接ベクトル空間
∑ i = 1 m a i ( ∂ ∂ x i ) p = ∑ i = 1 m a i { ∑ j = 1 m ∂ y j ∂ x i ( p ) ( ∂ ∂ y j ) p } = ∑ j = 1 m { ∑ i = 1 m a i ∂ y j ∂ x i ( p ) } ( ∂ ∂ y j ) p {\displaystyle \sum _{i=1}^{m}a_{i}\left({\frac {\partial }{\partial x_{i}}}\right)_{p}=\sum _{i=1}^{m}a_{i}\left\{\sum _{j=1}^{m}{\frac {\partial y_{j}}{\partial x_{i}}}(p)\left({\frac {\partial }{\partial y_{j}}}\right)_{p}\right\}=\sum _{j=1}^{m}\left\{\sum _{i=1}^{m}a_{i}{\frac {\partial y_{j}}{\partial x_{i}}}(p)\right\}\left({\frac {\partial }{\partial y_{j}}}\right)_{p}}
という計算によって
∑ i = 1 m b i ( ∂ ∂ y i ) p {\displaystyle \sum _{i=1}^{m}b_{i}\left({\frac {\partial }{\partial y_{i}}}\right)_{p}}
(ただし bi ∈ R (1 ≤ i ≤ m) )
の形に変換できる。