向量恆等式列表

在這篇文章內，向量與其量值分別用粗體與斜體表示；例如， | r | = r {\displaystyle \left|\mathbf {r} \right|=r\,\!} 。 這條目陳列一些常用的向量代數的恆等式。 A × ( B × C ) = ( C × B ) × A = B ( A ⋅ C ) − C ( A ⋅ B ) {\displaystyle \mathbf {A} \times (\mathbf {B} \times \mathbf {C} )=(\mathbf {C} \times \mathbf {B} )\times \mathbf {A} =\mathbf {B} (\mathbf {A} \cdot \mathbf {C} )-\mathbf {C} (\mathbf {A} \cdot \mathbf {B} )} A ⋅ ( B × C ) = B ⋅ ( C × A ) = C ⋅ ( A × B ) {\displaystyle \mathbf {A} \cdot (\mathbf {B} \times \mathbf {C} )=\mathbf {B} \cdot (\mathbf {C} \times \mathbf {A} )=\mathbf {C} \cdot (\mathbf {A} \times \mathbf {B} )} ( A × B ) ⋅ ( A × B ) = A 2 B 2 − ( A ⋅ B ) 2 = B ⋅ ( A × ( B × A ) ) {\displaystyle (\mathbf {A} \times \mathbf {B} )\cdot (\mathbf {A} \times \mathbf {B} )=A^{2}B^{2}-(\mathbf {A} \cdot \mathbf {B} )^{2}=\mathbf {B} \cdot (\mathbf {A} \times (\mathbf {B} \times \mathbf {A} ))} ( A × B ) × ( C × D ) = ( A ⋅ ( B × D ) ) C − ( A ⋅ ( B × C ) ) D {\displaystyle \mathbf {\left(A\times B\right)\times } \left(\mathbf {C} \times \mathbf {D} \right)=\left(\mathbf {A} \cdot (\mathbf {B\times D} )\right)\mathbf {C} -\left(\mathbf {A} \cdot (\mathbf {B\times C} )\right)\mathbf {D} } ∇ ( f g ) = f ( ∇ g ) + g ( ∇ f ) {\displaystyle \mathbf {\nabla } (fg)=f(\mathbf {\nabla } g)+g(\mathbf {\nabla } f)} ∇ ( A ⋅ B ) = A × ( ∇ × B ) + B × ( ∇ × A ) + ( A ⋅ ∇ ) B + ( B ⋅ ∇ ) A {\displaystyle \mathbf {\nabla } (\mathbf {A} \cdot \mathbf {B} )=\mathbf {A} \times (\mathbf {\nabla } \times \mathbf {B} )+\mathbf {B} \times (\mathbf {\nabla } \times \mathbf {A} )+(\mathbf {A} \cdot \mathbf {\nabla } )\mathbf {B} +(\mathbf {B} \cdot \mathbf {\nabla } )\mathbf {A} } ∇ ( A ⋅ B ) = ( A × ∇ ) × B + ( B × ∇ ) × A + A ( ∇ ⋅ B ) + B ( ∇ ⋅ A ) {\displaystyle \mathbf {\nabla } (\mathbf {A} \cdot \mathbf {B} )=(\mathbf {A} \times \mathbf {\nabla } )\times \mathbf {B} +(\mathbf {B} \times \mathbf {\nabla } )\times \mathbf {A} +\mathbf {A} (\mathbf {\nabla } \cdot \mathbf {B} )+\mathbf {B} (\mathbf {\nabla } \cdot \mathbf {A} )} ∇ ⋅ ( f A ) = f ( ∇ ⋅ A ) + A ⋅ ( ∇ f ) {\displaystyle \mathbf {\nabla } \cdot (f\mathbf {A} )=f(\mathbf {\nabla } \cdot \mathbf {A} )+\mathbf {A} \cdot (\mathbf {\nabla } f)} ∇ ⋅ ( 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 \nabla \times (f\mathbf {A} )=f(\nabla \times \mathbf {A} )+(\nabla f)\times \mathbf {A} } ∇ × ( A × B ) = ( B ⋅ ∇ ) A − ( A ⋅ ∇ ) B + A ( ∇ ⋅ B ) − B ( ∇ ⋅ A ) {\displaystyle \nabla \times (\mathbf {A} \times \mathbf {B} )=(\mathbf {B} \cdot \nabla )\mathbf {A} -(\mathbf {A} \cdot \nabla )\mathbf {B} +\mathbf {A} (\nabla \cdot \mathbf {B} )-\mathbf {B} (\nabla \cdot \mathbf {A} )} ∇ × ( A ⋅ B ) = A × ( ∇ × B ) − B × ( ∇ × A ) − ( A × ∇ ) × B + ( B × ∇ ) × A {\displaystyle \nabla \times (\mathbf {A} \cdot \mathbf {B} )=\mathbf {A} \times (\nabla \times \mathbf {B} )-\mathbf {B} \times (\nabla \times \mathbf {A} )-(\mathbf {A} \times \nabla )\times \mathbf {B} +(\mathbf {B} \times \nabla )\times \mathbf {A} } ∇ ( 1 | r − r ′ | ) = − ∇ ′ ( 1 | r − r ′ | ) = −   r − r ′ | r − r ′ | 3 {\displaystyle \nabla \left({\frac {1}{|\mathbf {r} -\mathbf {r} '|}}\right)=-\nabla '\left({\frac {1}{|\mathbf {r} -\mathbf {r} '|}}\right)=-\ {\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{3}}}\,\!} ∇ 2 ( 1 | r − r ′ | ) = − 4 π δ ( r − r ′ ) {\displaystyle \nabla ^{2}\left({\frac {1}{|\mathbf {r} -\mathbf {r} '|}}\right)=-4\pi \delta (\mathbf {r} -\mathbf {r} ')} ∇ ⋅ ( ∇ × A ) = 0 {\displaystyle \nabla \cdot (\nabla \times \mathbf {A} )=0} ∇ × ( ∇ f ) = 0 {\displaystyle \nabla \times (\nabla f)=\mathbf {0} } ∇ 2 ( ∇ ⋅ A ) = ∇ ⋅ ( ∇ 2 A ) {\displaystyle \nabla ^{2}(\nabla \cdot \mathbf {A} )=\nabla \cdot (\nabla ^{2}\mathbf {A} )} ∇ × ( ∇ × A ) = ∇ ( ∇ ⋅ A ) − ∇ 2 A {\displaystyle \nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A} } 這裏， ∇ 2 A {\displaystyle \nabla ^{2}\mathbf {A} } 應被理解爲對 A {\displaystyle \mathbf {A} } 的每個分量取拉普拉斯算子，卽向量值函数的拉普拉斯算子。 ∮ S A ⋅ d S = ∫ V ( ∇ ⋅ A ) d V {\displaystyle \oint _{\mathbb {S} }\mathbf {A} \cdot \mathrm {d} \mathbf {S} =\int _{\mathbb {V} }\left(\nabla \cdot \mathbf {A} \right)\mathrm {d} V} （散度定理） ∮ S ψ d S = ∫ V ∇ ψ d V {\displaystyle \oint _{\mathbb {S} }\psi \mathrm {d} \mathbf {S} =\int _{\mathbb {V} }\nabla \psi \,\mathrm {d} V} ∮ S ( n ^ × A ) ⋅ d S = ∫ V ( ∇ × A ) d V {\displaystyle \oint _{\mathbb {S} }\left({\hat {\mathbf {n} }}\times \mathbf {A} \right)\cdot \mathrm {d} S=\int _{\mathbb {V} }\left(\nabla \times \mathbf {A} \right)\mathrm {d} V} ∮ C A ⋅ d l = ∫ S ( ∇ × A ) ⋅ d S {\displaystyle \oint _{\mathbb {C} }\mathbf {A} \cdot d\mathbf {l} =\int _{\mathbb {S} }\left(\nabla \times \mathbf {A} \right)\cdot \mathrm {d} \mathbf {S} } （斯托克斯定理） ∮ C ψ d l = ∫ S ( n ^ × ∇ ψ ) d S {\displaystyle \oint _{\mathbb {C} }\psi d\mathbf {l} =\int _{\mathbb {S} }\left({\hat {\mathbf {n} }}\times \nabla \psi \right)\mathrm {d} S} 格林第一恆等式： ∫ U ( ψ ∇ 2 ϕ + ∇ ϕ ⋅ ∇ ψ ) d V = ∮ ∂ U ψ ∂ ϕ ∂ n d S {\displaystyle \int _{\mathbb {U} }(\psi \nabla ^{2}\phi +\nabla \phi \cdot \nabla \psi )\,\mathrm {d} V=\oint _{\partial \mathbb {U} }\psi {\partial \phi \over \partial n}\,\mathrm {d} S} 格林第二恆等式： ∫ U ( ψ ∇ 2 ϕ − ϕ ∇ 2 ψ ) d V = ∮ ∂ U ( ψ ∂ ϕ ∂ n − ϕ ∂ ψ ∂ n ) d S {\displaystyle \int _{\mathbb {U} }\left(\psi \nabla ^{2}\phi -\phi \nabla ^{2}\psi \right)\,\mathrm {d} V=\oint _{\partial \mathbb {U} }\left(\psi {\partial \phi \over \partial n}-\phi {\partial \psi \over \partial n}\right)\,\mathrm {d} S} 格林第三恆等式： ψ ( x ) − ∫ U [ G ( x , x ′ ) ∇ ′ 2 ψ ( x ′ ) ] d V ′ = ∮ ∂ U [ ψ ( x ′ ) ∂ G ( x , x ′ ) ∂ n ′ − G ( x , x ′ ) ∂ ψ ( x ′ ) ∂ n ′ ] d S ′ {\displaystyle \psi (\mathbf {x} )-\int _{\mathbb {U} }\left[G(\mathbf {x} ,\mathbf {x} ')\nabla '^{\,2}\psi (\mathbf {x} ')\right]\,\mathrm {d} V'=\oint _{\partial \mathbb {U} }\left[\psi (\mathbf {x} '){\partial G(\mathbf {x} ,\mathbf {x} ') \over \partial n'}-G(\mathbf {x} ,\mathbf {x} '){\partial \psi (\mathbf {x} ') \over \partial n'}\right]\,\mathrm {d} S'} 格林恆等式 數學恆等式列表 (List of mathematical identities) 向量微積分恆等式 (Vector calculus identities)