ベクトル解析の公式の一覧
∇ ⋅ ( A × B ) = B ⋅ ( ∇ × A ) − A ⋅ ( ∇ × B ) {\displaystyle \mathbf {\nabla } \cdot (\mathbf {A} \times \mathbf {B} )=\mathbf {B} \cdot (\mathbf {\nabla } \times \mathbf {A} )-\mathbf {A} \cdot (\mathbf {\nabla } \times \mathbf {B} )}
∇ × ( f A ) = ∇ f × A + f ∇ × A {\displaystyle \mathbf {\nabla } \times (f\mathbf {A} )=\mathbf {\nabla } f\times \mathbf {A} +f\mathbf {\nabla } \times \mathbf {A} }
∇ × ( A × B ) = ( B ⋅ ∇ ) A − ( A ⋅ ∇ ) B + A ( ∇ ⋅ B ) − B ( ∇ ⋅ A ) {\displaystyle \mathbf {\nabla } \times (\mathbf {A} \times \mathbf {B} )=(\mathbf {B} \cdot \mathbf {\nabla } )\mathbf {A} -(\mathbf {A} \cdot \mathbf {\nabla } )\mathbf {B} +\mathbf {A} (\mathbf {\nabla } \cdot \mathbf {B} )-\mathbf {B} (\mathbf {\nabla } \cdot \mathbf {A} )}
[ ( A ⋅ ∇ ) B ] r = ( A ⋅ ∇ ) B r − A θ B θ + A φ B φ r {\displaystyle [(\mathbf {A} \cdot \mathbf {\nabla } )\mathbf {B} ]_{r}=(\mathbf {A} \cdot \mathbf {\nabla } )B_{r}-{\frac {A_{\theta }B_{\theta }+A_{\phi }B_{\phi }}{r}}}
[ ( A ⋅ ∇ ) B ] θ = ( A ⋅ ∇ ) B θ + A θ B r r − A φ B φ cot  θ r {\displaystyle [(\mathbf {A} \cdot \mathbf {\nabla } )\mathbf {B} ]_{\theta }=(\mathbf {A} \cdot \mathbf {\nabla } )B_{\theta }+{\frac {A_{\theta }B_{r}}{r}}-{\frac {A_{\phi }B_{\phi }\cot \theta }{r}}}
[ ( A ⋅ ∇ ) B ] φ = ( A ⋅ ∇ ) B z + A φ B r r + A φ B θ cot  θ r {\displaystyle [(\mathbf {A} \cdot \mathbf {\nabla } )\mathbf {B} ]_{\phi }=(\mathbf {A} \cdot \mathbf {\nabla } )B_{z}+{\frac {A_{\phi }B_{r}}{r}}+{\frac {A_{\phi }B_{\theta }\cot \theta }{r}}}