3次元デカルト座標での定義式。
∇ = ( ∂ / ∂ x ∂ / ∂ y ∂ / ∂ z ) {\displaystyle \nabla ={\begin{pmatrix}{\partial /\partial x}\\{\partial /\partial y}\\{\partial /\partial z}\end{pmatrix}}}
g r a d ϕ = ∇ ϕ = ( ∂ ϕ / ∂ x ∂ ϕ / ∂ y ∂ ϕ / ∂ z ) {\displaystyle \mathrm {grad} \phi =\nabla \phi ={\begin{pmatrix}{\partial \phi /\partial x}\\{\partial \phi /\partial y}\\{\partial \phi /\partial z}\end{pmatrix}}}
d i v F = ∇ ⋅ F = ∂ F x / ∂ x + ∂ F y / ∂ y + ∂ F z / ∂ z {\displaystyle \mathrm {div} \mathbf {F} =\nabla \cdot \mathbf {F} =\partial F_{x}/\partial x+\partial F_{y}/\partial y+\partial F_{z}/\partial z}
c u r l F = r o t F = ∇ × F = ( ∂ F z / ∂ y − ∂ F y / ∂ z ∂ F x / ∂ z − ∂ F z / ∂ x ∂ F y / ∂ x − ∂ F x / ∂ y ) {\displaystyle \mathrm {curl} \mathbf {F} =\mathrm {rot} \mathbf {F} =\nabla \times \mathbf {F} ={\begin{pmatrix}{\partial F_{z}/\partial y}-{\partial F_{y}/\partial z}\\{\partial F_{x}/\partial z}-{\partial F_{z}/\partial x}\\{\partial F_{y}/\partial x}-{\partial F_{x}/\partial y}\end{pmatrix}}}
Δ = ∇ 2 = ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 + ∂ 2 ∂ z 2 {\displaystyle \Delta =\nabla ^{2}={\partial ^{2} \over \partial x^{2}}+{\partial ^{2} \over \partial y^{2}}+{\partial ^{2} \over \partial z^{2}}}
Δ ϕ = ∇ 2 ϕ = ∇ ⋅ ( ∇ ϕ ) = d i v ( g r a d ϕ ) {\displaystyle \Delta \phi =\nabla ^{2}\phi =\nabla \cdot (\nabla \phi )=\mathrm {div} (\mathrm {grad} \phi )}
div ( a F + b G ) = a div ( F ) + b div ( G ) {\displaystyle \operatorname {div} (a\mathbf {F} +b\mathbf {G} )=a\;\operatorname {div} (\mathbf {F} )+b\;\operatorname {div} (\mathbf {G} )}
div ( ϕ F ) = grad ( ϕ ) ⋅ F + ϕ div ( F ) {\displaystyle \operatorname {div} (\phi \mathbf {F} )=\operatorname {grad} (\phi )\cdot \mathbf {F} +\phi \;\operatorname {div} (\mathbf {F} )}
∇ ⋅ ( ϕ F ) = ( ∇ ϕ ) ⋅ F + ϕ ( ∇ ⋅ F ) {\displaystyle \nabla \cdot (\phi \mathbf {F} )=(\nabla \phi )\cdot \mathbf {F} +\phi \;(\nabla \cdot \mathbf {F} )}
div ( F × G ) = curl ( F ) ⋅ G − F ⋅ curl ( G ) {\displaystyle \operatorname {div} (\mathbf {F} \times \mathbf {G} )=\operatorname {curl} (\mathbf {F} )\cdot \mathbf {G} \;-\;\mathbf {F} \cdot \operatorname {curl} (\mathbf {G} )}
div ( curl F ) = div ( ∇ × F ) = curl ( ∇ ) ⋅ F − ∇ ⋅ curl ( F ) {\displaystyle \operatorname {div} (\operatorname {curl} \mathbf {F} )=\operatorname {div} (\nabla \times \mathbf {F} )=\operatorname {curl} (\nabla )\cdot \mathbf {F} -\nabla \cdot \operatorname {curl} (\mathbf {F} )}
ここで [ curl ( ∇ ) ] x = ∂ 2 ∂ z ∂ y − ∂ 2 ∂ y ∂ z = 0 {\displaystyle \left[\operatorname {curl} (\nabla )\right]_{x}={\frac {\partial ^{2}}{\partial z\partial y}}-{\frac {\partial ^{2}}{\partial y\partial z}}=0} (演算対象の関数が連続でなめらかな場合) であるので
div ( curl F ) = − ∇ ⋅ curl ( F ) = − div ( curl F ) {\displaystyle \operatorname {div} (\operatorname {curl} \mathbf {F} )=-\nabla \cdot \operatorname {curl} (\mathbf {F} )=-\operatorname {div} (\operatorname {curl} \mathbf {F} )}
結局 div ( curl F ) = 0 {\displaystyle \operatorname {div} (\operatorname {curl} \mathbf {F} )=0}
curl ( curl ( F ) ) = − Δ F + grad ( div F ) {\displaystyle \operatorname {curl} (\operatorname {curl} (\mathbf {F} ))=-\Delta \mathbf {F} +\operatorname {grad} (\operatorname {div} \mathbf {F} )}
x成分をとって証明する。
[ curl ( curl ( F ) ) ] x = [ ∇ × ( ∇ × F ) ] x = ∂ ∂ y [ ∇ × F ] z − ∂ ∂ z [ ∇ × F ] y {\displaystyle \left[\operatorname {curl} (\operatorname {curl} (\mathbf {F} ))\right]_{x}=\left[\nabla \times (\nabla \times \mathbf {F} )\right]_{x}={\frac {\partial }{\partial y}}\left[\nabla \times \mathbf {F} \right]_{z}-{\frac {\partial }{\partial z}}\left[\nabla \times \mathbf {F} \right]_{y}} = ∂ ∂ y ( ∂ F y ∂ x − ∂ F x ∂ y ) − ∂ ∂ z ( ∂ F x ∂ z − ∂ F z ∂ x ) {\displaystyle ={\frac {\partial }{\partial y}}({\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}})-{\frac {\partial }{\partial z}}({\frac {\partial F_{x}}{\partial z}}-{\frac {\partial F_{z}}{\partial x}})} = − ( ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 + ∂ 2 ∂ z 2 ) F x + ∂ ∂ x ( ∂ F x ∂ x + ∂ F y ∂ y + ∂ F z ∂ z ) {\displaystyle =-({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}})F_{x}+{\frac {\partial }{\partial x}}({\frac {\partial F_{x}}{\partial x}}+{\frac {\partial F_{y}}{\partial y}}+{\frac {\partial F_{z}}{\partial z}})} = − Δ F x + ∂ ∂ x div F = [ − Δ F + grad ( div F ) ] x {\displaystyle =-\Delta F_{x}+{\frac {\partial }{\partial x}}\operatorname {div} \mathbf {F} =\left[-\Delta \mathbf {F} +\operatorname {grad} (\operatorname {div} \mathbf {F} )\right]_{x}}
![{\displaystyle \left[\operatorname {curl} (\nabla )\right]_{x}={\frac {\partial ^{2}}{\partial z\partial y}}-{\frac {\partial ^{2}}{\partial y\partial z}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1650c48da63774117242ef8f225e1f731ca29ce)
(演算対象の関数が連続でなめらかな場合)
であるので
![{\displaystyle \left[\operatorname {curl} (\operatorname {curl} (\mathbf {F} ))\right]_{x}=\left[\nabla \times (\nabla \times \mathbf {F} )\right]_{x}={\frac {\partial }{\partial y}}\left[\nabla \times \mathbf {F} \right]_{z}-{\frac {\partial }{\partial z}}\left[\nabla \times \mathbf {F} \right]_{y}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e20a764b9c7676e777e58f446143f8150638899e)
![{\displaystyle =-\Delta F_{x}+{\frac {\partial }{\partial x}}\operatorname {div} \mathbf {F} =\left[-\Delta \mathbf {F} +\operatorname {grad} (\operatorname {div} \mathbf {F} )\right]_{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95749aa8f1c42f623313be909a1fb4a96e56d108)
