# 爱因斯坦求和约定

${\displaystyle y=c_{i}x^{i}\,\!}$

${\displaystyle y=\sum _{i=1}^{3}c_{i}x^{i}=c_{1}x^{1}+c_{2}x^{2}+c_{3}x^{3}\,\!}$

## 簡介

${\displaystyle y=c_{1}x^{1}+c_{2}x^{2}+c_{3}x^{3}+\cdots +c_{n}x^{n}\,\!}$

${\displaystyle y=\sum _{i=1}^{n}c_{i}x^{i}\,\!}$

${\displaystyle y=c_{i}x^{i}\,\,\!}$

## 向量的表示

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

${\displaystyle {\boldsymbol {\alpha }}=\alpha _{i}{\boldsymbol {\omega }}^{i}={\begin{bmatrix}\alpha _{1}&\alpha _{2}&\cdots &\alpha _{n}\end{bmatrix}}\,\!}$

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

${\displaystyle {\overline {\alpha }}_{i}={\frac {\partial x^{i}}{\partial {\overline {x}}^{j}}}\alpha _{j}\,\!}$

## 一般運算

### 內積

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

### 向量乘以矩陣

${\displaystyle b^{i}=A_{j}^{i}a^{j}\,\!}$

${\displaystyle \beta _{j}=B_{j}^{i}\alpha _{i}=\alpha _{i}B_{j}^{i}\,\!}$

### 矩陣乘法

${\displaystyle C_{k}^{i}=A_{j}^{i}\,B_{k}^{j}\,\!}$

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

### 跡

${\displaystyle t=A_{i}^{i}\,\!}$

### 外積

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

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

${\displaystyle A_{j}^{i}=a^{i}\,\alpha _{j}\,\!}$

## 向量的內積

${\displaystyle \mathbf {u} =u_{x}{\hat {\mathbf {i} }}+u_{y}{\hat {\mathbf {j} }}+u_{z}{\hat {\mathbf {k} }}\,\!}$

${\displaystyle \mathbf {u} =u_{1}{\hat {\mathbf {e} }}_{1}+u_{2}{\hat {\mathbf {e} }}_{2}+u_{3}{\hat {\mathbf {e} }}_{3}=\sum _{i=1}^{3}u_{i}{\hat {\mathbf {e} }}_{i}\,\!}$

${\displaystyle \mathbf {u} =u^{i}{\hat {\mathbf {e} }}_{i}=\sum _{i=1}^{3}u^{i}{\hat {\mathbf {e} }}_{i}\,\!}$

${\displaystyle \mathbf {u} =\sum _{i=1}^{3}u_{i}{\hat {\mathbf {e} }}_{i}\,\!}$

${\displaystyle \mathbf {u} \cdot \mathbf {v} =(u^{i}{\hat {\mathbf {e} }}_{i})\cdot (v^{j}{\hat {\mathbf {e} }}_{j})=\left(\sum _{i=1}^{3}u_{i}{\hat {\mathbf {e} }}_{i}\right)\cdot \left(\sum _{j=1}^{3}v_{j}\mathbf {e} _{j}\right)=\sum _{i=1}^{3}\sum _{j=1}^{3}u_{i}v_{j}({\hat {\mathbf {e} }}_{i}\cdot {\hat {\mathbf {e} }}_{j})\,\!}$

${\displaystyle {\hat {\mathbf {e} }}_{i}\cdot {\hat {\mathbf {e} }}_{j}=\delta _{ij}\,\!}$

${\displaystyle \mathbf {u} \cdot \mathbf {v} =\sum _{i=1}^{3}\sum _{j=1}^{3}u_{i}v_{j}\delta _{ij}=\sum _{i=1}^{3}u_{i}v_{i}\,\!}$

## 向量的叉積

${\displaystyle \mathbf {u} \times \mathbf {v} =(u^{j}{\hat {\mathbf {e} }}_{j})\times (v^{k}{\hat {\mathbf {e} }}_{k})=\left(\sum _{j=1}^{3}u_{j}{\hat {\mathbf {e} }}_{j}\right)\times \left(\sum _{k=1}^{3}v_{k}{\hat {\mathbf {e} }}_{k}\right)\,\!}$
${\displaystyle =\sum _{j=1}^{3}\sum _{k=1}^{3}u_{j}v_{k}(\mathbf {e} _{j}\times \mathbf {e} _{k})=\sum _{j=1}^{3}\sum _{k=1}^{3}u_{j}v_{k}\epsilon _{ijk}\mathbf {e} _{i}\,\!}$

${\displaystyle {\hat {\mathbf {e} }}_{j}\times {\hat {\mathbf {e} }}_{k}=\epsilon _{ijk}{\hat {\mathbf {e} }}_{i}\,\!}$

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

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

${\displaystyle w^{i}{\hat {\mathbf {e} }}_{i}=\epsilon ^{ijk}u_{j}v_{k}{\hat {\mathbf {e} }}_{i}\,\!}$

${\displaystyle \ w^{i}=\epsilon ^{ijk}u_{j}v_{k}\,\!}$

## 向量的共變分量和反變分量

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

${\displaystyle Y^{i}\cdot X_{j}=\delta _{j}^{i}\,\!}$

{\displaystyle {\begin{aligned}\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{aligned}}\,\!}

### 歐幾里得空間

${\displaystyle \mathbf {e} ^{1}={\frac {\mathbf {e} _{2}\times \mathbf {e} _{3}}{\tau }};\qquad \mathbf {e} ^{2}={\frac {\mathbf {e} _{3}\times \mathbf {e} _{1}}{\tau }};\qquad \mathbf {e} ^{3}={\frac {\mathbf {e} _{1}\times \mathbf {e} _{2}}{\tau }}\,\!}$

${\displaystyle \mathbf {e} _{1}={\frac {\mathbf {e} ^{2}\times \mathbf {e} ^{3}}{\tau '}};\qquad \mathbf {e} _{2}={\frac {\mathbf {e} ^{3}\times \mathbf {e} ^{1}}{\tau '}};\qquad \mathbf {e} _{3}={\frac {\mathbf {e} ^{1}\times \mathbf {e} ^{2}}{\tau '}}\,\!}$

${\displaystyle \mathbf {e} _{i}\cdot \mathbf {e} ^{j}=\delta _{i}^{j}\,\!}$

${\displaystyle \mathbf {e} ^{i}\cdot \mathbf {e} _{j}=\delta _{j}^{i}\,\!}$

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

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

${\displaystyle \mathbf {a} =a_{i}\mathbf {e} ^{i}=a_{1}\mathbf {e} ^{1}+a_{2}\mathbf {e} ^{2}+a_{3}\mathbf {e} ^{3}\,\!}$

${\displaystyle \mathbf {a} =a^{i}\mathbf {e} _{i}=a^{1}\mathbf {e} _{1}+a^{2}\mathbf {e} _{2}+a^{3}\mathbf {e} _{3}\,\!}$

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

${\displaystyle a_{i}=\mathbf {a} \cdot \mathbf {e} _{i}=(a^{j}\mathbf {e} _{j})\cdot \mathbf {e} _{i}=(\mathbf {e} _{j}\cdot \mathbf {e} _{i})a^{j}=g_{ji}a^{j}\,\!}$

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

## 抽象定義

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

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

${\displaystyle \mathbf {e} ^{i}\cdot \mathbf {e} _{j}=\delta _{j}^{i}\,\!}$

## 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：
${\displaystyle c=a^{\mu }b_{\mu }=a^{0}b_{0}+a^{1}b_{1}+a^{2}b_{2}+a^{3}b_{3}\,\!}$
• 方程式的右手邊有兩個項目：
${\displaystyle c^{\nu }=a^{\mu \nu }b_{\mu }+f^{\nu }=a^{0\nu }b_{0}+a^{1\nu }b_{1}+a^{2\nu }b_{2}+a^{3\nu }b_{3}+f^{\nu }\,\!}$

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

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

## 參考文獻

1. ^ Einstein, Albert, The Foundation of the General Theory of Relativity, Annalen der Physik, 1916 [2006-09-03], （原始内容 (PDF)存档于2007-07-22）
2. ^ Byron, Frederick; Fuller, Robert, Mathematics of classical and quantum physics, Courier Dover Publications: pp. 5, 1992, ISBN 9780486671642