本文是笔记向的矢量分析，非教学向的。本文中推导了常见的矢量恒等式，给出详细推导过程，步骤基本没省略。

如果你要观看文章内容，需要拥有基本的多元微积分基础。

本文的用加粗字母表示矢量函数，非加粗则表示标量函数

单位矢量采取e，求和皆采用爱因斯坦求和约定。

当然，为了使内容更完善，对基本概念和基本运算进行说明。一是为了帮需要的朋友快速回顾，二是可能以后会改进并写成教学向的文章吧。

即重复指标表示求和，求和号省略不写。如矢量表示：

\boldsymbol{a}=a_1\boldsymbol{e}_1+a_2\boldsymbol{e}_2+a_3\boldsymbol{e}_3=\sum_{i=1}^3{a_i\boldsymbol{e}_{\boldsymbol{i}}}=a_i\boldsymbol{e}_{\boldsymbol{i}}

如矢量点积：

\boldsymbol{a}\cdot \boldsymbol{b}=a_1b_1+a_2b_2+a_3b_3=\sum_{i=1}^3{a_ib_i}=a_ib_i

定义为

\delta _{ij}=\begin{cases} 1 \left( i=j \right)\\ 0 \left( i\ne j \right)\\ \end{cases}

配合爱因斯坦求和约定，矢量点积可写成：

\boldsymbol{a}=a_i\boldsymbol{e}_{\boldsymbol{i}}=a_i\boldsymbol{e}_j\delta _{ij}

\varepsilon _{ijk}\begin{cases} +1, \text{当}\left( i,j,k \right) =\left( 1,2,3 \right) \text{、}\left( 2,3,1 \right) \text{或}\left( 3,1,2 \right)\\ -1, \text{当}\left( i,j,k \right) =\left( 3,2,1 \right) \text{、}\left( 2,1,3 \right) \text{或}\left( 1,3,2 \right)\\ 0, \text{当}i=j\text{、}j=k\text{或}i=k\\ \end{cases}

借助Levi-Civita符号，叉积可以写为：

\boldsymbol{A}\times \boldsymbol{B}=\varepsilon _{ijk}A_iB_j\boldsymbol{e}_{\boldsymbol{k}}

Levi-Civita符号还可以用行列式表示：

\varepsilon _{ijk}=\left| \begin{matrix} \delta _{1i}& \delta _{1j}& \delta _{1k}\\ \delta _{2i}& \delta _{2j}& \delta _{2k}\\ \delta _{3i}& \delta _{3j}& \delta _{3k}\\ \end{matrix} \right|

两个Levi-Civita符号乘积满足行列式乘积规则：

\varepsilon _{ijk}\varepsilon _{lmn}=\left| \begin{matrix} \delta _{im}& \delta _{in}& \delta _{il}\\ \delta _{jm}& \delta _{jn}& \delta _{jk}\\ \delta _{km}& \delta _{kn}& \delta _{kl}\\ \end{matrix} \right|

后面常用的公式：

\varepsilon _{kij}\varepsilon _{lmj}=\left| \begin{matrix} \delta _{kl}& \delta _{km}& \delta _{kj}\\ \delta _{il}& \delta _{im}& \delta _{ij}\\ \delta _{jl}& \delta _{im}& \delta _{jj}\\ \end{matrix} \right|

要使结果不为零， 则 \delta _{jl},\delta _{im},\delta _{kj},\delta _{ij} 的指标不能相同，

否则 \varepsilon _{kij}\varepsilon _{lmj}=0 。

所以 \varepsilon _{kij}\varepsilon _{lmj}=\left| \begin{matrix} \delta _{kl}& \delta _{km}& 0\\ \delta _{il}& \delta _{im}& 0\\ 0& 0& 1\\ \end{matrix} \right|=\left( \delta _{kl}\delta _{im}-\delta _{km}\delta _{il} \right)

\nabla =\left( \frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z} \right)

grad\,\,T=\nabla T=\left( \frac{\partial T}{\partial x},\frac{\partial T}{\partial y},\frac{\partial T}{\partial z} \right) \\ \,\, =\frac{\partial T}{\partial x}\boldsymbol{e}_{\boldsymbol{i}}+\frac{\partial T}{\partial y}\boldsymbol{e}_{\boldsymbol{j}}+\frac{\partial T}{\partial z}\boldsymbol{e}_{\boldsymbol{k}}

div\,\,\boldsymbol{A}=\nabla \cdot \boldsymbol{A}=\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}

由于文章内容是我不同时间写的，使用了两套符号体系，下面两种完全等价。

curl\,\,\boldsymbol{A}=\nabla \times \boldsymbol{A}=\left| \begin{matrix} \boldsymbol{e}_{\boldsymbol{i}}& \boldsymbol{e}_{\boldsymbol{j}}& \boldsymbol{e}_{\boldsymbol{k}}\\ \partial _x& \partial _y& \partial _z\\ A_x& A_y& A_z\\ \end{matrix} \right|=\left| \begin{matrix} \boldsymbol{e}_1& \boldsymbol{e}_2& \boldsymbol{e}_3\\ \partial _1& \partial _2& \partial _3\\ A_1& A_2& A_3\\ \end{matrix} \right|

\boldsymbol{A}\times \left( \boldsymbol{B}\times \boldsymbol{C} \right) =\varepsilon _{ijk}\boldsymbol{A}_i\left( \varepsilon _{lmn}\boldsymbol{B}_l\boldsymbol{C}_m\boldsymbol{e}_n \right) _j\boldsymbol{e}_k \\ =\varepsilon _{ijk}\varepsilon _{lmj}\boldsymbol{A}_i\boldsymbol{B}_l\boldsymbol{C}_m\boldsymbol{e}_k \\ =\left( \delta _{kl}\delta _{im}-\delta _{km}\delta _{il} \right) \boldsymbol{A}_i\boldsymbol{B}_l\boldsymbol{C}_m\boldsymbol{e}_k \\ =\boldsymbol{A}_i\boldsymbol{B}_k\boldsymbol{C}_i\boldsymbol{e}_k-\boldsymbol{A}_i\boldsymbol{B}_i\boldsymbol{C}_k\boldsymbol{e}_k \\ =\left( \boldsymbol{A}\cdot \boldsymbol{C} \right) \boldsymbol{B}-\left( \boldsymbol{A}\cdot \boldsymbol{B} \right) \boldsymbol{C}

\boldsymbol{A}\cdot \left( \boldsymbol{B}\times \boldsymbol{C} \right) =\boldsymbol{A}_l\left( \varepsilon _{ijk}\boldsymbol{B}_i\boldsymbol{C}_j\boldsymbol{e}_k \right) _l \\ =\varepsilon _{ijl}\boldsymbol{A}_l\boldsymbol{B}_i\boldsymbol{C}_j \\ =\varepsilon _{jli}\boldsymbol{B}_i\boldsymbol{C}_j\boldsymbol{A}_l=\boldsymbol{B}\cdot \left( \boldsymbol{C}\times \boldsymbol{A} \right) \\ =\varepsilon _{lij}\boldsymbol{C}_j\boldsymbol{A}_l\boldsymbol{B}_i=\boldsymbol{C}\cdot \left( \boldsymbol{A}\times \boldsymbol{B} \right)

\left( \boldsymbol{A}\times \boldsymbol{B} \right) \cdot \left( \boldsymbol{A}\times \boldsymbol{B} \right) =\left( \varepsilon _{ijk}\boldsymbol{A}_i\boldsymbol{B}_j\boldsymbol{e}_k \right) _p\cdot \left( \varepsilon _{lmn}\boldsymbol{A}_l\boldsymbol{B}_m\boldsymbol{e}_n \right) _p \\ =\varepsilon _{ijp}\boldsymbol{A}_i\boldsymbol{B}_j\varepsilon _{lmp}\boldsymbol{A}_l\boldsymbol{B}_m \\ =\varepsilon _{ijp}\varepsilon _{lmp}\boldsymbol{A}_i\boldsymbol{B}_j\boldsymbol{A}_l\boldsymbol{B}_m \\ =\left( \delta _{il}\delta _{jp}-\delta _{im}\delta _{jl} \right) \boldsymbol{A}_i\boldsymbol{B}_j\boldsymbol{A}_l\boldsymbol{B}_m \\ =\boldsymbol{A}_i\boldsymbol{B}_j\boldsymbol{A}_i\boldsymbol{B}_j-\boldsymbol{A}_i\boldsymbol{B}_j\boldsymbol{A}_j\boldsymbol{B}_i \\ =\boldsymbol{A}^2\boldsymbol{B}^2-\left( \boldsymbol{A}\cdot \boldsymbol{B} \right) ^2

\left( \boldsymbol{A}\times \boldsymbol{B} \right) \times \left( \boldsymbol{C}\times \boldsymbol{D} \right) =\left( \varepsilon _{ijk}\boldsymbol{A}_i\boldsymbol{B}_j\boldsymbol{e}_k \right) \times \left( \varepsilon _{lmn}\boldsymbol{C}_l\boldsymbol{D}_m\boldsymbol{e}_n \right) \\ =\varepsilon _{opq}\left( \varepsilon _{ijk}\boldsymbol{A}_i\boldsymbol{B}_j\boldsymbol{e}_k \right) _o\left( \varepsilon _{lmn}\boldsymbol{C}_l\boldsymbol{D}_m\boldsymbol{e}_n \right) _p\boldsymbol{e}_q \\ =\varepsilon _{opq}\varepsilon _{ijo}\varepsilon _{lmp}\boldsymbol{A}_i\boldsymbol{B}_j\boldsymbol{C}_l\boldsymbol{D}_m\boldsymbol{e}_q\gets \text{此处有多种}\varepsilon \text{组合} \\ =\left( \delta _{pi}\delta _{qj}-\delta _{pj}\delta _{qi} \right) \varepsilon _{lmp}\boldsymbol{A}_i\boldsymbol{B}_j\boldsymbol{C}_l\boldsymbol{D}_m\boldsymbol{e}_q \\ =\varepsilon _{lmi}\boldsymbol{A}_i\boldsymbol{B}_j\boldsymbol{C}_l\boldsymbol{D}_m\boldsymbol{e}_j-\varepsilon _{lmj}\boldsymbol{A}_i\boldsymbol{B}_j\boldsymbol{C}_l\boldsymbol{D}_m\boldsymbol{e}_i \\ =\varepsilon _{lmi}\boldsymbol{C}_l\boldsymbol{D}_m\boldsymbol{A}_i\boldsymbol{B}_j\boldsymbol{e}_j-\varepsilon _{lmj}\boldsymbol{C}_l\boldsymbol{D}_m\boldsymbol{B}_j\boldsymbol{A}_i\boldsymbol{e}_i \\ =\left[ \left( \boldsymbol{C}\times \boldsymbol{D} \right) \cdot \boldsymbol{A} \right] \boldsymbol{B}-\left[ \left( \boldsymbol{C}\times \boldsymbol{D} \right) \cdot \boldsymbol{B} \right] \boldsymbol{A}

涉及两个函数，哈密顿算符会对两个函数起作用

\nabla \left( f\boldsymbol{A} \right) =\partial _i\left( f\boldsymbol{A} \right) _i \\ =\left( \partial _if \right) \boldsymbol{A}_i+f\left( \partial _i\boldsymbol{A}_i \right) \\ =\nabla f\cdot \boldsymbol{A}+f\left( \nabla \cdot \boldsymbol{A} \right)

\nabla \left( \boldsymbol{A}\cdot \boldsymbol{B} \right) =\boldsymbol{B}\times \left( \nabla \times \boldsymbol{A} \right) +\boldsymbol{A}\times \left( \nabla \times \boldsymbol{B} \right) +\left( \boldsymbol{B}\cdot \nabla \right) \boldsymbol{A}+\left( \boldsymbol{A}\cdot \nabla \right) \boldsymbol{B} \\ \\ Left=\nabla \left( \boldsymbol{A}_1\boldsymbol{B}_1 \right) +\nabla \left( \boldsymbol{A}_2\boldsymbol{B}_2 \right) +\nabla \left( \boldsymbol{A}_3\boldsymbol{B}_3 \right) \\ =\boldsymbol{B}_1\nabla \boldsymbol{A}_1+\boldsymbol{B}_2\nabla \boldsymbol{A}_2+\boldsymbol{B}_3\nabla \boldsymbol{A}_3+\boldsymbol{A}_1\nabla \boldsymbol{B}_1+\boldsymbol{A}_2\nabla \boldsymbol{B}_2+\boldsymbol{A}_3\nabla \boldsymbol{B}_3 \\ =\boldsymbol{B}_i\nabla \boldsymbol{A}_i+\boldsymbol{A}_i\nabla \boldsymbol{B}_i

Right1=\varepsilon _{ijk}\boldsymbol{B}_i\left( \varepsilon _{lmn}\partial _l\boldsymbol{A}_m\boldsymbol{e}_n \right) _j\boldsymbol{e}_k \\ =\varepsilon _{ijk}\boldsymbol{B}_i\varepsilon _{lmj}\partial _l\boldsymbol{A}_m\boldsymbol{e}_k \\ =\varepsilon _{kij}\varepsilon _{lmj}\boldsymbol{B}_i\partial _l\boldsymbol{A}_m\boldsymbol{e}_k \\ =\left( \delta _{kl}\delta _{im}-\delta _{km}\delta _{il} \right) \boldsymbol{B}_i\partial _l\boldsymbol{A}_m\boldsymbol{e}_k \\ =\boldsymbol{B}_i\partial _k\boldsymbol{A}_i\boldsymbol{e}_k-\boldsymbol{B}_i\partial _i\boldsymbol{A}_k\boldsymbol{e}_k \\ =\boldsymbol{B}_1\nabla \boldsymbol{A}_1+\boldsymbol{B}_2\nabla \boldsymbol{A}_2+\boldsymbol{B}_3\nabla \boldsymbol{A}_3-\left( \boldsymbol{B}\cdot \nabla \right) \boldsymbol{A} \\ =\boldsymbol{B}_i\nabla \boldsymbol{A}_i-Right3

Right2=\boldsymbol{A}_1\nabla \boldsymbol{B}_1+\boldsymbol{A}_2\nabla \boldsymbol{B}_2+\boldsymbol{A}_3\nabla \boldsymbol{B}_3-\left( \boldsymbol{A}\cdot \nabla \right) \boldsymbol{B} \\ =\boldsymbol{A}_i\nabla \boldsymbol{B}_i-Right4

Left=Right1+Right2+Right3+Right4 \\ \nabla \left( \boldsymbol{A}\cdot \boldsymbol{B} \right) =\boldsymbol{B}\times \left( \nabla \times \boldsymbol{A} \right) +\boldsymbol{A}\times \left( \nabla \times \boldsymbol{B} \right) +\left( \boldsymbol{B}\cdot \nabla \right) \boldsymbol{A}+\left( \boldsymbol{A}\cdot \nabla \right) \boldsymbol{B} \\

\nabla \cdot \left( f\boldsymbol{A} \right) =f\left( \nabla \cdot \boldsymbol{A} \right) +\boldsymbol{A}\cdot \left( \nabla f \right) \\ \\ \nabla \cdot \left( f\boldsymbol{A} \right) =\partial _i\left( f\boldsymbol{A}_{\boldsymbol{i}} \right) \\ =f\left( \partial _i\boldsymbol{A}_{\boldsymbol{i}} \right) +\boldsymbol{A}_{\boldsymbol{i}}\left( \partial _if \right) \\ =f\left( \nabla \cdot \boldsymbol{A} \right) +\boldsymbol{A}\cdot \left( \nabla f \right)

\nabla \cdot \left( \boldsymbol{A}\times \boldsymbol{B} \right) =\boldsymbol{B}\cdot \left( \nabla \times \boldsymbol{A} \right) -\boldsymbol{A}\cdot \left( \nabla \times \boldsymbol{B} \right) \\ Left=\partial _l\left( \varepsilon _{ijk}\boldsymbol{A}_i\boldsymbol{B}_j\boldsymbol{e}_k \right) _l \\ =\partial _l\varepsilon _{ijl}\left( \boldsymbol{A}_i\boldsymbol{B}_j \right) \\ =\varepsilon _{ijl}\left( \partial _l\boldsymbol{A}_i \right) \boldsymbol{B}_j+\varepsilon _{ijl}\boldsymbol{A}_i\left( \partial _l\boldsymbol{B}_j \right) \\ =\varepsilon _{ijl}\left( \partial _l\boldsymbol{A}_i \right) \boldsymbol{B}_j-\varepsilon _{ilj}\boldsymbol{A}_i\left( \partial _l\boldsymbol{B}_j \right) \\ =\varepsilon _{lij}\left( \partial _l\boldsymbol{A}_i \right) \boldsymbol{B}_j-\varepsilon _{lij}\boldsymbol{A}_l\left( \partial _i\boldsymbol{B}_j \right)

Right1=\boldsymbol{B}_l\cdot \left( \varepsilon _{ijk}\partial _i\boldsymbol{A}_j\boldsymbol{e}_k \right) _l \\ =\varepsilon _{ijl}\boldsymbol{B}_l\partial _i\boldsymbol{A}_j \\ =\varepsilon _{lij}\boldsymbol{B}_l\partial _i\boldsymbol{A}_j

Right2=-\boldsymbol{A}\cdot \left( \nabla \times \boldsymbol{B} \right) \\ =-\varepsilon _{ijl}\boldsymbol{A}_l\partial _i\boldsymbol{B}_j \\ =-\varepsilon _{lij}\boldsymbol{A}_l\partial _i\boldsymbol{B}_j

Left=Right1+Right2 \\ \nabla \cdot \left( \boldsymbol{A}\times \boldsymbol{B} \right) =\boldsymbol{B}\cdot \left( \nabla \times \boldsymbol{A} \right) -\boldsymbol{A}\cdot \left( \nabla \times \boldsymbol{B} \right)

\nabla \times \left( f\boldsymbol{A} \right) =f\left( \nabla \times \boldsymbol{A} \right) +\nabla f\times \boldsymbol{A} \\ \\ \nabla \times \left( f\boldsymbol{A} \right) =\varepsilon _{ijk}\partial _i\left( f\boldsymbol{A} \right) _j\boldsymbol{e}_k \\ =\varepsilon _{ijk}\partial _i\left( f\boldsymbol{A}_j \right) \boldsymbol{e}_k \\ =f\varepsilon _{ijk}\partial _i\boldsymbol{A}_j\boldsymbol{e}_k+\varepsilon _{ijk}\left( \partial _if \right) \boldsymbol{A}_j\boldsymbol{e}_k \\ =f\left( \nabla \times \boldsymbol{A} \right) +\nabla f\times \boldsymbol{A}

\nabla \times \left( \boldsymbol{A}\times \boldsymbol{B} \right) =\varepsilon _{ijk}\partial _i\left( \varepsilon _{lmn}\boldsymbol{A}_l\boldsymbol{B}_m\boldsymbol{e}_n \right) _j\boldsymbol{e}_k \\ =\varepsilon _{ijk}\partial _i\varepsilon _{lmj}\left( \boldsymbol{A}_l\boldsymbol{B}_m \right) \boldsymbol{e}_k \\ =\varepsilon _{ijk}\varepsilon _{lmj}\partial _i\left( \boldsymbol{A}_l\boldsymbol{B}_m \right) \boldsymbol{e}_k \\ =\left( \delta _{kl}\delta _{im}-\delta _{km}\delta _{il} \right) \partial _i\left( \boldsymbol{A}_l\boldsymbol{B}_m \right) \boldsymbol{e}_k \\ =\partial _i\left( \boldsymbol{A}_k\boldsymbol{B}_i \right) \boldsymbol{e}_k-\partial _i\left( \boldsymbol{A}_i\boldsymbol{B}_k \right) \boldsymbol{e}_k \\ =\left[ \boldsymbol{B}_i\left( \partial _i\boldsymbol{A}_k \right) \boldsymbol{e}_k+\boldsymbol{A}_k\left( \partial _i\boldsymbol{B}_i \right) \boldsymbol{e}_k \right] -\left[ \left( \partial _i\boldsymbol{A}_i \right) \boldsymbol{B}_k\boldsymbol{e}_k+\boldsymbol{A}_i\left( \partial _i\boldsymbol{B}_k \right) \boldsymbol{e}_k \right] \\ =\left( \boldsymbol{B}\cdot \nabla \right) \boldsymbol{A}+\boldsymbol{A}\left( \nabla \cdot \boldsymbol{B} \right) -\left( \nabla \cdot \boldsymbol{A} \right) \boldsymbol{B}-\left( \boldsymbol{A}\cdot \nabla \right) \boldsymbol{B}

\begin{cases} \boldsymbol{B}\times \left( \nabla \times \boldsymbol{A} \right) =\boldsymbol{B}_i\nabla \boldsymbol{A}_i-\left( \boldsymbol{B}\cdot \nabla \right) \boldsymbol{A}\\ \boldsymbol{A}\times \left( \nabla \times \boldsymbol{B} \right) =\boldsymbol{A}_i\nabla \boldsymbol{B}_i-\left( \boldsymbol{A}\cdot \nabla \right) \boldsymbol{B}\\ \end{cases}\gets \nabla \left( \boldsymbol{A}\cdot \boldsymbol{B} \right) \text{处推导过} \\ \begin{cases} \left( \boldsymbol{B}\times \nabla \right) \times \boldsymbol{A}=\boldsymbol{B}_i\nabla \boldsymbol{A}_i-\boldsymbol{B}\left( \nabla \cdot \boldsymbol{A} \right)\\ \left( \boldsymbol{A}\times \nabla \right) \times \boldsymbol{B}=\boldsymbol{A}_i\nabla \boldsymbol{B}_i-\boldsymbol{A}\left( \nabla \cdot \boldsymbol{B} \right)\\ \end{cases} \\ \nabla \times \left( \boldsymbol{A}\times \boldsymbol{B} \right) =\boldsymbol{A}\times \left( \nabla \times \boldsymbol{B} \right) -\boldsymbol{B}\times \left( \nabla \times \boldsymbol{A} \right) +\left( \boldsymbol{B}\times \nabla \right) \times \boldsymbol{A}-\left( \boldsymbol{A}\times \nabla \right) \times \boldsymbol{B}

\boldsymbol{r}=x\boldsymbol{e}_{\boldsymbol{i}}+y\boldsymbol{e}_{\boldsymbol{j}}+z\boldsymbol{e}_{\boldsymbol{k}},\boldsymbol{r}\prime=x\prime\boldsymbol{e}_{\boldsymbol{i}}+y\prime\boldsymbol{e}_{\boldsymbol{j}}+z\prime\boldsymbol{e}_{\boldsymbol{k}} \\

\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|=\sqrt{\left( x-x\prime \right) ^2+\left( y-y\prime \right) ^2+\left( z-z\prime \right) ^2} \\

\nabla \left( \frac{1}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|} \right) =\frac{\partial \frac{1}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|}}{\,\, \partial x}\boldsymbol{e}_{\boldsymbol{i}}+\frac{\partial \frac{1}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|}}{\partial y}\boldsymbol{e}_{\boldsymbol{j}}+\frac{\partial \frac{1}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|}}{\partial z}\boldsymbol{e}_{\boldsymbol{k}} \\ =\frac{\partial \frac{1}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|}}{\partial r}\frac{\partial r}{\partial x}\boldsymbol{e}_{\boldsymbol{i}}+\frac{\partial \frac{1}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|}}{\partial r}\frac{\partial r}{\partial y}\boldsymbol{e}_{\boldsymbol{j}}+\frac{\partial \frac{1}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|}}{\partial r}\frac{\partial r}{\partial z}\boldsymbol{e}_{\boldsymbol{k}} \\ =-\frac{1}{\left( \boldsymbol{r}-\boldsymbol{r}\prime \right) ^2}\cdot \frac{\frac{1}{2}\times 2\left( x-x\prime \right)}{\sqrt{\left( x-x\prime \right) ^2+\left( y-y\prime \right) ^2+\left( z-z\prime \right) ^2}}\boldsymbol{e}_{\boldsymbol{i}} \\ +-\frac{1}{\left( \boldsymbol{r}-\boldsymbol{r}\prime \right) ^2}\cdot \frac{\frac{1}{2}\times 2\left( y-y\prime \right)}{\sqrt{\left( x-x\prime \right) ^2+\left( y-y\prime \right) ^2+\left( z-z\prime \right) ^2}}\boldsymbol{e}_{\boldsymbol{j}} \\ +-\frac{1}{\left( \boldsymbol{r}-\boldsymbol{r}\prime \right) ^2}\cdot \frac{\frac{1}{2}\times 2\left( z-z\prime \right)}{\sqrt{\left( x-x\prime \right) ^2+\left( y-y\prime \right) ^2+\left( z-z\prime \right) ^2}}\boldsymbol{e}_{\boldsymbol{k}} \\ =-\frac{1}{\left( \boldsymbol{r}-\boldsymbol{r}\prime \right) ^2}\frac{\left( x-x\prime \right) \boldsymbol{e}_{\boldsymbol{i}}+\left( y-y\prime \right) \boldsymbol{e}_{\boldsymbol{j}}+\left( z-z\prime \right) \boldsymbol{e}_{\boldsymbol{k}}}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|} \\ =-\frac{\boldsymbol{r}-\boldsymbol{r}\prime}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^3}

\nabla ^2\left( \frac{1}{\boldsymbol{r}-\boldsymbol{r}\prime} \right) =\nabla \cdot \left[ \nabla \left( \frac{1}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|} \right) \right] \\ =-\nabla \cdot \left( \frac{\boldsymbol{r}-\boldsymbol{r}\prime}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^3} \right) \\ =-\left[ \frac{\partial \left( \frac{x-x\prime}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^3} \right)}{\partial x}+\frac{\partial \left( \frac{y-y\prime}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^3} \right)}{\partial y}+\frac{\partial \left( \frac{z-z\prime}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^3} \right)}{\partial z} \right] \downarrow \text{使用商的导数和链式求导法则，} \\ \frac{\partial \left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^3}{\partial x}=\frac{\partial \left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^3}{\partial r}\frac{\partial r}{\partial x}

\left[ \begin{array}{c} \frac{1\cdot \left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^3-3\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^2\frac{\left( x-x\prime \right)}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|}\cdot \left( x-x\prime \right)}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^6}\\ +\frac{1\cdot \left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^3-3\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^2\frac{\left( y-y\prime \right)}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|}\cdot \left( y-y\prime \right)}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^6}\\ +\frac{1\cdot \left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^3-3\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^2\frac{\left( z-z\prime \right)}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|}\cdot \left( z-z\prime \right)}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^6}\\ \end{array} \right] \\ =-\frac{3\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^3-3\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|\left[ \left( x-x\prime \right) ^2+\left( y-y\prime \right) ^2+\left( z-z\prime \right) ^2 \right]}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^6} \\ =-\frac{3\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^3-3\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^3}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^6} \\

\text{当}\boldsymbol{r}\ne \boldsymbol{r}\prime,-\frac{3\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^3-3\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^3}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^6}=0\Longrightarrow \nabla ^2\left( \frac{1}{\boldsymbol{r}-\boldsymbol{r}\prime} \right) =0

\text{当}\boldsymbol{r}=\boldsymbol{r}\prime\Longrightarrow \frac{0}{0}\text{有奇点，结果无解}?\text{实际上，我们可以使用高斯定律}: \\ \oint_{\boldsymbol{S}}{\boldsymbol{A}\cdot d\boldsymbol{S}}=\int_V{\left( \nabla \cdot \boldsymbol{A} \right) dV},\boldsymbol{S}=S\cdot \boldsymbol{n} \\ \boldsymbol{A}=\nabla \left( \frac{1}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|} \right)

\oint_{\boldsymbol{S}}{\left[ \nabla \left( \frac{1}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|} \right) \right] \cdot d\boldsymbol{S}}=\int_V{\left[ \nabla \cdot \nabla \left( \frac{1}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|} \right) \right] dV}\gets \text{积分内就是我们所求} \\ \text{当}\boldsymbol{r}\rightarrow \boldsymbol{r}\prime\text{时，}S\text{为一闭合曲面，}V\text{是}S\text{内的体积} \\ \oint_{\boldsymbol{S}}{\left[ \nabla \left( \frac{1}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|} \right) \right] \cdot d\boldsymbol{S}}=-\oint_{\boldsymbol{S}}{\left( \frac{1}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^2}\cdot \boldsymbol{e}_{\boldsymbol{r}} \right) \cdot d\boldsymbol{S}}\gets \boldsymbol{e}_{\boldsymbol{r}}=\frac{\boldsymbol{r}-\boldsymbol{r}\prime}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|} \\ =-\frac{1}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^2}\oint_S{\boldsymbol{e}_{\boldsymbol{r}}\cdot d\boldsymbol{S}}\gets \frac{\text{法向矢量做无限小闭合曲面的通量积分}}{\text{可取闭合曲面为球面，把}r\prime\text{位置选为原点}} \\ \left( \oint_S{\boldsymbol{e}_{\boldsymbol{r}}\cdot d\boldsymbol{S}}=\int_0^{4\pi R^2}{\boldsymbol{e}_{\boldsymbol{r}}\cdot \boldsymbol{n}dS}=4\pi R^2 \right) \\ =-\frac{1}{\left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^2}4\pi \left| \boldsymbol{r}-\boldsymbol{r}\prime \right|^2 \\ =-4\pi

\nabla ^2\left( \frac{1}{\boldsymbol{r}-\boldsymbol{r}\prime} \right) =-4\pi \delta \left( \boldsymbol{r}-\boldsymbol{r}\prime \right) \gets \text{利用狄拉克}\delta \text{函数}

虽然是二阶，但比“乘积规则”部分简单，只涉及一个函数。

\nabla \cdot \left( \nabla \times \boldsymbol{A} \right) =0 \\ =\partial _l\left( \varepsilon _{ijk}\partial _i\boldsymbol{A}_j\boldsymbol{e}_k \right) \\ =\varepsilon _{ijl}\partial _l\partial _i\boldsymbol{A}_j \\ =\partial _1\partial _2\boldsymbol{A}_3-\partial _2\partial _1\boldsymbol{A}_3 \\ +\partial _2\partial _3\boldsymbol{A}_1-\partial _3\partial _2\boldsymbol{A}_1 \\ +\partial _3\partial _1\boldsymbol{A}_2-\partial _1\partial _3\boldsymbol{A}_2 \\ =0

\nabla \times \left( \nabla f \right) =0 \\ =\varepsilon _{ijk}\partial _i\partial _jf\boldsymbol{e}_k \\ =\partial _1\partial _2f\boldsymbol{e}_3-\partial _2\partial _1f\boldsymbol{e}_3 \\ +\partial _2\partial _3f\boldsymbol{e}_1-\partial _3\partial _2f\boldsymbol{e}_1 \\ +\partial _3\partial _1f\boldsymbol{e}_2-\partial _1\partial _3f\boldsymbol{e}_2 \\ =0

\nabla ^2\left( \nabla \cdot \boldsymbol{A} \right) =\nabla \cdot \left( \nabla ^2\boldsymbol{A} \right) \\ =\nabla ^2\partial _i\boldsymbol{A}_i \\ =\partial _i\nabla ^2\boldsymbol{A}_i \\ =\nabla \cdot \left( \nabla ^2\boldsymbol{A} \right)

\nabla \times \left( \nabla \times \boldsymbol{A} \right) =\varepsilon _{ijk}\partial _i\left( \varepsilon _{lmn}\partial _l\boldsymbol{A}_m\boldsymbol{e}_n \right) _j\boldsymbol{e}_k \\ =\varepsilon _{ijk}\varepsilon _{lmj}\partial _i\partial _l\boldsymbol{A}_m\boldsymbol{e}_k \\ =\left( \delta _{kl}\delta _{im}-\delta _{km}\delta _{il} \right) \partial _i\partial _l\boldsymbol{A}_m\boldsymbol{e}_k \\ =\partial _k\partial _i\boldsymbol{A}_i\boldsymbol{e}_k-\partial _i\partial _i\boldsymbol{A}_k\boldsymbol{e}_k \\ =\nabla \left( \nabla \cdot \boldsymbol{A} \right) -\nabla ^2\boldsymbol{A}

以上是一些常用矢量恒等式的证明，对于这些式子，不要去背诵，尝试利用爱因斯坦求和约定，Kronecker符号和Levi-Civita符号去把式子拆解化简再合并。