斜交座標系
g i j = ( e → 1 ⋅ e → 1 e → 1 ⋅ e → 2 e → 2 ⋅ e → 1 e → 2 ⋅ e → 2 ) = ( 1 cos  ( φ − θ ) cos  ( φ − θ ) 1 ) , g i j = ( e → 1 ⋅ e → 1 e → 1 ⋅ e → 2 e → 2 ⋅ e → 1 e → 2 ⋅ e → 2 ) = 1 sin 2  ( φ − θ ) ( 1 − cos  ( φ − θ ) − cos  ( φ − θ ) 1 ) = ( g i j ) − 1 {\displaystyle {\begin{aligned}g_{ij}&={\begin{pmatrix}{\vec {e}}_{1}\cdot {\vec {e}}_{1}&{\vec {e}}_{1}\cdot {\vec {e}}_{2}\\{\vec {e}}_{2}\cdot {\vec {e}}_{1}&{\vec {e}}_{2}\cdot {\vec {e}}_{2}\end{pmatrix}}={\begin{pmatrix}1&\cos(\phi -\theta )\\\cos(\phi -\theta )&1\end{pmatrix}},\\g^{ij}&={\begin{pmatrix}{\vec {e}}^{1}\cdot {\vec {e}}^{1}&{\vec {e}}^{1}\cdot {\vec {e}}^{2}\\{\vec {e}}^{2}\cdot {\vec {e}}^{1}&{\vec {e}}^{2}\cdot {\vec {e}}^{2}\end{pmatrix}}={\frac {1}{\sin ^{2}(\phi -\theta )}}{\begin{pmatrix}1&-\cos(\phi -\theta )\\-\cos(\phi -\theta )&1\end{pmatrix}}=(g_{ij})^{-1}\end{aligned}}}
となる。また反変成分と共変成分の変換は
u i = g i j u j , u i = g i j u j {\displaystyle u_{i}=g_{ij}u^{j},\quad u^{i}=g^{ij}u_{j}}
とシンプルに表すことができる.