_{1}

P: For every coordinate system, there is no immediate reason for preferring certain systems of co-ordinates to others. If we don’t recognize that P is establishment, we must recognize to existence of the absolute coordinate system. Therefore, we must recognize that P is establishment. Nevertheless, I got conclusion that P isn’t es-tablishment for all coordinate systems . If P is establishment, this is the trouble. As against, I got conclusion that if we consider “Binary Law” for all coordinate systems , P is establishment for all coordinate systems . If we consider Binary Law for all coordinate systems , we must consider Binary Law for the coordinate systems using into Tensor, too. So, I decided to report for the Tensor which satisfied Binary Law.

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

Definition 3.

Definition 4.

Definition 5.

Definition 6. Convariant and contravariant tensor of the first rank

Definition 7. Tensor of rank zero

Definition 8. If tensor

Definition 9. Convariant differentiation for Convariant Bector

Definition 10.

Definition 11. Convariant differentiation for contravariant bector

Definition 12. Convariant differentiation for Scalar

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

I think that I change the coordinate systems of the standard

by

-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

Proof: I get

from (1), (2) if all coordinate systems

(3) is equal with (4) here. In other words, (2) is equal with (1) if all coordinate sys- tems

-End Proof

Proposition 3. If all coordinate systems

Proof: If all coordinate systems

-End Proof

Proposition 4. If

Proof: I get

from (5), (7) if I assume establishment of

when (5) is established. Because (6) includes contradiction,

is established when (5) is established.

-End Proof

Proposition 5. If

Proof: When (5) is established, (8) is established from Proposition 4. Therefore, I get

from (8), (10) if I assume establishment of

here. When (5) is established, I get

from Definition 3. Because (9) includes contradiction for (11),

is established when (5) is established.

Similary, I get

from (8), (14) if I assume establishment of

here. When (5) is established, I get

from Definition 4. Because (13) includes contradiction for (15),

is established when (5) is established.

Similary, I get

from (8), (18) if I assume establishment of

here. When (5) is established, I get

from Definition 5. Because (17) includes contradiction for (19),

is established when (5) is established. And, I get

from (12), (16), (20).

-End Proof

We will have to think about adaptation of the establishment of Binary Law for the coordinate systems

Proposition 6. If all coordinate systems

Proof: I get

from Definition 6 if all coordinate systems

-End Proof

Proposition 7. Tensor of the second rank becomes Symmetric Tensor if all coor- dinate systems

Proof: I get

from Definition 7 if all coordinate systems

Then, I get

from (23),(24). And we can rewrite (23) by using (20), (21) for

Then, I get

from (26). Therefore, Tensor of the second rank becomes Symmetric Tensor than consideration of Definition 8 when all coordinate systems

-End Proof

Proposition 8. If all coordinate systems

Proof: I get

from Definition 10 if all coordinate systems

from Definition 9 if all coordinate systems

I decide not to handle (33) by consideration of (28) here. Well, I get conclution from (32) that if all coordinate systems

-End Proof

Proposition 9. If all coordinate systems

Proof: I get

from Definition 11 if all coordinate systems

And, I can get

from (37) for consideration of (28). And we can rewrite (38) by using (21) for

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

-End Proof

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

establishment and this is constant. And,

About Proposition 9:

In (39), we can handle

Ichidayama, K. (2017) Introduction of the Tensor Which Satisfied Binary Law. Journal of Modern Physics, 8, 126-132. http://dx.doi.org/10.4236/jmp.2017.81011