Therefore, I get
(9)
from (8), (10) if I assume establishment of
when (5) is established. I can rewrite
as
(10)
here. When (5) is established, I get
(11)
from Definition 3. Because (9) includes contradiction for (11),
(12)
is established when (5) is established.
Similary, I get
(13)
from (8), (14) if I assume establishment of
when (5) is established. I can rewrite
as
(14)
here. When (5) is established, I get
(15)
from Definition 4. Because (13) includes contradiction for (15),
(16)
is established when (5) is established.
Similary, I get
(17)
from (8), (18) if I assume establishment of
when (5) is established. I can rewrite
as
(18)
here. When (5) is established, I get
(19)
from Definition 5. Because (17) includes contradiction for (19),
(20)
is established when (5) is established. And, I get
(21)
from (12), (16), (20).
-End Proof
3. About the Tensor Which Satisfied Binary Law
We will have to think about adaptation of the establishment of Binary Law for the coordinate systems
in the tensor if we think about establishment of Binary Law for all coordinate systems
. Therefore, I decided to report Tensor when all coordinate systems
satisfied Binary Law.
Proposition 6. If all coordinate systems
satisfied
, Convariant and Contravariant Tensor of the first rank does not change the form of the equation.
Proof: I get
(22)
from Definition 6 if all coordinate systems
satisfies (5). Definition 6 and (22) are equal here. Therefore, if all coordinate systems
satisfied (5), Convariant and Contravariant Tensor of the first rank does not change the form of the equation.
-End Proof
Proposition 7. Tensor of the second rank becomes Symmetric Tensor if all coor- dinate systems
satisfies ![]()
Proof: I get
(23)
from Definition 7 if all coordinate systems
satisfies (5). Definition 7 and (23) are equal here. We can use (12), (16), (20), (21) for (23) by considering Pro- position 5 here. And we can rewrite (23) by using (12), (16) for
(24)
Then, I get
(25)
from (23),(24). And we can rewrite (23) by using (20), (21) for
(26)
Then, I get
(27)
from (26). Therefore, Tensor of the second rank becomes Symmetric Tensor than consideration of Definition 8 when all coordinate systems
satisfies (5).
-End Proof
Proposition 8. If all coordinate systems
satisfied
, The distance of two points be able to change oneself in connection with the metric of space.
Proof: I get
(28)
from Definition 10 if all coordinate systems
satisfies (5). I get
![]()
(29)
(30)
(31)
from Definition 9 if all coordinate systems
satisfies (5). By the way, we cannot handle (30), (31) according to Proposition 3. We can use (12), (16), (20), (21) for (29) by considering Proposition 5 here. And we must rewrite (29) by using (16) for
(32)
(33)
I decide not to handle (33) by consideration of (28) here. Well, I get conclution from (32) that if all coordinate systems
satisfied (5), Scalar quantity be able to change oneself in connection with the metric of space. Here, This Scalar quantity expressed the all of quantity expressed as Scalar. Therefore, I get conclution that the distance of two points be able to change oneself in connection with the metric of space.
-End Proof
Proposition 9. If all coordinate systems
satisfied
, convariant differentiation for Contravariant Bector
behave like a convariant differentiation for Scalar ![]()
Proof: I get
![]()
(34)
(35)
(36)
from Definition 11 if all coordinate systems
satisfies (5). By the way, we cannot handle (35), (36) according to Proposition 3. We can use (12), (16), (20), (21) for (34) by considering Proposition 5 here. And we must rewrite (34) by using (21) for
![]()
(37)
And, I can get
(38)
from (37) for consideration of (28). And we can rewrite (38) by using (21) for
(39)
Because the second term of the right side of (38) does not exist here, we may adopt (38) and (39) description form of which. Well, I get conclution from (39), Definition 12 that if all coordinate systems
satisfied (5), Convariant differentiation for Contravariant Bector
behave like a Convariant differentiation for Scalar
.
-End Proof
4. Discussion
About Definition 2:
I named (5) “Binary Law” by Proposition 3.
About Proposition 6:
Convariant and contravariant tensor of the first rank don’t change the formula whether it’s satisfied (5) or not.
About Proposition 8:
In (32), we can think that
expressed the distance of two points in
is
establishment and this is constant. And,
expresses the distance of two points in general and this is not constant.
About Proposition 9:
In (39), we can handle
as tensor similarly
.