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Contraction of a tensor

  1. Oct 7, 2007 #1
    Let Y[tex]_{1}[/tex],..,Y[tex]_{k}[/tex] be vector fields and let A be a tensor field of type [tex]^{k}_{1}[/tex]. Could you explain how applying k contractions to A[tex]\otimes[/tex]Y[tex]_{1}[/tex][tex]\otimes[/tex]...Y[tex]_{k}[/tex] yields A(Y[tex]_{1}[/tex]...Y[tex]_{k}[/tex])?

    Actually, could you first explain why contraction of w[tex]\otimes[/tex]Y is equal to w(Y)?
    Here, w is a 1-form and Y is a vector field.
    Thank you.
     
  2. jcsd
  3. Oct 7, 2007 #2

    HallsofIvy

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    Isn't that pretty much the definition of "contraction"?
     
  4. Oct 7, 2007 #3
  5. Oct 7, 2007 #4
    [tex]w \otimes Y[/tex] has incides [tex]w_{i} \otimes Y^{j}[/tex]. Contract the indices to make [tex]w \otimes Y[/tex] into a scalar gives [tex]w_{i} \otimes Y^{i}[/tex]. This is the definition of w(Y).

    Similarly for everything else.
     
  6. Oct 7, 2007 #5
    a special case is the dot product of two vectors, this is how everyone really things about contraction anyway
     
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