Christoffel Symbol of the First Kind

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

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

where $g_{mk}$ is the Metric Tensor and

$$\vec e_i \equiv {\partial \vec r\over\partial q^i} = h_i \hat e_i.$$ (2)

But

$${\partial g_{ij}\over\partial q^k} = {\partial\over\partial q^k}(\vec e_i\cdot\vec e_j) = {\partial\ve... ...er\partial q^k}\cdot \vec e_j+\vec e_i\cdot{\partial \vec e_j\over\partial q^k}$$

$$= [ik,j]+[jk,i],$$ (3)

so

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

References

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