実際，つぎが成り立つ： \[ \begin{align*} \nabla'_j a'_i \ =\ \frac{\partial a'_i}{\partial X'^j} - \sum_k {\it \Gamma}^{\,'k}_{\ ij} a'_k \\ =\ \sum_{k,l} \frac{\partial x^l}{\partial x'^i} \frac{\partial X^k}{\partial X'^j} \nabla_k a_l \end{align*} \] 以下，これの計算過程を示す。
 「共変座標」)
\[ \Gamma^{\,'k}_{ij} = \sum_l \sum_m \frac{\partial^2 {X'}^m}{\partial {x'}^l \partial {x'}^i } \frac{\partial {x'}^l}{\partial {X'}^j} \frac{\partial {x'}^k}{\partial {X'}^m} \] \[ \begin{align*} \frac{\partial^2 {X'}^m}{\partial {x'}^l \partial {x'}^i } &= \frac{\partial}{\partial {x'}^l } \frac{\partial {X'}^m}{\partial {x'}^i } \\&= \sum_u \frac{\partial x^u}{\partial {x'}^l } \frac{\partial}{\partial x^u} \left( \sum_t \frac{\partial X^t}{\partial {x'}^i } \frac{\partial {X'}^m}{\partial {X}^t } \right) \\&= \sum_u \frac{\partial x^u}{\partial {x'}^l } \frac{\partial}{\partial x^u} \left( \sum_t \left( \sum_s \frac{\partial x^s}{\partial {x'}^i } \frac{\partial X^t}{\partial x^s } \right) \frac{\partial {X'}^m}{\partial {X}^t } \right) \\&= \sum_{s,t,u} \frac{\partial x^u}{\partial {x'}^l } \frac{\partial}{\partial x^u} \left( \frac{\partial x^s}{\partial {x'}^i } \frac{\partial X^t}{\partial x^s } \frac{\partial {X'}^m}{\partial {X}^t } \right) \\&= \sum_{s,t,u} \frac{\partial x^u}{\partial {x'}^l } \frac{\partial {X'}^m}{\partial {X}^t } \frac{\partial}{\partial x^u} \left( \frac{\partial X^t}{\partial x^s } \frac{\partial x^s}{\partial {x'}^i } \right) \\&= \sum_{s,t,u} \frac{\partial x^u}{\partial {x'}^l } \frac{\partial {X'}^m}{\partial {X}^t } \left( \frac{\partial^2 X^t}{\partial x^u \partial x^s } \frac{\partial x^s}{\partial {x'}^i } + \frac{\partial X^t}{\partial x^s } \frac{\partial^2 x^s}{\partial x^u \partial {x'}^i } \right) \\&= \sum_{s,t,u} \frac{\partial x^u}{\partial {x'}^l } \frac{\partial {X'}^m}{\partial {X}^t } \frac{\partial^2 X^t}{\partial x^u \partial x^s } \frac{\partial x^s}{\partial {x'}^i } + \sum_{s,t,u} \frac{\partial x^u}{\partial {x'}^l } \frac{\partial {X'}^m}{\partial {X}^t } \frac{\partial X^t}{\partial x^s } \frac{\partial^2 x^s}{\partial x^u \partial {x'}^i } \\&= \sum_{s,t,u} \frac{\partial^2 X^t}{\partial x^u \partial x^s } \frac{\partial x^s}{\partial {x'}^i } \frac{\partial x^u}{\partial {x'}^l } \frac{\partial {X'}^m}{\partial {X}^t } + \sum_{s} \left( \sum_{u} \frac{\partial x^u}{\partial {x'}^l } \frac{\partial}{\partial x^u} \frac{\partial x^s}{\partial {x'}^i } \right) \left( \sum_{t} \frac{\partial X^t}{\partial x^s } \frac{\partial {X'}^m}{\partial {X}^t } \right) \\&= \sum_{s,t,u} \frac{\partial^2 X^t}{\partial x^u \partial x^s } \frac{\partial x^s}{\partial {x'}^i } \frac{\partial x^u}{\partial {x'}^l } \frac{\partial {X'}^m}{\partial {X}^t } + \sum_{s} \frac{\partial^2 x^s}{\partial {x'}^l \partial {x'}^i } \frac{\partial {X'}^m}{\partial x^s } \end{align*} \] \[ \frac{\partial {x'}^l}{\partial {X'}^j} = \sum_v \frac{\partial X^v }{\partial {X'}^j} \frac{\partial {x'}^l}{\partial X^v} \] \[ \frac{\partial {x'}^k}{\partial {X'}^m} = \sum_w \frac{\partial X^w }{\partial {X'}^m} \frac{\partial {x'}^k}{\partial X^w} \] よって， \[ \Gamma^{\,'k}_{ij} = \sum_l \sum_m \frac{\partial^2 {X'}^m}{\partial {x'}^l \partial {x'}^i } \frac{\partial {x'}^l}{\partial {X'}^j} \frac{\partial {x'}^k}{\partial {X'}^m} \\ \\= \sum_{l,m,s,t,u,v,w} \left( \frac{\partial^2 X^t}{\partial x^u \partial x^s } \frac{\partial x^s}{\partial {x'}^i } \frac{\partial x^u}{\partial {x'}^l } \frac{\partial {X'}^m}{\partial {X}^t } \right) \left( \frac{\partial X^v }{\partial {X'}^j} \frac{\partial {x'}^l}{\partial X^v} \right) \left( \frac{\partial X^w }{\partial {X'}^m} \frac{\partial {x'}^k}{\partial X^w} \right) \\+ \sum_{l,m,s} \left( \frac{\partial^2 x^s}{\partial {x'}^l \partial {x'}^i } \frac{\partial {X'}^m}{\partial x^s } \right) \frac{\partial {x'}^l}{\partial {X'}^j} \frac{\partial {x'}^k}{\partial {X'}^m} \\ \\= \sum_{s,t,u,v,w} \frac{\partial^2 X^t}{\partial x^u \partial x^s } \left( \sum_{l} \frac{\partial x^u}{\partial {x'}^l } \frac{\partial {x'}^l}{\partial X^v} \right) \left( \sum_{m} \frac{\partial X^w }{\partial {X'}^m} \frac{\partial {X'}^m}{\partial {X}^t } \right) \frac{\partial x^s}{\partial {x'}^i } \frac{\partial X^v }{\partial {X'}^j} \frac{\partial {x'}^k}{\partial X^w} \\+ \sum_{s} \left( \sum_{l} \frac{\partial {x'}^l}{\partial {X'}^j} \frac{\partial}{\partial {x'}^l} \frac{\partial x^s}{\partial {x'}^i } \right) \left( \sum_{m} \frac{\partial {X'}^m}{\partial x^s } \frac{\partial {x'}^k}{\partial {X'}^m} \right) \\ \\= \sum_{s,t,u,v,w} \frac{\partial^2 X^t}{\partial x^u \partial x^s } \frac{\partial x^u}{\partial X^v} \frac{\partial X^w }{\partial {X}^t } \frac{\partial x^s}{\partial {x'}^i } \frac{\partial X^v }{\partial {X'}^j} \frac{\partial {x'}^k}{\partial X^w} \\+ \sum_{s} \frac{\partial^2 x^s}{\partial {X'}^j \partial {x'}^i } \frac{\partial {x'}^k}{\partial x^s} \\ \\= \sum_{s,v,w} \left( \sum_{t,u} \frac{\partial^2 X^t}{\partial x^u \partial x^s } \frac{\partial x^u}{\partial X^v} \frac{\partial X^w }{\partial {X}^t } \right) \frac{\partial x^s}{\partial {x'}^i } \frac{\partial X^v }{\partial {X'}^j} \frac{\partial {x'}^k}{\partial X^w} \\+ \sum_{s} \frac{\partial {x'}^k}{\partial x^s} \frac{\partial^2 x^s}{\partial {X'}^j \partial {x'}^i } \\ \\= \sum_{s,v,w} \Gamma^{w}_{sv}\, \frac{\partial x^s}{\partial {x'}^i } \frac{\partial X^v }{\partial {X'}^j} \frac{\partial {x'}^k}{\partial X^w} \\+ \sum_{s} \frac{\partial {x'}^k}{\partial x^s} \frac{\partial^2 x^s}{\partial {X'}^j \partial {x'}^i } \\ \\= \sum_{s,v,w} \frac{\partial x^s}{\partial {x'}^i } \frac{\partial X^v }{\partial {X'}^j} \frac{\partial {x'}^k}{\partial X^w} \,\Gamma^{w}_{sv} + \sum_{s} \frac{\partial {x'}^k}{\partial x^s} \frac{\partial^2 x^s}{\partial {X'}^j \partial {x'}^i } \] |