# 共變和反變

## 轉換方式

### 向量：反變轉換

• 標記法說明：向量 ${\displaystyle \mathbf {v} \,\!}$向量空間 ${\displaystyle V\,\!}$ 的元素。向量基底 ${\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},...,\mathbf {e} _{n}\,\!}$ 構成了向量空間的一個基底，其座標系統為${\displaystyle x^{1},x^{2},...,x^{n}\,\!}$。對應這個基底，向量${\displaystyle \mathbf {v} \,\!}$的分量為${\displaystyle v^{1},v^{2},...,v^{n}\,\!}$，即${\displaystyle \textstyle \mathbf {v} =\sum _{i}v^{i}\mathbf {e} _{i}}$

（註：${\displaystyle v^{2}\,\!}$ 這符號中的上標${\displaystyle 2}$不代表平方，而是代表第二個坐標，在較基礎的數學上，常寫作 ${\displaystyle v_{2}\,\!}$ ，但是，在张量分析领域，指标写作上标或下标牵涉到对张量性质的提示，以及愛因斯坦求和約定。）

${\displaystyle {\bar {v}}^{\mu }={\frac {\partial {\bar {x}}^{\mu }}{\partial x^{1}}}v^{1}+{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{2}}}v^{2}+...+{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{n}}}v^{n}\,\!}$

${\displaystyle {\bar {v}}^{\mu }={\frac {\partial {\bar {x}}^{\mu }}{\partial x^{i}}}v^{i}\,\!}$

### 餘向量：共變轉換

${\displaystyle {\bar {\mathbf {\eta } }}_{\mu }={\frac {\partial x^{1}}{\partial {\bar {x}}^{\mu }}}\mathbf {\eta } _{1}+{\frac {\partial x^{2}}{\partial {\bar {x}}^{\mu }}}\mathbf {\eta } _{2}+...+{\frac {\partial x^{n}}{\partial {\bar {x}}^{\mu }}}\mathbf {\eta } _{n}\,\!}$

${\displaystyle {\bar {\mathbf {\eta } }}_{\mu }={\frac {\partial x^{i}}{\partial {\bar {x}}^{\mu }}}\mathbf {\eta } _{i}\,\!}$

## 向量的共變分量和反變分量

${\displaystyle \alpha (\mathbf {w} )=\mathbf {v} \cdot \mathbf {w} \,\!}$

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

{\displaystyle {\begin{aligned}v&=\sum _{i}v^{i}[{\mathfrak {f}}]X_{i}={\mathfrak {f}}\,\mathbf {v} [{\mathfrak {f}}]\\&=\sum _{i}v_{i}[{\mathfrak {f}}]Y^{i}={\mathfrak {f}}^{\sharp }\,\mathbf {v} [{\mathfrak {f}}^{\sharp }]\end{aligned}}\,\!}

### 歐幾里得空間

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

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

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

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

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

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

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

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

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

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

## 参考来源

1. ^ Goldstein, Herbert, Classical Mechanics 3rd, United States of America: Addison Wesley, 1980, ISBN 0201657023 （英语）