向量恆等式

: 在這篇文章內，向量與其量值分別用粗體與斜體表示；例如， $\left|\mathbf {r} \right|=r\,\!$ 。

這條目陳列一些常用的向量代數的恆等式。

## 三重積

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

## 其他乘積

 * $(\mathbf {A} \times \mathbf {B} )\cdot (\mathbf {A} \times \mathbf {B} )=A^{2}B^{2}-(\mathbf {A} \cdot \mathbf {B} )^{2}=\mathbf {B} \cdot (\mathbf {A} \times (\mathbf {B} \times \mathbf {A} ))$
* $\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} $

## 乘積定則

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

## 二次微分

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

　
: : 這裏， $\nabla ^{2}\mathbf {A} $ 應被理解爲對 $\mathbf {A} $ 的每個分量取拉普拉斯算子，卽向量值函數的拉普拉斯算子。

## 積分

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

### 格林恆等式

* 格林第一恆等式： $\int _{\mathbb {U} }(\psi \nabla ^{2}\phi +\nabla \phi \cdot \nabla \psi )\,\mathrm {d} V=\oint _{\partial \mathbb {U} }\psi {\partial \phi  \over \partial n}\,\mathrm {d} S$
* 格林第二恆等式： $\int _{\mathbb {U} }\left(\psi \nabla ^{2}\phi -\phi \nabla ^{2}\psi \right)\,\mathrm {d} V=\oint _{\partial \mathbb {U} }\left(\psi {\partial \phi  \over \partial n}-\phi {\partial \psi  \over \partial n}\right)\,\mathrm {d} S$
* 格林第三恆等式： $\psi (\mathbf {x} )-\int _{\mathbb {U} }\left[G(\mathbf {x} ,\mathbf {x} ')\nabla '^{\,2}\psi (\mathbf {x} ')\right]\,\mathrm {d} V'=\oint _{\partial \mathbb {U} }\left[\psi (\mathbf {x} '){\partial G(\mathbf {x} ,\mathbf {x} ') \over \partial n'}-G(\mathbf {x} ,\mathbf {x} '){\partial \psi (\mathbf {x} ') \over \partial n'}\right]\,\mathrm {d} S'$

## 參閱

* 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)