\textbf{e}^i are the reciprocal basis. Ok, now I've understood my mistake; the key is to see the scalar product between a vector and a covector as a tensor of rank (1,1), that's the application of a covector to a vector or vice versa; while the scalar product between two vectors as a tensor of...
Yes sorry, my bad! i was focused on dual basis and i missed the partial derivative of the vector component! Also, you're right again; it's better using the notation of the affine connection \nabla.
So am i right? i think there's an abuse of notation. For the first expression...
right,so i think that my mistake is an abuse of notation... in fact in the first case the operator between \omega^i and \textbf{e}_j is a bilinear operator on \textbf{V}^*\times \textbf{V} whose properties resemble the properties of an inner product. In the second case, i use the inner product...
Hello all!
I've just started to study general relativity and I'm a bit confused about dual basis vectors.
If we have a vector space \textbf{V} and a basis \{\textbf{e}_i\}, I can define a dual basis \{\omega^i\} in \textbf{V}^* such that: \omega^i(\textbf{e}_j) = \delta^i_j But in some pdf and...