Can you check my proof?

  1. learningphysics

    learningphysics 4,124
    Homework Helper

    An antisymmetric tensor of rank two [tex]E_{ij}[/tex] has [tex]E_{ij}=-E_{ji}[/tex] for all i and j right? Meaning for i=j, [tex]E_{ij}=0 [/tex]... Just wanted to make sure of this.

    Here's what I'm trying to prove. Given [tex]E_{ij}[/tex] is an antisymmetric tensor. Is [tex]E_{ij,k}[/tex] a tensor?

    I get yes. Here's my proof:

    [tex]E_{i'j',k'} =\sum_i \sum_j \sum_k (\frac{\partial}{x^k}(E_{ij}\frac{\partial x^i}{\partial x^{i'}}\frac{\partial x^j}{\partial x^{j'}}))\frac{\partial x^k}{\partial x^{k'}}[/tex]

    This reduces to:
    [tex]\sum_i \sum_j \sum_k E_{ij,k}\frac{\partial x^i}{\partial x^{i'}}\frac{\partial x^j}{\partial x^{j'}}\frac{\partial x^k}{\partial x^{k'}} + \sum_i \sum_j \sum_k E_{ij}(\frac{\partial}{x^k}(\frac{\partial x^i}{\partial x^{i'}}\frac{\partial x^j}{\partial x^{j'}}))\frac{\partial x^k}{\partial x^{k'}}[/tex]

    Finally after rewriting the second part:
    [tex]\sum_i \sum_j \sum_k E_{ij,k}\frac{\partial x^i}{\partial x^{i'}}\frac{\partial x^j}{\partial x^{j'}}\frac{\partial x^k}{\partial x^{k'}} + \sum_i \sum_j \sum_k E_{ij}(\frac{{\partial}^2 x^i}{\partial x^k \partial x^{i'}}\frac{\partial x^j}{\partial x^{j'}} + \frac{\partial x^i}{\partial x^{i'}}\frac{{\partial}^2 x^j}{\partial x^k \partial x^{j'}})\frac{\partial x^k}{\partial x^{k'}}[/tex]

    The expression [tex]E_{ij}[/tex] is multiplied by in the second triple summation is symmetric about i and j... so if I write one element of the second summation as:
    [tex]E_{ij}X_{ijk}[/tex]... so we add two elements of this sum [tex]i_1[/tex] and [tex]j_1[/tex] are not equal:

    [tex]E_{i_1 j_1}X_{i_1 j_1 k_1} + E_{j_1 i_1}X_{j_1 i_1 k_1} = E_{i_1 j_1}X_{i_1 j_1 k_1} - E_{i_1 j_1}X_{i_1 j_1 k_1} =0[/tex] since E is antisymmetric, and X is symmetric about i and j...

    So in this manner each element with indices [tex]i_1,j_1,k_1[/tex] where the i and j elements are different, cancels with the element with indices [tex]j_1,i_1,k_1[/tex].

    And when [tex]i_1 = j_1[/tex] we have [tex]E_{i_1,j_1} =0 [/tex] so that element in the summation is 0 anyway.

    So the second summation goes to zero. And we have:

    [tex]E_{i'j',k'} = \sum_i \sum_j \sum_k E_{ij,k}\frac{\partial x^i}{\partial x^{i'}}\frac{\partial x^j}{\partial x^{j'}}\frac{\partial x^k}{\partial x^{k'}}[/tex] so [tex]E_{ij,k}[/tex] is a tensor?

    Does this look right? Thanks a bunch. :smile:
     
  2. jcsd
  3. dextercioby

    dextercioby 12,309
    Science Advisor
    Homework Helper

    Of course it's incorrect.

    This part

    [tex] \frac{\partial^{2}x^{i}}{\partial x^{k'}\partial x^{i'}}\frac{\partial x^{j}}{\partial x^{j'}}+\frac{\partial x^{i}}{\partial x^{i'}}\frac{\partial^{2}x^{j}}{\partial x^{k'}\partial x^{j'}} [/tex]

    is obviously nonsymmetric in "i" and "j" and nonantisymmetric in <<j'>> and <<k'>>.

    Daniel.
     
  4. what could you do to make it a tensor?

    Hint:

    [tex]A_{i,j}[/tex] is not a tensor but
    [tex]A_{i,j} - A_{j,i}[/tex] is a tensor
     
  5. learningphysics

    learningphysics 4,124
    Homework Helper

    Thanks dexter.
     
  6. learningphysics

    learningphysics 4,124
    Homework Helper


    I just proved that [tex]E_{ij,k} + E_{jk,i} + E_{ki,j}[/tex] is a tensor where [tex]E_{ij}[/tex] is antisymmetric. Is that what you were referring to?
     
  7. yes :smile:
     
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