Home > Mathematics > Christoffel symbols of the first kind

Christoffel symbols of the first kind

\mathbf{e_i}=\frac{\partial \mathbf{r}}{\partial q^i}

\mathbf{g_{ij}}=\mathbf{e_i}\cdot\mathbf{e_j}

\Gamma_{ijk} =\frac{\partial\mathbf{e_i}}{\partial q^j}\cdot\mathbf{e_k}=\frac{\partial ^2\mathbf{r}}{\partial q^i\partial q^j}\cdot \mathbf{e_k}=\frac{\partial\mathbf{ e_j}}{\partial q^i}\cdot\mathbf{e_k}=\Gamma_{jik}

\begin{array}{lll}\frac{\partial g_{ij}}{\partial q^k}=g_{ijk}&=&\frac{\partial (\mathbf{e_i}\cdot\mathbf{e_j})}{\partial q^k}\\&=&\frac{\partial \mathbf{e_i}}{\partial q^k}\cdot\mathbf{e_j}+\frac{\partial\mathbf{e_j}}{\partial q^k}\cdot\mathbf{e_i}\\&=&\Gamma_{ikj}+\Gamma_{jki}\end{array}

g_{jki}+ g_{kij}-g_{ijk} = \Gamma_{jik}+\Gamma_{kij} +\Gamma_{kji}+\Gamma_{ijk}-\Gamma_{ikj}-\Gamma_{jki}=2\Gamma_{ijk}

\Gamma_{ijk} =\frac{1}{2} (g_{jki}+g_{kij}-g_{ijk})

Advertisements
Categories: Mathematics
  1. No comments yet.
  1. No trackbacks yet.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: