科学网—爱因斯坦求和约定

一、求和约定和哑指标

$S=a_1x_1+a_2x_2+\cdots +a_nx_n$

$S=\sum _{i=1} ^n a_ix_i=\sum _{j=1} ^n a_jx_j$

约定

$S= a_ix_i$

$\sum_{i=1}^3 \sum_{j=1}^3 A_{ij}x_iy_j$

$A_{ij}x_iy_j= \\A_{11}x_1y_1+A_{12}x_1y_2+A_{13}x_1y_3+ \\A_{21}x_2y_1+A_{22}x_2y_2+A_{23}x_2y_3+ \\A_{31}x_3y_1+A_{32}x_3y_2+A_{33}x_3y_3$

$A_{ijk}x_iy_jz_k$ 表示27项求和

二、自由指标

$A_{11}x_1+A_{12}x_2+A_{13}x_3=b_1\\ A_{21}x_1+A_{22}x_2+A_{23}x_3=b_2\\ A_{31}x_1+A_{32}x_2+A_{33}x_3=b_3\\$

简记为

$b_i=A_{ij}x_j\\$

其中 i 为自由指标，j 为哑指标。

三、Kronecker-delta符号和置换符号(Ricci符号)

$\delta_{ij}=\left\{\begin{matrix}0, i\ne{j} \\ 1, i=j \end{matrix}\right.\\$

$\delta_{ij}a_j=\delta_{i1}a_1+\delta_{i2}a_2+\delta_{i3}a_3=a_i\\$

$\delta_{im}A_{mj}=A_{ij}\\$

Ricci符号

$\epsilon_{ijk}=\left\{\begin{matrix}1, (ijk)=(1,2,3)/(2,3,1)/(3,1,2) 顺序 \\ -1, (ijk)=(1,3,2)/(2,1,3)/(3,2,1) 反序 \\0, 其它 \end{matrix}\right.\\$

$l=k \\ \epsilon_{ijk}\epsilon_{lmn}=\begin{vmatrix} \delta_{im} \delta_{in} \\ \delta_{jm} \delta_{jn} \end{vmatrix}=\delta_{im}\delta_{jn}- \delta_{in}\delta_{jm} \\$

$e_{i j k}=\left| \begin{array}{ccc}{\delta_{i 1}} & {\delta_{i 2}} & {\delta_{i 3}} \\ {\delta_{j 1}} & {\delta_{j 2}} & {\delta_{j 3}} \\ {\delta_{k 1}} & {\delta_{k 2}} & {\delta_{k 3}}\end{array}\right|=\left| \begin{array}{ccc}{\delta_{i 1}} & {\delta_{j 1}} & {\delta_{k 1}} \\ {\delta_{i 2}} & {\delta_{j 2}} & {\delta_{k 2}} \\ {\delta_{i 3}} & {\delta_{j 3}} & {\delta_{k 3}}\end{array}\right| \\$

$e_{321}=\left| \begin{array}{ccc}{\delta_{31}} & {\delta_{32}} & {\delta_{33}} \\ {\delta_{21}} & {\delta_{22}} & {\delta_{23}} \\ {\delta_{11}} & {\delta_{12}} & {\delta_{13}}\end{array}\right|=\left| \begin{array}{ccc}{0} & {0} & {1} \\ {0} & {1} & {0} \\ {1} & {0} & {0}\end{array}\right|=-1\\$

$e_{i j k} e_{p q r}=\left| \begin{array}{ccc}{\delta_{i 1}} & {\delta_{i 2}} & {\delta_{i 3}} \\ {\delta_{j 1}} & {\delta_{j 2}} & {\delta_{j 3}} \\ {\delta_{k 1}} & {\delta_{k 2}} & {\delta_{k 3}}\end{array}\right| \cdot \left| \begin{array}{ccc}{\delta_{p 1}} & {\delta_{q 1}} & {\delta_{r 1}} \\ {\delta_{p 2}} & {\delta_{q 2}} & {\delta_{r 2}} \\ {\delta_{p 3}} & {\delta_{q 3}} & {\delta_{r 3}}\end{array}\right| \\$

$\begin{array}{l}{p=k} \\ {e_{i j k} e_{k q r}=\left| \begin{array}{ll}{\delta_{i q}} & {\delta_{i r}} \\ {\delta_{j q}} & {\delta_{j r}}\end{array}\right|=\delta_{i q} \delta_{j r}-\delta_{i r} \delta_{j q}}\end{array}\\$

$\begin{array}{l}{|A|=\left| \begin{array}{ccc}{a_{11}} & {a_{12}} & {a_{13}} \\ {a_{21}} & {a_{22}} & {a_{23}} \\ {a_{31}} & {a_{32}} & {a_{33}}\end{array}\right|=a_{11} a_{22} a_{33}+a_{12} a_{23} a_{31}+} \\ {a_{13} a_{21} a_{32}-a_{13} a_{22} a_{31}-a_{12} a_{21} a_{33}-a_{11} a_{23} a_{32}} \\ {=e_{i j k} a_{1 i} a_{2 j} a_{3 k}=e_{i j k} a_{i 1} a_{j 2} a_{k 3}}\end{array} \\$

矢量的基本运算

$a=a_{1} e_{1}+a_{2} e_{2}+a_{3} e_{3}=a_{i} e_{i} \\$

一、矢量点积

$e_{i} \cdot e_{j}=\delta_{i j}$

$\begin{aligned} a \cdot b &=a_{i} e_{i} \cdot b_{j} e_{j}=a_{i} b_{j} \delta_{i j} \\ a_{i} b_{i} &=a_{j} b_{j} \end{aligned}$

二、矢量叉积 $e_{i} \times e_{j}=e_{i j k} e_{k}$

$\begin{aligned} e_{i} &=\delta_{i k} e_{k} \\ e_{j} &=\delta_{j k} e_{k} \end{aligned}\\$ $\begin{array}{c}{e_{i} \times e_{j}=\left| \begin{array}{ccc}{\delta_{i 1}} & {\delta_{i 2}} & {\delta_{i 3}} \\ {\delta_{j 1}} & {\delta_{j 2}} & {\delta_{j 3}} \\ {e_{1}} & {e_{2}} & {e_{3}}\end{array}\right|} \\ {=e_{r s t} \delta_{i r} \delta_{j s} e_{t}} & {=e_{i j t} e_{t}=e_{i j k} e_{k}}\end{array}$

$\begin{aligned} a \times b \cdot c=& e_{i j k} a_{i} b_{j} e_{k} \cdot c_{r} e_{r} \\ &=e_{i j k} a_{i} b_{j} c_{r} \delta_{k r} \\ &=e_{i j k} a_{i} b_{j} c_{k} \end{aligned}$

$\begin{aligned} e_{i} \times e_{j} \cdot e_{k} &=e_{i j r} e_{r} \cdot e_{k} \\ &=e_{i j r} \delta_{r k}=e_{i j k} \end{aligned}\\$

四、矢量的并乘(并矢)

howking的个人博客分享 http://blog.sciencenet.cn/u/howking

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爱因斯坦求和约定

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