ベクトル解析の公式の一覧
∇ × A = 1 r sin  θ [ ∂ ∂ θ ( sin  θ A φ ) − ∂ A θ ∂ φ ] e r + 1 r [ 1 sin  θ ∂ A r ∂ φ − ∂ ∂ r ( r A φ ) ] e θ + 1 r [ ∂ ∂ r ( r A θ ) − ∂ A r ∂ θ ] e φ {\displaystyle \mathbf {\nabla } \times \mathbf {A} ={\frac {1}{r\sin \theta }}\left[{\frac {\partial }{\partial \theta }}(\sin \theta A_{\phi })-{\frac {\partial A_{\theta }}{\partial \phi }}\right]\mathbf {e} _{r}+{\frac {1}{r}}\left[{\frac {1}{\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}-{\frac {\partial }{\partial r}}(rA_{\phi })\right]\mathbf {e} _{\theta }+{\frac {1}{r}}\left[{\frac {\partial }{\partial r}}(rA_{\theta })-{\frac {\partial A_{r}}{\partial \theta }}\right]\mathbf {e} _{\phi }}
ラプラシアン (スカラー場)
∇ 2 f = 1 r 2 ∂ ∂ r ( r 2 ∂ f ∂ r ) + 1 r 2 sin  θ ∂ ∂ θ ( sin  θ ∂ f ∂ θ ) + 1 r 2 sin 2  θ ∂ 2 f ∂ φ 2 {\displaystyle \mathbf {\nabla } ^{2}f={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \phi ^{2}}}}
∇ × ∇ f = 0 {\displaystyle \mathbf {\nabla } \times \mathbf {\nabla } f=0}
∇ ⋅ ( ∇ × A ) = 0 {\displaystyle \mathbf {\nabla } \cdot (\mathbf {\nabla } \times \mathbf {A} )=0}
∇ × ( ∇ × A ) = ∇ ( ∇ ⋅ A ) − ∇ 2 A {\displaystyle \mathbf {\nabla } \times (\mathbf {\nabla } \times \mathbf {A} )=\mathbf {\nabla } (\mathbf {\nabla } \cdot \mathbf {A} )-\mathbf {\nabla } ^{2}\mathbf {A} }
ヘルムホルツ分解
B = ∇ f + ∇ × A {\displaystyle \mathbf {B} =\mathbf {\nabla } f+\mathbf {\nabla } \times \mathbf {A} }