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