- #1
kiuhnm
- 66
- 1
I hope you can understand my notation. The Christoffel symbol can be defined through the relation$$
\frac{\partial \pmb{Z}_i} {\partial Z^k} = \Gamma_{ik}^j \pmb{Z}_j
$$ I can solve for the Christoffel symbol this way: $$
\frac{\partial \pmb{Z}_i} {\partial Z^k} \cdot \pmb{Z}^m = \Gamma_{ik}^j \pmb{Z}_j \cdot \pmb{Z}^m = \Gamma_{ik}^j \delta^m_j = \Gamma_{ik}^m
$$
This might be a stupid question, but how can I go from the last relation back to the first one? Probably by multiplying by ##\pmb{Z}_m##: $$
\left( \frac{\partial \pmb{Z}_i} {\partial Z^k} \cdot \pmb{Z}^m \right) \pmb{Z}_m = \ldots\
$$ Now, either I need to rewrite the dot product more explicitly or there's some propriety I don't know of.
\frac{\partial \pmb{Z}_i} {\partial Z^k} = \Gamma_{ik}^j \pmb{Z}_j
$$ I can solve for the Christoffel symbol this way: $$
\frac{\partial \pmb{Z}_i} {\partial Z^k} \cdot \pmb{Z}^m = \Gamma_{ik}^j \pmb{Z}_j \cdot \pmb{Z}^m = \Gamma_{ik}^j \delta^m_j = \Gamma_{ik}^m
$$
This might be a stupid question, but how can I go from the last relation back to the first one? Probably by multiplying by ##\pmb{Z}_m##: $$
\left( \frac{\partial \pmb{Z}_i} {\partial Z^k} \cdot \pmb{Z}^m \right) \pmb{Z}_m = \ldots\
$$ Now, either I need to rewrite the dot product more explicitly or there's some propriety I don't know of.