向量恒等式列表

这条目陈列一些常用的向量代数的恒等式。

在这篇文章内，向量与其量值分别用粗体与斜体表示；例如，${\displaystyle \left|\mathbf {r} \right|=r\,\!}$ 。

三重积

主条目：三重积

• ${\displaystyle \mathbf {A} \times (\mathbf {B} \times \mathbf {C} )=(\mathbf {C} \times \mathbf {B} )\times \mathbf {A} =\mathbf {B} (\mathbf {A} \cdot \mathbf {C} )-\mathbf {C} (\mathbf {A} \cdot \mathbf {B} )}$
• ${\displaystyle \mathbf {A} \cdot (\mathbf {B} \times \mathbf {C} )=\mathbf {B} \cdot (\mathbf {C} \times \mathbf {A} )=\mathbf {C} \cdot (\mathbf {A} \times \mathbf {B} )}$

其他乘积

• ${\displaystyle (\mathbf {A} \times \mathbf {B} )\cdot (\mathbf {A} \times \mathbf {B} )=A^{2}B^{2}-(\mathbf {A} \cdot \mathbf {B} )^{2}=\mathbf {B} \cdot (\mathbf {A} \times (\mathbf {B} \times \mathbf {A} ))}$
• ${\displaystyle \mathbf {\left(A\times B\right)\times } \left(\mathbf {C} \times \mathbf {D} \right)=\left(\mathbf {A} \cdot (\mathbf {B\times D} )\right)\mathbf {C} -\left(\mathbf {A} \cdot (\mathbf {B\times C} )\right)\mathbf {D} }$

乘积定则

• ${\displaystyle \mathbf {\nabla } (fg)=f(\mathbf {\nabla } g)+g(\mathbf {\nabla } f)}$
• ${\displaystyle \mathbf {\nabla } (\mathbf {A} \cdot \mathbf {B} )=\mathbf {A} \times (\mathbf {\nabla } \times \mathbf {B} )+\mathbf {B} \times (\mathbf {\nabla } \times \mathbf {A} )+(\mathbf {A} \cdot \mathbf {\nabla } )\mathbf {B} +(\mathbf {B} \cdot \mathbf {\nabla } )\mathbf {A} }$
• ${\displaystyle \mathbf {\nabla } (\mathbf {A} \cdot \mathbf {B} )=(\mathbf {A} \times \mathbf {\nabla } )\times \mathbf {B} +(\mathbf {B} \times \mathbf {\nabla } )\times \mathbf {A} +\mathbf {A} (\mathbf {\nabla } \cdot \mathbf {B} )+\mathbf {B} (\mathbf {\nabla } \cdot \mathbf {A} )}$
• ${\displaystyle \mathbf {\nabla } \cdot (f\mathbf {A} )=f(\mathbf {\nabla } \cdot \mathbf {A} )+\mathbf {A} \cdot (\mathbf {\nabla } f)}$
• ${\displaystyle \mathbf {\nabla } \cdot (\mathbf {A} \times \mathbf {B} )=\mathbf {B} \cdot (\mathbf {\nabla } \times \mathbf {A} )-\mathbf {A} \cdot (\mathbf {\nabla } \times \mathbf {B} )}$
• ${\displaystyle \nabla \times (f\mathbf {A} )=f(\nabla \times \mathbf {A} )+(\nabla f)\times \mathbf {A} }$
• ${\displaystyle \nabla \times (\mathbf {A} \times \mathbf {B} )=(\mathbf {B} \cdot \nabla )\mathbf {A} -(\mathbf {A} \cdot \nabla )\mathbf {B} +\mathbf {A} (\nabla \cdot \mathbf {B} )-\mathbf {B} (\nabla \cdot \mathbf {A} )}$
• ${\displaystyle \nabla \times (\mathbf {A} \cdot \mathbf {B} )=\mathbf {A} \times (\nabla \times \mathbf {B} )-\mathbf {B} \times (\nabla \times \mathbf {A} )-(\mathbf {A} \times \nabla )\times \mathbf {B} +(\mathbf {B} \times \nabla )\times \mathbf {A} }$
• ${\displaystyle \nabla \left({\frac {1}{|\mathbf {r} -\mathbf {r} '|}}\right)=-\nabla '\left({\frac {1}{|\mathbf {r} -\mathbf {r} '|}}\right)=-\ {\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{3}}}\,\!}$
• ${\displaystyle \nabla ^{2}\left({\frac {1}{|\mathbf {r} -\mathbf {r} '|}}\right)=-4\pi \delta (\mathbf {r} -\mathbf {r} ')}$

二次微分

• ${\displaystyle \nabla \cdot (\nabla \times \mathbf {A} )=0}$
• ${\displaystyle \nabla \times (\nabla f)=\mathbf {0} }$
• ${\displaystyle \nabla ^{2}(\nabla \cdot \mathbf {A} )=\nabla \cdot (\nabla ^{2}\mathbf {A} )}$
• ${\displaystyle \nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A} }$

这里，${\displaystyle \nabla ^{2}\mathbf {A} }$ 应被理解为对 ${\displaystyle \mathbf {A} }$ 的每个分量取拉普拉斯算子，即向量值函数的拉普拉斯算子。

积分

• ${\displaystyle \oint _{\mathbb {S} }\mathbf {A} \cdot \mathrm {d} \mathbf {S} =\int _{\mathbb {V} }\left(\nabla \cdot \mathbf {A} \right)\mathrm {d} V}$ （散度定理）
• ${\displaystyle \oint _{\mathbb {S} }\psi \mathrm {d} \mathbf {S} =\int _{\mathbb {V} }\nabla \psi \,\mathrm {d} V}$
• ${\displaystyle \oint _{\mathbb {S} }\left({\hat {\mathbf {n} }}\times \mathbf {A} \right)\cdot \mathrm {d} S=\int _{\mathbb {V} }\left(\nabla \times \mathbf {A} \right)\mathrm {d} V}$
• ${\displaystyle \oint _{\mathbb {C} }\mathbf {A} \cdot d\mathbf {l} =\int _{\mathbb {S} }\left(\nabla \times \mathbf {A} \right)\cdot \mathrm {d} \mathbf {S} }$ （斯托克斯定理）

• ${\displaystyle \oint _{\mathbb {C} }\psi d\mathbf {l} =\int _{\mathbb {S} }\left({\hat {\mathbf {n} }}\times \nabla \psi \right)\mathrm {d} S}$

格林恒等式

• 格林第一恒等式： ${\displaystyle \int _{\mathbb {U} }(\psi \nabla ^{2}\phi +\nabla \phi \cdot \nabla \psi )\,\mathrm {d} V=\oint _{\partial \mathbb {U} }\psi {\partial \phi \over \partial n}\,\mathrm {d} S}$
• 格林第二恒等式：${\displaystyle \int _{\mathbb {U} }\left(\psi \nabla ^{2}\phi -\phi \nabla ^{2}\psi \right)\,\mathrm {d} V=\oint _{\partial \mathbb {U} }\left(\psi {\partial \phi \over \partial n}-\phi {\partial \psi \over \partial n}\right)\,\mathrm {d} S}$
• 格林第三恒等式：${\displaystyle \psi (\mathbf {x} )-\int _{\mathbb {U} }\left[G(\mathbf {x} ,\mathbf {x} ')\nabla '^{\,2}\psi (\mathbf {x} ')\right]\,\mathrm {d} V'=\oint _{\partial \mathbb {U} }\left[\psi (\mathbf {x} '){\partial G(\mathbf {x} ,\mathbf {x} ') \over \partial n'}-G(\mathbf {x} ,\mathbf {x} '){\partial \psi (\mathbf {x} ') \over \partial n'}\right]\,\mathrm {d} S'}$

参阅

• 格林恒等式
• 数学恒等式列表 (List of mathematical identities)
• 向量微积分恒等式 (Vector calculus identities)