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Can a Physical Law Formulated by One component Tensor ?

  1. Jul 26, 2004 #1
    Can a Physical Law Formulated by One component Tensor ???

    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.
     
    Last edited: Jul 28, 2004
  2. jcsd
  3. Jul 27, 2004 #2
    Electromagnetism (rank 1) and gravity (rank 2)would be equal in 1D.
     
  4. Jul 27, 2004 #3
    That's right. They have the same number of component.
     
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