# 爱因斯坦求和约定

$y = c_i x^i\,\!$

$y = \sum_{i=1}^3 c_i x^i = c_1 x^1 + c_2 x^2 + c_3 x^3\,\!$

## 簡介

$y = c_1 x^1+c_2x^2+c_3x^3+ \cdots + c_nx^n\,\!$

$y = \sum_{i=1}^n c_ix^i\,\!$

$y = c_i x^i \,\,\!$

## 向量的表示

$\mathbf{a}= a^i \mathbf{e}_i= \begin{bmatrix}a^1\\a^2\\\vdots\\a^n\end{bmatrix}\,\!$

$\boldsymbol{\alpha}= \alpha_i \boldsymbol{\omega}^i= \begin{bmatrix}\alpha_1 & \alpha_2 & \cdots & \alpha_n\end{bmatrix}\,\!$

${\overline{a}}^{i}= \frac{\partial {\overline{x}}^{i}}{\partial x^j} a^j\,\!$

$\overline{\alpha}_i= \frac{\partial x^i}{\partial {\overline{x}}^{j}} \alpha_j\,\!$

## 一般運算

### 內積

$\mathbf{a}\cdot\boldsymbol{\alpha}=a^i \alpha_i\,\!$

### 向量乘以矩陣

$b^i=A^i_j a^j\,\!$

$\beta_j=B^i_j \alpha_i=\alpha_i B^i_j\,\!$

### 矩陣乘法

$C^i_k = A^i_j \, B^j_k \,\!$

$C_{ik} = (A \, B)_{ik} = \sum_{j=1}^N A_{ij} B_{jk}\,\!$

### 跡

$t=A^i_i\,\!$

### 外積

M維向量$\mathbf{a}\,\!$和N維餘向量$\boldsymbol{\alpha}\,\!$外積是一個M×N矩陣$A\,\!$

$A= \mathbf{a} \, \boldsymbol{\alpha} \,\!$

$A^i_j = a^i \, \alpha_j\,\!$

## 向量的內積

$\mathbf{u} = u_x\hat{\mathbf{i}} + u_y\hat{\mathbf{j}} + u_z\hat{\mathbf{k}}\,\!$

$\mathbf{u} = u_1 \hat{\mathbf{e}}_1 + u_2 \hat{\mathbf{e}}_2 + u_3 \hat{\mathbf{e}}_3 = \sum_{i = 1}^3 u_i \hat{\mathbf{e}}_i\,\!$

$\mathbf{u} =u^i \hat{\mathbf{e}}_i = \sum_{i = 1}^3 u^i \hat{\mathbf{e}}_i \,\!$

$\mathbf{u}= \sum_{i = 1}^3 u_i \hat{\mathbf{e}}_i \,\!$

$\mathbf{u} \cdot \mathbf{v} = (u^i\hat{\mathbf{e}}_i) \cdot (v^j \hat{\mathbf{e}}_j) = \left( \sum_{i = 1}^3 u_i\hat{\mathbf{e}}_i \right) \cdot \left( \sum_{j = 1}^3 v_j \mathbf{e}_j \right)=\sum_{i = 1}^3 \sum_{j = 1}^3 u_i v_j ( \hat{\mathbf{e}}_i \cdot \hat{\mathbf{e}}_j ) \,\!$

$\hat{\mathbf{e}}_i \cdot \hat{\mathbf{e}}_j = \delta_{ij} \,\!$

$\mathbf{u} \cdot \mathbf{v} =\sum_{i = 1}^3 \sum_{j = 1}^3 u_i v_j\delta_{ij}= \sum_{i = 1}^3 u_i v_i \,\!$

## 向量的叉積

$\mathbf{u} \times \mathbf{v}= (u^j \hat{\mathbf{e}}_j ) \times (v^k \hat{\mathbf{e}}_k)= \left( \sum_{j = 1}^3 u_j \hat{\mathbf{e}}_j \right) \times \left( \sum_{k = 1}^3 v_k \hat{\mathbf{e}}_k \right) \,\!$
$=\sum_{j = 1}^3 \sum_{k = 1}^3 u_j v_k (\mathbf{e}_j \times \mathbf{e}_k ) = \sum_{j = 1}^3 \sum_{k = 1}^3 u_j v_k\epsilon_{ijk} \mathbf{e}_i \,\!$

$\hat{\mathbf{e}}_j \times \hat{\mathbf{e}}_k = \epsilon_{ijk} \hat{\mathbf{e}}_i\,\!$

 $\epsilon_{ijk} = \epsilon^{ijk}\ \stackrel{def}{=} \begin{cases} +1 \\ -1 \\ 0 \end{cases} \,\!$ ，若$(i,j,k)=\,\!$ $\{1,2,3\}\,\!$、$\{2,3,1\}\,\!$或$\{3,1,2\}\,\!$ （偶置換） ，若$(i,j,k)=\,\!$ $\{3,2,1\}\,\!$、$\{2,1,3\}\,\!$或$\{1,3,2\}\,\!$（奇置換） ，若 $i=j\,\!$、$j=k\,\!$或$i=k\,\!$

$\mathbf{u} \times \mathbf{v} = (u^2 v^3 - u^3 v^2) \hat{\mathbf{e}}_1 + (u^3 v^1 - u^1 v^3) \hat{\mathbf{e}}_2 + (u^1 v^2 - u^2 v^1) \hat{\mathbf{e}}_3\,\!$

$w^i \hat{\mathbf{e}}_i= \epsilon^{ijk} u_j v_k\hat{\mathbf{e}}_i \,\!$

$\ w^i = \epsilon^{ijk} u_j v_k \,\!$

## 向量的共變分量和反變分量

$\boldsymbol{\alpha}(\mathbf{b})=\mathbf{a}\cdot\mathbf{b}\,\!$

$Y^i \cdot X_j = \delta^i_j\,\!$

\begin{align} \mathbf{a} &= \sum_i a^i[\mathfrak{f}]X_i = \mathfrak{f}\,\mathbf{a}[\mathfrak{f}]\\ &=\sum_i a_i[\mathfrak{f}]Y^i = \mathfrak{f}^\sharp\,\mathbf{a}[\mathfrak{f}^\sharp] \end{align} \,\!

### 歐幾里得空間

$\mathbf{e}^1 = \frac{\mathbf{e}_2 \times \mathbf{e}_3}{\tau} ; \qquad \mathbf{e}^2 = \frac{\mathbf{e}_3 \times \mathbf{e}_1}{\tau}; \qquad \mathbf{e}^3 = \frac{\mathbf{e}_1 \times \mathbf{e}_2}{\tau}\,\!$

$\mathbf{e}_1 = \frac{\mathbf{e}^2 \times \mathbf{e}^3}{\tau'} ; \qquad \mathbf{e}_2 = \frac{\mathbf{e}^3 \times \mathbf{e}^1}{\tau'}; \qquad \mathbf{e}_3 = \frac{\mathbf{e}^1 \times \mathbf{e}^2}{\tau'}\,\!$

$\mathbf{e}_i \cdot \mathbf{e}^j = \delta_i^j\,\!$

$\mathbf{e}^i \cdot \mathbf{e}_j = \delta^i_j\,\!$

$a^1 = \mathbf{a} \cdot \mathbf{e}^1; \qquad a^2 = \mathbf{a} \cdot \mathbf{e}^2; \qquad a^3 = \mathbf{a} \cdot \mathbf{e}^3\,\!$

$a_1 = \mathbf{a} \cdot \mathbf{e}_1; \qquad a_2 = \mathbf{a} \cdot \mathbf{e}_2; \qquad a_3 = \mathbf{a} \cdot \mathbf{e}_3\,\!$

$\mathbf{a} = a_i \mathbf{e}^i = a_1 \mathbf{e}^1 + a_2 \mathbf{e}^2 + a_3 \mathbf{e}^3 \,\!$

$\mathbf{a} = a^i \mathbf{e}_i = a^1 \mathbf{e}_1 + a^2 \mathbf{e}_2 + a^3 \mathbf{e}_3\,\!$

$\mathbf{a} = (\mathbf{a} \cdot \mathbf{e}_i) \mathbf{e}^i = (\mathbf{a} \cdot \mathbf{e}^i) \mathbf{e}_i \,\!$

$a_i = \mathbf{a}\cdot \mathbf{e}_i = (a^j \mathbf{e}_j)\cdot \mathbf{e}_i = (\mathbf{e}_j\cdot\mathbf{e}_i) a^j=g_{ji}a^j\,\!$

$a^i = \mathbf{a}\cdot \mathbf{e}^i = (a_j \mathbf{e}^j)\cdot \mathbf{e}^i = (\mathbf{e}^j\cdot\mathbf{e}^i) a_j =g^{ji}a_j\,\!$ ;

## 抽象定義

$\mathbf{v} = v^i\mathbf{e}_i.\,\!$

$\mathbf{T} = T^{ij}\mathbf{e}_{ij}\,\!$

$\mathbf{e}^i \cdot\mathbf{e}_j = \delta^i_j\,\!$

## 範例

• 思考四維時空，標號的值是從0到3。兩個張量，經過張量縮併tensor contraction）運算後，變為一個純量：
$c=a^\mu b_\mu = a^0 b_0 + a^1 b_1 + a^2 b_2 + a^3 b_3\,\!$
• 方程式的右手邊有兩個項目：
$c^\nu=a^{\mu\nu} b_\mu +f^\nu= a^{0\nu} b_0 + a^{1\nu} b_1 + a^{2\nu} b_2 + a^{3\nu} b_3+f^\nu\,\!$

• 思考在黎曼空間的弧線元素長度$ds\,\!$
$ds^2=g_{ij}dx^i dx^j=g_{0j}dx^0 dx^j+g_{1j}dx^1 dx^j+g_{2j}dx^2 dx^j+g_{3j}dx^3 dx^j\,\!$。請將這兩種標號跟自由變量和約束變量相比較。

$ds^2=g_{00}dx^0 dx^0+g_{10}dx^1 dx^0+g_{20}dx^2 dx^0+g_{30}dx^3 dx^0\,\!$
$\qquad +g_{01}dx^0 dx^1+g_{11}dx^1 dx^1+g_{21}dx^2 dx^1+g_{31}dx^3 dx^1\,\!$
$\qquad +g_{02}dx^0 dx^2+g_{12}dx^1 dx^2+g_{22}dx^2 dx^2+g_{32}dx^3 dx^2\,\!$
$\qquad +g_{03}dx^0 dx^3+g_{13}dx^1 dx^3+g_{23}dx^2 dx^3+g_{33}dx^3 dx^3\,\!$

## 參考文獻

1. ^ Einstein, Albert, The Foundation of the General Theory of Relativity (PDF), Annalen der Physik, 1916 [2006-09-03]
2. ^ Byron, Frederick; Fuller, Robert, Mathematics of classical and quantum physics, Courier Dover Publications, pp. 5, 1992, ISBN 9780486671642