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The number of component of a tensor of any rank is given by

[tex] c = d^r [/tex]

where c is the number of component, d is the dimension of the tensor, r is the rank of the tensor.

For r=0, the tensors are the scalars. For r=1, the tensors are the vectors.

For r=0

[tex] 0^0 = 1 [/tex]

[tex] 1^0 = 1 [/tex]

[tex] 2^0 = 1 [/tex]

[tex] 3^0 = 1 [/tex]

[tex] 4^0 = 1 [/tex]

the above show that for scalar tensors, there is only one component for any dimension. And for scalar tensors even the zero dimension has one component.

For r=1

[tex] 0^1 = 0 [/tex]

[tex] 1^1 = 1 [/tex]

[tex] 2^1 = 2 [/tex]

[tex] 3^1 = 3 [/tex]

[tex] 4^1 = 4 [/tex]

the above show that for vector tensors, the number of component is the same as the dimension.

For r=2

[tex] 0^2 = 0 [/tex]

[tex] 1^2 = 1 [/tex]

[tex] 2^2 = 4 [/tex]

[tex] 3^2 = 9 [/tex]

[tex] 4^2 = 16 [/tex]

For r=3

[tex] 0^3 = 0 [/tex]

[tex] 1^3 = 1 [/tex]

[tex] 2^3 = 8 [/tex]

[tex] 3^3 = 27 [/tex]

[tex] 4^3 = 64 [/tex]

For r=4

[tex] 0^4 = 0 [/tex]

[tex] 1^4 = 1 [/tex]

[tex] 2^4 = 16 [/tex]

[tex] 3^4 = 81 [/tex]

[tex] 4^4 = 256 [/tex]

From these, it can be noted that only in 1D is the number of component equals 1 for any tensor. So when a physical law is formulated in one dimension, it can represent tensor of any rank and no transformation is needed hence a coordinate system is not necessary.

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# Can a Physical Law Formulated by One component Tensor ?

**Physics Forums | Science Articles, Homework Help, Discussion**