# What's the difference?

Homework Helper
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## Main Question or Discussion Point

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

$$\mathbf{T}=T^i_{ \ j}e_i\otimes e^j$$

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 $T_j^{ \ i}$, T would have acted on V x V*.

Now my question.

If $\Gamma$ is a function from vector spaces V to W of respective bases {$e_i$} and {$\tilde{e}_i$}, and if we define the components of $\Gamma$ as the numbers $\Gamma_i^{ \ j}$ 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 $\Gamma^{j}_{ \ i}$???

Thanks.

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Chris Hillman
Hi, quasar987, have you looked at Geroch, Geometry of Physics or Nakayama, Geometry, Topology and Physics? These should answer your questions.

George Jones
Staff Emeritus
Gold Member
Hi, quasar987, have you looked at Geroch, Geometry of Physics ...
Frankel?

Presumably, although Geroch's Mathematical Physics, if I recall correctly, also discusses this.

dextercioby