# 向量恆等式列表

## 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

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

## 二次微分

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

## 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：$\psi(\mathbf{x} ) - \int_\mathbb{U} \left[ G(\mathbf{x},\mathbf{x}' ) \nabla'^{\,2} \psi(\mathbf{x}')\right]\, \mathrm{d}V'= \oint_{\partial \mathbb{U}} \left[\psi(\mathbf{x}') {\partial G(\mathbf{x},\mathbf{x}' ) \over \partial n'} - G(\mathbf{x},\mathbf{x}' ) {\partial \psi(\mathbf{x}') \over \partial n'} \right] \, \mathrm{d}S'$