# 列維-奇維塔符號

${\displaystyle \varepsilon _{a_{1}a_{2}\cdots a_{n}}}$

${\displaystyle \varepsilon _{\cdots a_{p}\cdots a_{p}\cdots }=0}$

${\displaystyle \varepsilon _{a_{1}a_{2}\cdots a_{n}}=(-1)^{p}\varepsilon _{12\cdots n}}$

${\displaystyle \varepsilon _{12\cdots n}=+1}$

${\displaystyle \varepsilon _{\cdots a_{p}\cdots a_{q}\cdots }=-\varepsilon _{\cdots a_{q}\cdots a_{p}\cdots }}$

n 維列維-奇維塔符號”一詞是指符號上的指標數 n ，和所討論的向量空間維度相符，其中可指歐幾里得空間非歐幾里得空間，例如 R3n = 3閔可夫斯基空間n = 4

## 定義

### 二維

 ${\displaystyle \varepsilon _{ij}={\begin{cases}+1\\-1\\0\end{cases}}\,}$ 當 ${\displaystyle \left(i,j\right)=\left(1,2\right)}$ 當 ${\displaystyle \left(i,j\right)=\left(2,1\right)}$ 當 ${\displaystyle i=j}$

${\displaystyle {\begin{pmatrix}\varepsilon _{11}&\varepsilon _{12}\\\varepsilon _{21}&\varepsilon _{22}\end{pmatrix}}={\begin{pmatrix}0&1\\-1&0\end{pmatrix}}}$

### 三維

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

{\displaystyle {\begin{aligned}\varepsilon _{\color {BrickRed}{1}\color {Violet}{3}\color {Orange}{2}}=-\varepsilon _{\color {BrickRed}{1}\color {Orange}{2}\color {Violet}{3}}&=-1\\\varepsilon _{\color {Violet}{3}\color {BrickRed}{1}\color {Orange}{2}}=-\varepsilon _{\color {Orange}{2}\color {BrickRed}{1}\color {Violet}{3}}&=-(-\varepsilon _{\color {BrickRed}{1}\color {Orange}{2}\color {Violet}{3}})=1\\\varepsilon _{\color {Orange}{2}\color {Violet}{3}\color {BrickRed}{1}}=-\varepsilon _{\color {BrickRed}{1}\color {Violet}{3}\color {Orange}{2}}&=-(-\varepsilon _{\color {BrickRed}{1}\color {Orange}{2}\color {Violet}{3}})=1\\\varepsilon _{\color {Orange}{2}\color {Violet}{3}\color {Orange}{2}}=-\varepsilon _{\color {Orange}{2}\color {Violet}{3}\color {Orange}{2}}&=0\end{aligned}}}

### 四維

 ${\displaystyle \varepsilon _{ijkl}={\begin{cases}+1\\-1\\0\end{cases}}\,}$ 當 ${\displaystyle \left(i,j,k,l\right)=\left(1,2,3,4\right)}$  的偶排列 當 ${\displaystyle \left(i,j,k,l\right)=\left(1,2,3,4\right)}$  的奇排列 其餘情況，即任意兩個指標相等

{\displaystyle {\begin{aligned}\varepsilon _{\color {BrickRed}{1}\color {RedViolet}{4}\color {Violet}{3}\color {Orange}{\color {Orange}{2}}}=-\varepsilon _{\color {BrickRed}{1}\color {Orange}{\color {Orange}{2}}\color {Violet}{3}\color {RedViolet}{4}}&=-1\\\varepsilon _{\color {Orange}{\color {Orange}{2}}\color {BrickRed}{1}\color {Violet}{3}\color {RedViolet}{4}}=-\varepsilon _{\color {BrickRed}{1}\color {Orange}{\color {Orange}{2}}\color {Violet}{3}\color {RedViolet}{4}}&=-1\\\varepsilon _{\color {RedViolet}{4}\color {Violet}{3}\color {Orange}{\color {Orange}{2}}\color {BrickRed}{1}}=-\varepsilon _{\color {BrickRed}{1}\color {Violet}{3}\color {Orange}{\color {Orange}{2}}\color {RedViolet}{4}}&=-(-\varepsilon _{\color {BrickRed}{1}\color {Orange}{\color {Orange}{2}}\color {Violet}{3}\color {RedViolet}{4}})=1\\\varepsilon _{\color {Violet}{3}\color {Orange}{\color {Orange}{2}}\color {RedViolet}{4}\color {Violet}{3}}=-\varepsilon _{\color {Violet}{3}\color {Orange}{\color {Orange}{2}}\color {RedViolet}{4}\color {Violet}{3}}&=0\end{aligned}}}

### 推廣到高維

 ${\displaystyle \varepsilon _{a_{1}a_{2}a_{3}\ldots a_{n}}={\begin{cases}+1\\-1\\0\end{cases}}}$ 當 ${\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{n})}$  是 ${\displaystyle (1,2,3,\dots ,n)}$  的偶排列 當 ${\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{n})}$  是 ${\displaystyle (1,2,3,\dots ,n)}$  的奇排列 其餘情況，即任意兩個指標相等

{\displaystyle {\begin{aligned}\varepsilon _{a_{1}a_{2}a_{3}\ldots a_{n}}&=\prod _{1\leq i

${\displaystyle \varepsilon _{ijk\dots }\varepsilon _{mnl\dots }={\begin{vmatrix}\delta _{im}&\delta _{in}&\delta _{il}&\dots \\\delta _{jm}&\delta _{jn}&\delta _{jl}&\dots \\\delta _{km}&\delta _{kn}&\delta _{kl}&\dots \\\vdots &\vdots &\vdots \\\end{vmatrix}}}$

## 應用和範例

### 行列式

${\displaystyle A={\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}}}$

${\displaystyle \det(A)=\sum _{i,j,k=1}^{3}\varepsilon _{ijk}\,a_{1i}\,a_{2j}\,a_{3k}}$

${\displaystyle \det(A)=\sum _{a_{1},a_{2},\cdots ,a_{n}=1}^{n}\varepsilon _{a_{1}a_{2}\cdots a_{n}}\,a_{1a_{1}}\,a_{2a_{2}}\,\cdots \,a_{na_{n}},}$

### 向量的叉積

${\displaystyle {\boldsymbol {a}}\times {\boldsymbol {b}}={\begin{vmatrix}{\boldsymbol {e}}_{1}&{\boldsymbol {e}}_{2}&{\boldsymbol {e}}_{3}\\a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\\end{vmatrix}}=\sum _{1\leq i,j,k\leq 3}\varepsilon _{ijk}\,a_{i}b_{j}\,{\boldsymbol {e}}_{k}}$

${\displaystyle {\boldsymbol {a}}\cdot ({\boldsymbol {b}}\times {\boldsymbol {c}})={\begin{vmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\\\end{vmatrix}}=\sum _{1\leq i,j,k\leq 3}\varepsilon _{ijk}\,a_{i}b_{j}c_{k}}$

## 性質

${\displaystyle \varepsilon ^{ij\dots k}=\varepsilon _{ij\dots k}.}$

${\displaystyle \varepsilon _{ijk}\varepsilon ^{imn}\equiv \sum _{i=1,2,3}\varepsilon _{ijk}\varepsilon ^{imn}}$ .

### 二維

${\displaystyle \varepsilon _{ij}\varepsilon ^{mn}={\delta _{i}}^{m}{\delta _{j}}^{n}-{\delta _{i}}^{n}{\delta _{j}}^{m}}$

(1)

${\displaystyle \varepsilon _{ij}\varepsilon ^{in}={\delta _{j}}^{n}}$

(2)

${\displaystyle \varepsilon _{ij}\varepsilon ^{ij}=2.}$

(3)

### 三維

#### 指標和符號值

${\displaystyle \varepsilon _{ijk}\varepsilon ^{imn}=\delta _{j}{}^{m}\delta _{k}{}^{n}-\delta _{j}{}^{n}\delta _{k}{}^{m}}$

(4)

${\displaystyle \varepsilon _{jmn}\varepsilon ^{imn}=2{\delta _{j}}^{i}}$

(5)

${\displaystyle \varepsilon _{ijk}\varepsilon ^{ijk}=6.}$

(6)

#### 乘積

{\displaystyle {\begin{aligned}\varepsilon _{ijk}\varepsilon _{lmn}&={\begin{vmatrix}\delta _{il}&\delta _{im}&\delta _{in}\\\delta _{jl}&\delta _{jm}&\delta _{jn}\\\delta _{kl}&\delta _{km}&\delta _{kn}\\\end{vmatrix}}\\[6pt]&=\delta _{il}\left(\delta _{jm}\delta _{kn}-\delta _{jn}\delta _{km}\right)-\delta _{im}\left(\delta _{jl}\delta _{kn}-\delta _{jn}\delta _{kl}\right)+\delta _{in}\left(\delta _{jl}\delta _{km}-\delta _{jm}\delta _{kl}\right).\end{aligned}}}

${\displaystyle \sum _{i=1}^{3}\varepsilon _{ijk}\varepsilon _{imn}=\delta _{jm}\delta _{kn}-\delta _{jn}\delta _{km}}$

${\displaystyle \sum _{i=1}^{3}\sum _{j=1}^{3}\varepsilon _{ijk}\varepsilon _{ijn}=2\delta _{kn}}$

### n維

#### 指標和符號值

n維中，當所有${\displaystyle i_{1},\ldots ,i_{n},j_{1},\ldots ,j_{n}}$ take values${\displaystyle 1,2,\ldots ,n}$

${\displaystyle \varepsilon _{i_{1}\dots i_{n}}\varepsilon ^{j_{1}\dots j_{n}}=n!\delta _{[i_{1}}^{j_{1}}\dots \delta _{i_{n}]}^{j_{n}}=\delta _{i_{1}\dots i_{n}}^{j_{1}\dots j_{n}}}$

(7)

${\displaystyle \varepsilon _{i_{1}\dots i_{k}~i_{k+1}\dots i_{n}}\varepsilon ^{i_{1}\dots i_{k}~j_{k+1}\dots j_{n}}=k!(n-k)!~\delta _{[i_{k+1}}^{j_{k+1}}\dots \delta _{i_{n}]}^{j_{n}}=k!~\delta _{i_{k+1}\dots i_{n}}^{j_{k+1}\dots j_{n}}}$

(8)

${\displaystyle \varepsilon _{i_{1}\dots i_{n}}\varepsilon ^{i_{1}\dots i_{n}}=n!}$

(9)

${\displaystyle \sum _{i,j,k,\dots =1}^{n}\varepsilon _{ijk\dots }\varepsilon _{ijk\dots }=n!}$

• 每個排列是偶排列或奇排列，
• ${\displaystyle (+1)^{2}=(-1)^{2}=1}$ ，與
• 任何n-元素集合的排列數正好是${\displaystyle n!}$

#### 乘積

${\displaystyle \varepsilon _{i_{1}i_{2}\dots i_{n}}\varepsilon _{j_{1}j_{2}\dots j_{n}}={\begin{vmatrix}\delta _{i_{1}j_{1}}&\delta _{i_{1}j_{2}}&\dots &\delta _{i_{1}j_{n}}\\\delta _{i_{2}j_{1}}&\delta _{i_{2}j_{2}}&\dots &\delta _{i_{2}j_{n}}\\\vdots &\vdots &\ddots &\vdots \\\delta _{i_{n}j_{1}}&\delta _{i_{n}j_{2}}&\dots &\delta _{i_{n}j_{n}}\\\end{vmatrix}}}$