from Definision 11 if all coordinate systems satisfies Definision 2. By the way, we cannot handle (33), (34) according to Definision 7. I simplify (33) here and get

(35)

However, (35) can rewrite

(36)

if and of (35) are changeable to or each. Because index doesn’t exist at all in the third term of the right side of (35), I can change dummy index of (35) to dummy index. Furthermore, (36) can rewrite

(37)

Because index doesn’t exist at all in the second term of the right side of (36), I can change dummy index of (36) to dummy index. And we can

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

(38)

(39)

(40)

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

(41)

(42)

(43)

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

(44)

in consideration of establishment of from (42), (43) here. I get

(45)

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

(46)

(47)

(48)

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

(49)

Because the second, third, term of the right side of (46), (47), (48) does not exist here, we may adopt (46), (47), (48) and (49) description form of which. Similarly, we must rewrite (34) to

(50)

(51)

(52)

by using Definision 4, Definision 6 with considering (38), (39). Because (51) includes here, I don’t handle (51). Therefore, I get (46), (48) from (50), (52) in consideration of (1).

―End Proof―

Proposition 4 If is established,

is established.

Proof: I get

(53)

(54)

from Definision 12 if all coordinate systems satisfies Definision 2. By the way, we cannot handle (53), (54) according to Definision 7. I simplify (53) here and get

(55)

However, (55) can rewrite

(56)

if and of (55) are changeable to or each. Because index doesn’t exist at all in the fourth term of the right side of (55), I can change dummy index of (55) to dummy index. Furthermore, (56) can rewrite

(57)

Because index doesn’t exist at all in the third term of the right side of (56), I can change dummy index of (56) to dummy index. Furthermore, (56) can rewrite

(58)

Because index doesn’t exist at all in the second term of the right side of (57), I can change dummy index of (57) to dummy index. And we can

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

(59)

(60)

(61)

(62)

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

(63)

(64)

(65)

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

(66)

in consideration of establishment of from (64), (65) here. I get

(67)

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

(68)

(69)

(70)

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

(71)

Because the second, third, term of the right side of (68), (69), (70) does not exist here, we may adopt (68), (69), (70) and (71) description form of which. Similarly, we must rewrite (54) to

(72)

(73)

(74)

by using Definision 4, Definision 6 with considering (60). Because (72) includes here, I don’t handle (72). I get

(75)

in consideration of establishment of from (73), (74) here. I get

(76)

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

―End Proof―

Proposition 5 If is established,

is established.

Proof: I get

(77)

from if all coordinate systems satisfies Definision 2. And I get

(78)

from (77), Proposition 2, Proposition 4.

―End Proof―

Because (78) is established, I decide not to handle the fifth-order, covariant derivative of the contravariant Vector.

Proposition 6 If is established,

is established.

Proof: I get

(79)

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

(80)

from (79). And I get

(81)

from (80). I get

(82)

from (81). I get

(83)

from (82), Definision 5. And I get

(84)

from (83). I get

(85)

from (84). And I get

(86)

from (85).

―End Proof―

3. Discussion

About Proposition 1

In (22), we can handle as Tensor similarly. Furthermore,

is established. I do not handle the derived function of a higher order because derived function is already 0.

About Proposition 3

In (49), we can handle as Tensor similarly.

About Proposition 4

In (71), we can handle as Tensor similarly.

Furthermore, is established.

About Proposition 5

In (78), is established. I do not handle the derived function of a higher order because derived function is already 0.

About Proposition 6

If is established in (71), can’t have a wave-like property. However, has a wave-like property if is established in (71).

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

Cite this paper

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

Ichidayama, K. (2017) Property of Tensor Satisfying Binary Law.

References

[1] Einstein, A. (1916) Annalen der Physik, 354, 769-822.

https://doi.org/10.1002/andp.19163540702

[2] Ichidayama, K. (2017) Journal of Modern Physics, 8.

[3] Dirac, P.A.M. (1975) General Theory of Relativity. John Wiley and Sons, Inc.

[4] Fleisch, D. (2012) A Student’s Guide to Vectors and Tensors. Cambridge University Press.

[1] Einstein, A. (1916) Annalen der Physik, 354, 769-822.

https://doi.org/10.1002/andp.19163540702

[2] Ichidayama, K. (2017) Journal of Modern Physics, 8.

[3] Dirac, P.A.M. (1975) General Theory of Relativity. John Wiley and Sons, Inc.

[4] Fleisch, D. (2012) A Student’s Guide to Vectors and Tensors. Cambridge University Press.