What's the difference?

  • Thread starter quasar987
  • Start date
quasar987
Science Advisor
Homework Helper
Gold Member
4,771
7

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 [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:

Answers and Replies

Chris Hillman
Science Advisor
2,337
8
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
Science Advisor
Gold Member
7,231
785
216
1
dextercioby
Science Advisor
Homework Helper
Insights Author
12,965
536
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.
 

Related Threads for: What's the difference?

Top