広田微分
D x l D y m D z n ⋯ f ⋅ g = ( ∂ ∂ x − ∂ ∂ x ′ ) l ( ∂ ∂ y − ∂ ∂ y ′ ) m ( ∂ ∂ z − ∂ ∂ z ′ ) n ⋯ f ( x , y , z , ⋯ ) g ( x ′ , y ′ , z ′ , ⋯ ) | x ′ = x , y ′ = y , z ′ = z , ⋯ {\displaystyle D_{x}{}^{l}D_{y}{}^{m}D_{z}{}^{n}\cdots f\cdot g=\left.{\biggl (}{\frac {\partial }{\partial x}}-{\frac {\partial }{\partial x'}}{\biggr )}^{l}{\biggl (}{\frac {\partial }{\partial y}}-{\frac {\partial }{\partial y'}}{\biggr )}^{m}{\biggl (}{\frac {\partial }{\partial z}}-{\frac {\partial }{\partial z'}}{\biggr )}^{n}\cdots f(x,y,z,\cdots )g(x',y',z',\cdots )\right|_{x'=x,y'=y,z'=z,\cdots }} = ∂ l ∂ r l ∂ m ∂ s m ∂ n ∂ t n ⋯ f ( x + r , y + s , z + t , ⋯ ) g ( x − r , y − s , z − t , ⋯ ) | r , s , t , ⋯ = 0 {\displaystyle =\left.{\frac {\partial ^{l}}{\partial r^{l}}}{\frac {\partial ^{m}}{\partial s^{m}}}{\frac {\partial ^{n}}{\partial t^{n}}}\cdots f(x+r,y+s,z+t,\cdots )g(x-r,y-s,z-t,\cdots )\right|_{r,s,t,\cdots =0}}
で定義される。
実際の計算例は次のようになる。
D x f ⋅ g = f x g − f g x {\displaystyle D_{x}f\cdot g=f_{x}g-fg_{x}}