How to prove the following identity

[amitech]

Homework Statement

e_j=g_(jk)e^k

where e_j is a covariant vector base
e^k is a a contravariant vector base
g_(jk) is the covariant metric

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

What is the definition of the "covariant metric". How do you go from a covariant to the corresponding contravariant vector and vice-versa?

 

[amitech]

Hi, by saying "covariant metic" I mean writing the metric in its covariant form - basically g(contra)=[g(cov)]^(-1)
you can use the metric to lower/raise an index on a vector component but here e_j and e^k are not components but the actual unit vectors.

 

[amitech]

Hi, by saying "covariant metic" I mean writing the metric in its covariant form - basically g(contra)=[g(cov)]^(-1)
you can use the metric to lower/raise an index on a vector component but here e_j and e^k are not components but the actual unit vectors.