# 潘洛斯圖形符號

## 特殊張量表象

### 度規張量

 度規張量${\displaystyle g^{ab}\,}$ 度規張量${\displaystyle g_{ab}\,}$

### 列維-奇維塔張量

 ${\displaystyle \varepsilon _{ab\ldots n}}$ ${\displaystyle \epsilon ^{ab\ldots n}}$ ${\displaystyle \varepsilon _{ab\ldots n}\,\epsilon ^{ab\ldots n}}$${\displaystyle =n!}$

### 結構常數

 結構常數${\displaystyle {\gamma _{\alpha \beta }}^{\chi }=-{\gamma _{\beta \alpha }}^{\chi }}$

## 張量運算

### 指標縮併

 克羅內克δ函數 ${\displaystyle \delta _{b}^{a}}$ 點積 ${\displaystyle \beta _{a}\,\xi ^{a}}$ ${\displaystyle g_{ab}\,g^{bc}=\delta _{a}^{c}=g^{cb}\,g_{ba}}$

### 對稱化

 對稱化${\displaystyle Q^{(ab\ldots n)}}$（其中${\displaystyle {}_{Q^{ab}=Q^{[ab]}+Q^{(ab)}}}$）

### 反對稱化

 反對稱化${\displaystyle E_{[ab\ldots n]}}$（其中${\displaystyle {}_{E_{ab}=E_{[ab]}+E_{(ab)}}}$）

## 行列式

 行列式${\displaystyle \det \mathbf {T} =\det \left(T_{\ b}^{a}\right)}$ 逆矩陣${\displaystyle \mathbf {T} ^{-1}=\left(T_{\ b}^{a}\right)^{-1}}$

### 協變導數

 協變導數${\displaystyle 12\nabla _{a}\left\{\xi ^{f}\,\lambda _{fb[c}^{(d}\,D_{gh]}^{e)b}\right\}}$

## 張量操作

${\displaystyle \varepsilon _{a...c}\epsilon ^{a...c}=n!}$

### 黎曼曲率張量

 黎曼曲率張量的符號 里奇張量 ${\displaystyle R_{ab}=R_{acb}^{\ \ \ c}}$ 里奇恆等式${\displaystyle (\nabla _{a}\,\nabla _{b}-\nabla _{b}\,\nabla _{a})\,\mathbf {\xi } ^{d}}$${\displaystyle =R_{abc}^{\ \ \ d}\,\mathbf {\xi } ^{c}}$ 比安基恆等式${\displaystyle \nabla _{[a}R_{bc]d}^{\ \ \ e}=0}$

## 參考文獻

1. ^ see e.g. Quantum invariants of knots and 3-manifolds" by V. G. Turaev (1994), page 71
2. ^ Predrag Cvitanović. Group Theory: Birdtracks, Lie's, and Exceptional Groups. Princeton University Press. 2008.
3. ^ Penrose, R.; Rindler, W. Spinors and Space-Time: Vol I, Two-Spinor Calculus and Relativistic Fields. Cambridge University Press. 1984: 424–434. ISBN 0-521-24527-3.
4. ^ Penrose, R.; Rindler, W. Spinors and Space-Time: Vol. II, Spinor and Twistor Methods in Space-Time Geometry. Cambridge University Press. 1986. ISBN 0-521-25267-9.