クリストッフェル記号
( ∂ x i ∂ u h ) 0 = δ h i , ( ∂ 2 x i ∂ u l ∂ u h ) 0 = − { i h l } 0 {\displaystyle \left({\frac {\partial x^{i}}{\partial u^{h}}}\right)_{0}=\delta _{h}^{i},\;\;\;\left({\frac {\partial ^{2}x^{i}}{\partial u^{l}\partial u^{h}}}\right)_{0}=-\left\{{{i} \atop {hl}}\right\}_{0}}
を得る。よって、ある一点 x0 におけるクリストッフェル記号の変数変換式が
( ∂ x i ∂ u k ) 0 { k h l ̄ } 0 = ( ∂ x j ∂ u h ) 0 ( ∂ x k ∂ u l ) 0 { i j k } 0 + ( ∂ 2 x i ∂ u h ∂ u l ) 0 = δ h j δ l k { i j k } 0 − { i h l } 0 = 0 {\displaystyle \left({\frac {\partial x^{i}}{\partial u^{k}}}\right)_{0}\left\{{\overline {{k} \atop {h\;l}}}\right\}_{0}=\left({\frac {\partial x^{j}}{\partial u^{h}}}\right)_{0}\left({\frac {\partial x^{k}}{\partial u^{l}}}\right)_{0}\left\{{{i} \atop {j\;k}}\right\}_{0}+\left({\frac {\partial ^{2}x^{i}}{\partial u^{h}\partial u^{l}}}\right)_{0}=\delta _{h}^{j}\delta _{l}^{k}\left\{{{i} \atop {j\;k}}\right\}_{0}-\left\{{{i} \atop {h\;l}}\right\}_{0}=0}