_{1}

I report the reason why Tensor satisfying Binary Law has relations toward physics in this article. Q: The
*n* th-order covariant derivative of the Vector
{A_{μ}, A^{μ}}:(n=1) satisfying Binary Law. R: The
*n* th-order covariant derivative of the Vector
{A_{μ}, A^{μ}}:(n≥2) satisfying Binary Law. I have reported in other articles about Q. I report R in this article. I obtained the following results in this. I got the conclusion that derived function became 0. The derived function becoming 0 in the n th-order covariant derivative of the covariant vector
A_{μ}_{}
_{} here in the case of
*n*=2. Similarly, in the
*n* th-order covariant derivative of the contravariant vector
A^{μ}^{}^{} in the case of
*n*
=4 .

Definition 1

is established [

Definition 2

I named

Definition 3 If

Definition 4 If

Definition 5 If

Definition 6 If

Definition 7 If all coordinate systems

Definition 8

is established [

Definition 9

Definition 10

Definition 11

is established [

Definition 12

is established [

Definition 13 For every coordinate systems, there is no immediate reason for preferring certain systems of co-ordinates to others.

Definition 14 The physical law is invariable for all coordinate systems [

Definition 15 “All coordinate systems satisfies Definision 13” is established if Definision 14 is established.

Definition 16 Definision 14 is established if “The physical law is described in Tensor” is established [

Definition 17 “All coordinate systems satisfies Definision 13” is established if “All coordinate systems satisfies Binary Law” is established [

A.Einstein required establishment of Definision 14 approximately 100 years ago [

Proposition 1 If

Proof: I get

from Definision 10 if all coordinate systems

from Definision 1 if all coordinate systems

However, (3) can rewrite

if

Because index

handle (5) according to Definision 7. The possible rewrite by

according to Definision 4, Definision 6. Because three covariant Vector of the same index exists in one term, I don’t handle (6). Two sets are dummy index among three same index in (7), (8). Therefore, we must rewrite (2) to

by using Definision 4, Definision 6 with considering (7), (8). I get

in consideration of establishment of

in consideration of (1) for (12). And I get

from (13). I get

from

from (14), (15). Therefore, I get

from (9), (10), (11) in consideration of (1), (13), (16). And we can rewrite (17), (18), (19) by using Definision 4, Definision 6 for

Because the second, third,

in consideration of Proposition 2. And I rewrite (21) by Definision 4 and get

―End Proof―

Because (22) is established, I decide not to handle the third-order,

Proposition 2 If

Proof: I get

from Definision 8 if all coordinate systems

from

from (24). I get

from (25), Definision 4. Therefore, I get

from (23), (26). I get

from Definision 9 if all coordinate systems

from

from (29). I get

from (28), (30). I get

from covariant derivative of (31). I get

―End Proof―

Proposition 3 If

Proof: I get

from Definision 11 if all coordinate systems

However, (35) can rewrite

if

Because index

handle (37) according to Definision 7. The possible rewrite by

according to Definision 4, Definision 6. Because three contravariant Vector of the same index exists in one term, I don’t handle (40). Two sets are dummy index among three same index in (38), (39). Therefore, we must rewrite (33) to

by using Definision 4, Definision 6 with considering (38), (39). I get

in consideration of establishment of

in consideration of (1) for (44). Therefore, I get

from (41), (42), (43) in consideration of (1), (45). And we can rewrite (46), (47), (48) by using Definision 4, Definision 6 for

Because the second, third,

by using Definision 4, Definision 6 with considering (38), (39). Because (51) includes

―End Proof―

Proposition 4 If

Proof: I get

from Definision 12 if all coordinate systems

However, (55) can rewrite

if

Because index

Because index

handle (58) according to Definision 7. The possible rewrite by

according to Definision 4, Definision 6. Because two covariant Vector of the same index exists in one term, I don’t handle (59). Because two contravariant Vector of the same index exists in one term, I don’t handle (61). Because four contravariant Vector of the same index exists in one term, I don’t handle (62). Therefore, we must rewrite (53) to

by using Definision 4, Definision 6 with considering (60). I get

in consideration of establishment of

in consideration of (1) for (66). Therefore, I get

from (63), (64), (65) in consideration of (1), (67). And we can rewrite (68), (69), (70) by using Definision 4, Definision 6 for

Because the second, third,

by using Definision 4, Definision 6 with considering (60). Because (72) includes

in consideration of establishment of

in consideration of (1) for (75). Therefore, I get (69), (70) from (73), (74) in consideration of (1), (76).

―End Proof―

Proposition 5 If

Proof: I get

from

from (77), Proposition 2, Proposition 4.

―End Proof―

Because (78) is established, I decide not to handle the fifth-order,

Proposition 6 If

Proof: I get

from (71) if a dimensional number is 2. I get

from (79). And I get

from (80). I get

from (81). I get

from (82), Definision 5. And I get

from (83). I get

from (84). And I get

from (85).

―End Proof―

About Proposition 1

In (22), we can handle

About Proposition 3

In (49), we can handle

About Proposition 4

In (71), we can handle

Furthermore,

About Proposition 5

In (78),

About Proposition 6

If

These remind me of the matter wave in the quantum theory.

Ichidayama, K. (2017) Property of Tensor Satisfying Binary Law. Journal of Modern Physics, 8, 944-963. https://doi.org/10.4236/jmp.2017.86060