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What's the difference?

  1. Jan 20, 2007 #1

    quasar987

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    Say V is a vector space with base {e_i}, V* is it's dual with dual basis {e^i}. If someone says that [itex]T^i_{ \ j}[/itex] are the components of a tensor, then I know this means that the actual tensor is

    [tex]\mathbf{T}=T^i_{ \ j}e_i\otimes e^j[/tex]

    The order of the indices of the components of T indicates on which set is T acting. In this case, V* x V. Were the components [itex]T_j^{ \ i}[/itex], T would have acted on V x V*.

    Now my question.

    If [itex]\Gamma[/itex] is a function from vector spaces V to W of respective bases {[itex]e_i[/itex]} and {[itex]\tilde{e}_i[/itex]}, and if we define the components of [itex]\Gamma[/itex] as the numbers [itex]\Gamma_i^{ \ j}[/itex] such that

    [tex]\Gamma(e_i)=\Gamma_i^{ \ j}\tilde{e}_j[/itex],

    is there a meaning to the order of the indiced, or could I have just as well noted the coefficients as [itex]\Gamma^{j}_{ \ i}[/itex]???

    Thanks.
     
    Last edited: Jan 20, 2007
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  3. Jan 20, 2007 #2

    Chris Hillman

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    Hi, quasar987, have you looked at Geroch, Geometry of Physics or Nakayama, Geometry, Topology and Physics? These should answer your questions.
     
  4. Jan 21, 2007 #3

    George Jones

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    Frankel? :smile:
     
  5. Jan 21, 2007 #4
    Presumably, although Geroch's Mathematical Physics, if I recall correctly, also discusses this.
     
  6. Jan 22, 2007 #5

    dextercioby

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    Nakahara, maybe ? :rolleyes:

    Daniel.
     
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