ベクトル解析の公式の一覧
ここで A {\displaystyle \mathbf {A} } , B {\displaystyle \mathbf {B} } は任意のベクトル場, f {\displaystyle f} は任意のスカラー場である。
∇ ⋅ ( f A ) = ∇ f ⋅ A + f ∇ ⋅ A {\displaystyle \mathbf {\nabla } \cdot (f\mathbf {A} )=\mathbf {\nabla } f\cdot \mathbf {A} +f\mathbf {\nabla } \cdot \mathbf {A} }
∇ ( A ⋅ B ) = ( B ⋅ ∇ ) A + ( A ⋅ ∇ ) B + A × ( ∇ × B ) + B × ( ∇ × A ) {\displaystyle \mathbf {\nabla } (\mathbf {A} \cdot \mathbf {B} )=(\mathbf {B} \cdot \mathbf {\nabla } )\mathbf {A} +(\mathbf {A} \cdot \mathbf {\nabla } )\mathbf {B} +\mathbf {A} \times (\mathbf {\nabla } \times \mathbf {B} )+\mathbf {B} \times (\mathbf {\nabla } \times \mathbf {A} )}
ガウスの発散定理および関連する公式(最後の等式はグリーンの定理である)
∫ V ∇ ⋅ A d V = ∮ ∂ V A ⋅ d S {\displaystyle \int _{V}\mathbf {\nabla } \cdot \mathbf {A} \,dV=\oint _{\partial V}\mathbf {A} \cdot d\mathbf {S} }
∫ V ∇ f d V = ∮ ∂ V f d S {\displaystyle \int _{V}\mathbf {\nabla } f\,dV=\oint _{\partial V}f\,d\mathbf {S} }
∫ V ∇ × A d V = ∮ ∂ V d S × A {\displaystyle \int _{V}\mathbf {\nabla } \times \mathbf {A} \,dV=\oint _{\partial V}d\mathbf {S} \times \mathbf {A} }
∫ V ( f ∇ 2 g − g ∇ 2 f ) d V = ∮ ∂ V ( f ∇ g − g ∇ f ) ⋅ d S {\displaystyle \int _{V}(f\mathbf {\nabla } ^{2}g-g\mathbf {\nabla } ^{2}f)dV=\oint _{\partial V}(f\mathbf {\nabla } g-g\mathbf {\nabla } f)\cdot d\mathbf {S} }
ラプラシアン (ベクトル場)
[ ∇ 2 A ] r = ∇ 2 A r − 2 r 2 ∂ A θ ∂ θ − 2 r 2 sin  θ ∂ A φ ∂ φ − 2 A r r 2 − 2 cot  θ A θ r 2 {\displaystyle [\mathbf {\nabla } ^{2}\mathbf {A} ]_{r}=\mathbf {\nabla } ^{2}A_{r}-{\frac {2}{r^{2}}}{\frac {\partial A_{\theta }}{\partial \theta }}-{\frac {2}{r^{2}\sin \theta }}{\frac {\partial A_{\phi }}{\partial \phi }}-{\frac {2A_{r}}{r^{2}}}-{\frac {2\cot \theta A_{\theta }}{r^{2}}}}