Introduction of the Tensor Which Satisfied Binary Law

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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.