ベクトル解析の公式の一覧
∇ 2 f = ∑ i = 1 3 1 h ∂ ∂ x i ( h h i 2 ∂ f ∂ x i ) {\displaystyle \mathbf {\nabla } ^{2}f=\sum _{i=1}^{3}{\frac {1}{h}}{\frac {\partial }{\partial x^{i}}}\left({\frac {h}{h_{i}^{2}}}{\frac {\partial f}{\partial x^{i}}}\right)}
物質微分
[ ( A ⋅ ∇ ) B ] i = ∑ k = 1 3 [ A k h k ∂ B i ∂ x k + ( A i ∂ h i ∂ x k − A k ∂ h k ∂ x i ) B k h k h i ] {\displaystyle [(\mathbf {A} \cdot \mathbf {\nabla } )\mathbf {B} ]_{i}=\sum _{k=1}^{3}\left[{\frac {A_{k}}{h_{k}}}{\frac {\partial B_{i}}{\partial x_{k}}}+\left(A_{i}{\frac {\partial h_{i}}{\partial x_{k}}}-A_{k}{\frac {\partial h_{k}}{\partial x_{i}}}\right){\frac {B_{k}}{h_{k}h_{i}}}\right]}
[ ∇ 2 A ] θ = ∇ 2 A θ + 2 r 2 ∂ A r ∂ θ − 2 cot  θ r 2 sin  θ ∂ A φ ∂ φ − A θ r 2 sin 2  θ {\displaystyle [\mathbf {\nabla } ^{2}\mathbf {A} ]_{\theta }=\mathbf {\nabla } ^{2}A_{\theta }+{\frac {2}{r^{2}}}{\frac {\partial A_{r}}{\partial \theta }}-{\frac {2\cot \theta }{r^{2}\sin \theta }}{\frac {\partial A_{\phi }}{\partial \phi }}-{\frac {A_{\theta }}{r^{2}\sin ^{2}\theta }}}
[ ∇ 2 A ] φ = ∇ 2 A φ + 2 r 2 sin  θ ∂ A r ∂ φ + 2 cot  θ r 2 sin  θ ∂ A θ ∂ φ − A φ r 2 sin 2  θ {\displaystyle [\mathbf {\nabla } ^{2}\mathbf {A} ]_{\phi }=\mathbf {\nabla } ^{2}A_{\phi }+{\frac {2}{r^{2}\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}+{\frac {2\cot \theta }{r^{2}\sin \theta }}{\frac {\partial A_{\theta }}{\partial \phi }}-{\frac {A_{\phi }}{r^{2}\sin ^{2}\theta }}}
物質微分
A × B = ( ε i j k A j B k ) e i = ( A y B z − A z B y ) e x + ( A z B x − A x B z ) e y + ( A x B y − A y B x ) e z {\displaystyle \mathbf {A} \times \mathbf {B} =(\epsilon _{ijk}A_{j}B_{k})\mathbf {e} _{i}=(A_{y}B_{z}-A_{z}B_{y})\mathbf {e} _{x}+(A_{z}B_{x}-A_{x}B_{z})\mathbf {e} _{y}+(A_{x}B_{y}-A_{y}B_{x})\mathbf {e} _{z}}
A × B = − B × A {\displaystyle \mathbf {A} \times \mathbf {B} =-\mathbf {B} \times \mathbf {A} }
| A × B | = | A | B | sin  θ {\displaystyle |\mathbf {A} \times \mathbf {B} |=|\mathbf {A} |\mathbf {B} |\sin \theta }
スカラー三重積
四重積
( A × B ) ⋅ ( C × D ) = ( A ⋅ C ) ( B ⋅ D ) − ( A ⋅ D ) ( B ⋅ C ) {\displaystyle (\mathbf {A} \times \mathbf {B} )\cdot (\mathbf {C} \times \mathbf {D} )=(\mathbf {A} \cdot \mathbf {C} )(\mathbf {B} \cdot \mathbf {D} )-(\mathbf {A} \cdot \mathbf {D} )(\mathbf {B} \cdot \mathbf {C} )}
( A × B ) × ( C × D ) = [ A ⋅ ( C × D ) ] B − [ B ⋅ ( C × D ) ] A {\displaystyle (\mathbf {A} \times \mathbf {B} )\times (\mathbf {C} \times \mathbf {D} )=[\mathbf {A} \cdot (\mathbf {C} \times \mathbf {D} )]\mathbf {B} -[\mathbf {B} \cdot (\mathbf {C} \times \mathbf {D} )]\mathbf {A} }