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*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.