有限変形理論
曲線座標系での変形テンソル > 変形尺度とクリストッフェル記号の間の関係

\mathbf {X} ={X^{1},X^{2},X^{3}}から \mathbf {x} ={x^{1},x^{2},x^{3}}への一対一の写像を考え、以下の条件を満たす2つの正定値で対称な2階のテンソル場 {\boldsymbol {G}}と {\boldsymbol {g}}が存在すると仮定する。
 G_{ij}={\frac {\partial X^{\alpha }}{\partial x^{i}}}~{\frac {\partial X^{\beta }}{\partial x^{j}}}~g_{\alpha \beta }
すると、
 {\frac {\partial G_{ij}}{\partial x^{k}}}=\left({\frac {\partial ^{2}X^{\alpha }}{\partial x^{i}\partial x^{k}}}~{\frac {\partial X^{\beta }}{\partial x^{j}}}+{\frac {\partial X^{\alpha }}{\partial x^{i}}}~{\frac {\partial ^{2}X^{\beta }}{\partial x^{j}\partial x^{k}}}\right)~g_{\alpha \beta }+{\frac {\partial X^{\alpha }}{\partial x^{i}}}~{\frac {\partial X^{\beta }}{\partial x^{j}}}~{\frac {\partial g_{\alpha \beta }}{\partial x^{k}}}
以下に注意すると
 {\frac {\partial g_{\alpha \beta }}{\partial x^{k}}}={\frac {\partial X^{\gamma }}{\partial x^{k}}}~{\frac {\partial g_{\alpha \beta }}{\partial X^{\gamma }}}

 g_{\alpha \beta }=g_{\beta \alpha }
我々は、以下を得る。
 {\begin{aligned}{\frac {\partial G_{ij}}{\partial x^{k}}}&=\left({\frac {\partial ^{2}X^{\alpha }}{\partial x^{i}\partial x^{k}}}~{\frac {\partial X^{\beta }}{\partial x^{j}}}+{\frac {\partial ^{2}X^{\alpha }}{\partial x^{j}\partial x^{k}}}~{\frac {\partial X^{\beta }}{\partial x^{i}}}\right)~g_{\alpha \beta }+{\frac {\partial X^{\alpha }}{\partial x^{i}}}~{\frac {\partial X^{\beta }}{\partial x^{j}}}~{\frac {\partial X^{\gamma }}{\partial x^{k}}}~{\frac {\partial g_{\alpha \beta }}{\partial X^{\gamma }}}\\{\frac {\partial G_{ik}}{\partial x^{j}}}&=\left({\frac {\partial ^{2}X^{\alpha }}{\partial x^{i}\partial x^{j}}}~{\frac {\partial X^{\beta }}{\partial x^{k}}}+{\frac {\partial ^{2}X^{\alpha }}{\partial x^{j}\partial x^{k}}}~{\frac {\partial X^{\beta }}{\partial x^{i}}}\right)~g_{\alpha \beta }+{\frac {\partial X^{\alpha }}{\partial x^{i}}}~{\frac {\partial X^{\beta }}{\partial x^{k}}}~{\frac {\partial X^{\gamma }}{\partial x^{j}}}~{\frac {\partial g_{\alpha \beta }}{\partial X^{\gamma }}}\\{\frac {\partial G_{jk}}{\partial x^{i}}}&=\left({\frac {\partial ^{2}X^{\alpha }}{\partial x^{i}\partial x^{j}}}~{\frac {\partial X^{\beta }}{\partial x^{k}}}+{\frac {\partial ^{2}X^{\alpha }}{\partial x^{i}\partial x^{k}}}~{\frac {\partial X^{\beta }}{\partial x^{j}}}\right)~g_{\alpha \beta }+{\frac {\partial X^{\alpha }}{\partial x^{j}}}~{\frac {\partial X^{\beta }}{\partial x^{k}}}~{\frac {\partial X^{\gamma }}{\partial x^{i}}}~{\frac {\partial g_{\alpha \beta }}{\partial X^{\gamma }}}\end{aligned}}
以下を定義する。
 {\begin{aligned}_{(x)}\Gamma _{ijk}&:={\frac {1}{2}}\left({\frac {\partial G_{ik}}{\partial x^{j}}}+{\frac {\partial G_{jk}}{\partial x^{i}}}-{\frac {\partial G_{ij}}{\partial x^{k}}}\right)\\_{(X)}\Gamma _{\alpha \beta \gamma }&:={\frac {1}{2}}\left({\frac {\partial g_{\alpha \gamma }}{\partial X^{\beta }}}+{\frac {\partial g_{\beta \gamma }}{\partial X^{\alpha }}}-{\frac {\partial g_{\alpha \beta }}{\partial X^{\gamma }}}\right)\\\end{aligned}}
従って、
 _{(x)}\Gamma _{ijk}={\frac {\partial X^{\alpha }}{\partial x^{i}}}~{\frac {\partial X^{\beta }}{\partial x^{j}}}~{\frac {\partial X^{\gamma }}{\partial x^{k}}}\,_{(X)}\Gamma _{\alpha \beta \gamma }+{\frac {\partial ^{2}X^{\alpha }}{\partial x^{i}\partial x^{j}}}~{\frac {\partial X^{\beta }}{\partial x^{k}}}~g_{\alpha \beta }
以下を定義する。
 [G^{ij}]=[G_{ij}]^{-1}~;~~[g^{\alpha \beta }]=[g_{\alpha \beta }]^{-1}
すると
 G^{ij}={\frac {\partial x^{i}}{\partial X^{\alpha }}}~{\frac {\partial x^{j}}{\partial X^{\beta }}}~g^{\alpha \beta }
Define the Christoffel symbols of the second kind as
 _{(x)}\Gamma _{ij}^{m}:=G^{mk}\,_{(x)}\Gamma _{ijk}~;~~_{(X)}\Gamma _{\alpha \beta }^{\nu }:=g^{\nu \gamma }\,_{(X)}\Gamma _{\alpha \beta \gamma }
すると、
 {\begin{aligned}_{(x)}\Gamma _{ij}^{m}&=G^{mk}~{\frac {\partial X^{\alpha }}{\partial x^{i}}}~{\frac {\partial X^{\beta }}{\partial x^{j}}}~{\frac {\partial X^{\gamma }}{\partial x^{k}}}\,_{(X)}\Gamma _{\alpha \beta \gamma }+G^{mk}~{\frac {\partial ^{2}X^{\alpha }}{\partial x^{i}\partial x^{j}}}~{\frac {\partial X^{\beta }}{\partial x^{k}}}~g_{\alpha \beta }\\&={\frac {\partial x^{m}}{\partial X^{\nu }}}~{\frac {\partial x^{k}}{\partial X^{\rho }}}~g^{\nu \rho }~{\frac {\partial X^{\alpha }}{\partial x^{i}}}~{\frac {\partial X^{\beta }}{\partial x^{j}}}~{\frac {\partial X^{\gamma }}{\partial x^{k}}}\,_{(X)}\Gamma _{\alpha \beta \gamma }+{\frac {\partial x^{m}}{\partial X^{\nu }}}~{\frac {\partial x^{k}}{\partial X^{\rho }}}~g^{\nu \rho }~{\frac {\partial ^{2}X^{\alpha }}{\partial x^{i}\partial x^{j}}}~{\frac {\partial X^{\beta }}{\partial x^{k}}}~g_{\alpha \beta }\\&={\frac {\partial x^{m}}{\partial X^{\nu }}}~\delta _{\rho }^{\gamma }~g^{\nu \rho }~{\frac {\partial X^{\alpha }}{\partial x^{i}}}~{\frac {\partial X^{\beta }}{\partial x^{j}}}\,_{(X)}\Gamma _{\alpha \beta \gamma }+{\frac {\partial x^{m}}{\partial X^{\nu }}}~\delta _{\rho }^{\beta }~g^{\nu \rho }~{\frac {\partial ^{2}X^{\alpha }}{\partial x^{i}\partial x^{j}}}~g_{\alpha \beta }\\&={\frac {\partial x^{m}}{\partial X^{\nu }}}~g^{\nu \gamma }~{\frac {\partial X^{\alpha }}{\partial x^{i}}}~{\frac {\partial X^{\beta }}{\partial x^{j}}}\,_{(X)}\Gamma _{\alpha \beta \gamma }+{\frac {\partial x^{m}}{\partial X^{\nu }}}~g^{\nu \beta }~{\frac {\partial ^{2}X^{\alpha }}{\partial x^{i}\partial x^{j}}}~g_{\alpha \beta }\\&={\frac {\partial x^{m}}{\partial X^{\nu }}}~{\frac {\partial X^{\alpha }}{\partial x^{i}}}~{\frac {\partial X^{\beta }}{\partial x^{j}}}\,_{(X)}\Gamma _{\alpha \beta }^{\nu }+{\frac {\partial x^{m}}{\partial X^{\nu }}}~\delta _{\alpha }^{\nu }~{\frac {\partial ^{2}X^{\alpha }}{\partial x^{i}\partial x^{j}}}\end{aligned}}
従って、
 _{(x)}\Gamma _{ij}^{m}={\frac {\partial x^{m}}{\partial X^{\nu }}}~{\frac {\partial X^{\alpha }}{\partial x^{i}}}~{\frac {\partial X^{\beta }}{\partial x^{j}}}\,_{(X)}\Gamma _{\alpha \beta }^{\nu }+{\frac {\partial x^{m}}{\partial X^{\alpha }}}~{\frac {\partial ^{2}X^{\alpha }}{\partial x^{i}\partial x^{j}}}
The invertibility of the mapping implies that
 {\begin{aligned}{\frac {\partial X^{\mu }}{\partial x^{m}}}\,_{(x)}\Gamma _{ij}^{m}&={\frac {\partial X^{\mu }}{\partial x^{m}}}~{\frac {\partial x^{m}}{\partial X^{\nu }}}~{\frac {\partial X^{\alpha }}{\partial x^{i}}}~{\frac {\partial X^{\beta }}{\partial x^{j}}}\,_{(X)}\Gamma _{\alpha \beta }^{\nu }+{\frac {\partial X^{\mu }}{\partial x^{m}}}~{\frac {\partial x^{m}}{\partial X^{\alpha }}}~{\frac {\partial ^{2}X^{\alpha }}{\partial x^{i}\partial x^{j}}}\\&=\delta _{\nu }^{\mu }~{\frac {\partial X^{\alpha }}{\partial x^{i}}}~{\frac {\partial X^{\beta }}{\partial x^{j}}}\,_{(X)}\Gamma _{\alpha \beta }^{\nu }+\delta _{\alpha }^{\mu }~{\frac {\partial ^{2}X^{\alpha }}{\partial x^{i}\partial x^{j}}}\\&={\frac {\partial X^{\alpha }}{\partial x^{i}}}~{\frac {\partial X^{\beta }}{\partial x^{j}}}\,_{(X)}\Gamma _{\alpha \beta }^{\mu }+{\frac {\partial ^{2}X^{\mu }}{\partial x^{i}\partial x^{j}}}\end{aligned}}
同様の結果を xに関する微分で表すこともできる。したがって、
 {\begin{aligned}{\frac {\partial ^{2}X^{\mu }}{\partial x^{i}\partial x^{j}}}&={\frac {\partial X^{\mu }}{\partial x^{m}}}\,_{(x)}\Gamma _{ij}^{m}-{\frac {\partial X^{\alpha }}{\partial x^{i}}}~{\frac {\partial X^{\beta }}{\partial x^{j}}}\,_{(X)}\Gamma _{\alpha \beta }^{\mu }\\{\frac {\partial ^{2}x^{m}}{\partial X^{\alpha }\partial X^{\beta }}}&={\frac {\partial x^{m}}{\partial X^{\mu }}}\,_{(X)}\Gamma _{\alpha \beta }^{\mu }-{\frac {\partial x^{i}}{\partial X^{\alpha }}}~{\frac {\partial x^{j}}{\partial X^{\beta }}}\,_{(x)}\Gamma _{ij}^{m}\end{aligned}}