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Checking validity of simple tensor equations

  1. Apr 19, 2009 #1

    trv

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    Can someone help me sort out the LaTeX firstly?

    Hoping someone can check that my solutions are correct/not. Also do comment on any impreciseness in any of my reasons.

    1. The problem statement, all variables and given/known data
    Determine which of the following equations is a valid tensor equation. Describe the errors in the other equations.

    2. Relevant equations

    [tex]g=A^{cc}B_{dd}[/tex]

    Not valid. Since summing over two raised indices or two lower indices, does not result in scalars.

    [tex]
    P^{ab}=Q^{ab}-Q^{ba}
    [/tex]

    Valid. (Almost a guess here.)

    [tex]
    R^a_b=S^cT^a_cU^cV_cb
    [/tex]

    Not valid. Four "c"s on the right term. Therefore unclear on how to sum over the repeated "c"s.

    [tex]
    D^a_{cb}=Q^a_c+M^a_b
    [/tex]

    Not valid. Firstly, can't subtract a rank 2 tensor from another rank 2 tensor to get a rank 3 tensor. Also, can't add the two tensors on the right, since although they are of similar rank, their indices are different.

    [tex]
    f=G^a_bK^d_aH^a_cL^a_d
    [/tex]

    Not valid. 4"a"s on right term so summing over indices ill defined. Also, b and c can't be summed over, so we wouldn't get a scalar.

    [tex]
    P_a=A_a^bB_b+U_cV^dW_d
    [/tex]

    Not valid. The two terms on the right reduce to covariant vectors, but the resultants would have a and d indices. These being different, the two resultant terms can't be added to get a single term.

    [tex]
    X_{ab}=Q^c_{bca}+U_bW_a
    [/tex]

    Valid.

    [tex]
    h=\delta^aV^a-\delta^b\delta_cZ^c_b
    [/tex]

    Not valid. The first term on the right can't be reduced to a scalar, since the two repeated indices are both raised.

    [tex]
    A_{ab}=B_a+C_b+D_{ab}
    [/tex]

    Not valid. The three terms on the right aren't all of rank 2 as they should be if they are to be added and result in a rank two term(the left term).

    [tex]
    M^{ab}=G^{ab}+Q^aR^b
    [/tex]

    valid

    [tex]
    J_b=T^c_aF_{bc}Z^{ad}_d
    [/tex]

    valid.

    [tex]
    K^c_b=Y_aZ^{ac}_aX^a_b
    [/tex]

    Not valid. The "a" index is repeated 4 times in the right term, while makes summation ill-defined.

    3. The attempt at a solution

    Below each equation.
     
    Last edited: Apr 19, 2009
  2. jcsd
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