Understanding Tensor Components: The Role of Index Order in Vector Spaces

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In summary, the conversation discusses the components of a tensor in a vector space V and its dual V*, as well as the meaning of the order of the indices in the components. It is suggested to consult sources such as Geroch's "Geometry of Physics" or Nakayama's "Geometry, Topology and Physics" for further understanding.
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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.
 
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  • #2
Hi, quasar987, have you looked at Geroch, Geometry of Physics or Nakayama, Geometry, Topology and Physics? These should answer your questions.
 
  • #3
Chris Hillman said:
Hi, quasar987, have you looked at Geroch, Geometry of Physics ...

Frankel? :smile:
 
  • #4
George Jones said:
Frankel? :smile:

Presumably, although Geroch's Mathematical Physics, if I recall correctly, also discusses this.
 
  • #5
Chris Hillman said:
Hi, quasar987, have you looked at Geroch, Geometry of Physics or Nakayama, Geometry, Topology and Physics? These should answer your questions.

Nakahara, maybe ? :rolleyes:

Daniel.
 

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