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Christoffel Symbol of the First Kind

Variously denoted $[ij,k]$, $\left[{i\quad j\atop k}\right]$, $\Gamma_{abc}$, or $\{ab,c\}$.

\begin{displaymath}[ij,k]= \left[{i\quad j\atop k}\right]\equiv g_{mk}\Gamma^m_{...
...ial q^i} = \vec e_k\cdot{\partial \vec e_i\over \partial q^j},
\end{displaymath} (1)

where $g_{mk}$ is the Metric Tensor and
\begin{displaymath}
\vec e_i \equiv {\partial \vec r\over\partial q^i} = h_i \hat e_i.
\end{displaymath} (2)

But
$\displaystyle {\partial g_{ij}\over\partial q^k}$ $\textstyle =$ $\displaystyle {\partial\over\partial q^k}(\vec e_i\cdot\vec e_j) = {\partial\ve...
...er\partial q^k}\cdot \vec e_j+\vec e_i\cdot{\partial \vec e_j\over\partial q^k}$  
  $\textstyle =$ $\displaystyle [ik,j]+[jk,i],$ (3)

so
\begin{displaymath}[ab,c]={\textstyle{1\over 2}}(g_{ac,b}+g_{bc,a}-g_{ab,c}).
\end{displaymath} (4)


References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 160-167, 1985.




© 1996-9 Eric W. Weisstein
1999-05-26