ベクトル解析の公式の一覧
[ ∇ 2 A ] z = ∇ 2 A z {\displaystyle [\mathbf {\nabla } ^{2}\mathbf {A} ]_{z}=\mathbf {\nabla } ^{2}A_{z}}
物質微分
[ ( A ⋅ ∇ ) B ] r = ( A ⋅ ∇ ) B r − A θ B θ r {\displaystyle [(\mathbf {A} \cdot \mathbf {\nabla } )\mathbf {B} ]_{r}=(\mathbf {A} \cdot \mathbf {\nabla } )B_{r}-{\frac {A_{\theta }B_{\theta }}{r}}}
[ ( A ⋅ ∇ ) B ] θ = ( A ⋅ ∇ ) B θ + A θ B r r {\displaystyle [(\mathbf {A} \cdot \mathbf {\nabla } )\mathbf {B} ]_{\theta }=(\mathbf {A} \cdot \mathbf {\nabla } )B_{\theta }+{\frac {A_{\theta }B_{r}}{r}}}
[ ( A ⋅ ∇ ) B ] z = ( A ⋅ ∇ ) B z {\displaystyle [(\mathbf {A} \cdot \mathbf {\nabla } )\mathbf {B} ]_{z}=(\mathbf {A} \cdot \mathbf {\nabla } )B_{z}}
∇ 2 f = 1 r ∂ ∂ r ( r ∂ f ∂ r ) + 1 r 2 ∂ 2 f ∂ θ 2 + ∂ 2 f ∂ z 2 {\displaystyle \mathbf {\nabla } ^{2}f={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \theta ^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}}
ラプラシアン (ベクトル場)
[ ∇ 2 A ] r = ∇ 2 A r − 2 r 2 ∂ A θ ∂ θ − A r 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 {A_{r}}{r^{2}}}}
[ ∇ 2 A ] θ = ∇ 2 A θ + 2 r 2 ∂ A r ∂ θ − A θ r 2 {\displaystyle [\mathbf {\nabla } ^{2}\mathbf {A} ]_{\theta }=\mathbf {\nabla } ^{2}A_{\theta }+{\frac {2}{r^{2}}}{\frac {\partial A_{r}}{\partial \theta }}-{\frac {A_{\theta }}{r^{2}}}}
発散
∇ ⋅ A = 1 r 2 ∂ ∂ r ( r 2 A r ) + 1 r sin  θ ∂ ∂ θ ( sin  θ A θ ) + 1 r sin  θ ∂ A φ ∂ φ {\displaystyle \mathbf {\nabla } \cdot \mathbf {A} ={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}(r^{2}A_{r})+{\frac {1}{r\sin \theta }}{\frac {\partial }{\partial \theta }}(\sin \theta A_{\theta })+{\frac {1}{r\sin \theta }}{\frac {\partial A_{\phi }}{\partial \phi }}}
回転