텐서 표기법(Indicial notation)

 

 

벡터와 텐서의 표기법인 Indicial notation에 대해서 정리하였습니다.

 

기본 표기법
Vector	$$\overrightarrow{v}(x,y,z)=v_i$$	$i:\text{free index}$
Tensor	$$\underset{―}{\underset{―}{A}}=A_{ij}$$	$i,j:\text{free index}$
벡터의 경우 1개의 free index, 2차원 텐서의 경우 2개의 free index로 표기

 

 

Einstein summation convention(아인슈타인의 합규약)
$$\overrightarrow{a}\cdot \overrightarrow{b}={\displaystyle \sum _{i=1}^3}{a}_i{b}_i=a_i{b}_i$$		$i:\text{dummy index}$
중복된 인덱스는 dummy index 로 $\sum$ 기호가 생략된 것

 

 

Kronecker Delta(크로네커 델타)
$${\delta }_{ij}=\{\begin{array}{ll}0& \text{if }i\ne j,\\ 1& \text{if }i=j.\end{array}$$		$${\delta }_{ij}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$ $$i\in \{1,2,3\},j\in \{1,2,3\}$$
$${\delta }_{ij}={\delta }_{ji}$$		교환법칙 성립
$${\delta }_{ii}=3$$		$i:\text{dummy index}$ 이므로 합규약에 의해
선형대수에서  Identity matrix ${\displaystyle \mathbf{I}}$ 와 동일한 개념

 

 

Levi-Civita epsilon (레비-치비타 기호)
$${ϵ}_{ijk}=\{\begin{array}{ll}0& ijk\text{ 가 그 외 다른값을 가질 때},(21개)\\ +1& ijk\in \{123,231,312\},(3개)\\ -1& ijk\in \{132,321,213\}.(3개)\end{array}$$
$${ϵ}_{ijk}=-{ϵ}_{ikj}$$
$${ϵ}_{ijk}{ϵ}_{ilm}={\delta }_{jl}{\delta }_{km}-{\delta }_{jm}{\delta }_{kl}$$		$\delta$ 와 $ϵ$ 의 관계
↓
$${ϵ}_{ijk}{ϵ}_{pjk}=2{\delta }_{ip}$$
↓
$${ϵ}_{ijk}{ϵ}_{ijk}=6$$

 

 

 

Vector operations
Dot product (내적)
$$\overrightarrow{a}\cdot \overrightarrow{b}=|a||b|cos\theta$$		$$=a_i{b}_i$$
Cross product (외적)
$$\overrightarrow{a}\times \overrightarrow{b}=|a||b|sin\theta =\left|\begin{array}{ccc}\hat{\mathit{ı}}& \hat{\mathit{ȷ}}& \hat{\mathit{k}}\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\right|$$		$$={ϵ}_{ijk}{a}_j{b}_k$$
Magnitude (크기)
$$|\overrightarrow{a}{|}^2$$		$$=a_i{a}_i$$
Scalar triple product (스칼라 삼중곱)
$$\overrightarrow{a}\cdot (\overrightarrow{b}\times \overrightarrow{c})=\overrightarrow{b}\cdot (\overrightarrow{c}\times \overrightarrow{a})=\overrightarrow{c}\cdot (\overrightarrow{a}\times \overrightarrow{b})$$		$$={ϵ}_{ijk}{a}_i{b}_j{c}_k$$
Vector triple product (벡터 삼중곱) [BAC-CAB:백캡]
$$\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=\overrightarrow{b}(\overrightarrow{a}\cdot \overrightarrow{c})-\overrightarrow{c}(\overrightarrow{a}\cdot \overrightarrow{b})$$		$$={ϵ}_{ijk}{a}_j{ϵ}_{klm}{b}_l{c}_m$$

 

 

Matrix Operation
Double-dot product
$$\begin{array}{c}\mathbf{A}:\mathbf{B}\\ =A_{11}{B}_{11}+A_{12}{B}_{12}+A_{13}{B}_{13}\\ +A_{21}{B}_{21}+A_{22}{B}_{22}+A_{23}{B}_{23}\\ +A_{31}{B}_{31}+A_{32}{B}_{32}+A_{33}{B}_{33}\end{array}$$	$$=A_{ij}{B}_{ij}$$
Norm
$$|\mathbf{A}|=\sqrt{\mathbf{A}:\mathbf{A}}$$	$$=\sqrt{A_{ij}{A}_{ij}}$$
Trace
$$\text{tr}(\mathbf{A})$$	$$=A_{ii}$$

 

 

 

Gradient Operation
scalar
$$\mathrm{\nabla }f=\left[\begin{array}{c}{\displaystyle \frac{\partial f}{\partial x_1}}\\ {\displaystyle \frac{\partial f}{\partial x_2}}\\ {\displaystyle \frac{\partial f}{\partial x_3}}\end{array}\right]$$	$$=f_{,i}$$
vector
$$\mathrm{\nabla }\overrightarrow{v}=\left[\begin{array}{ccc}{\displaystyle \frac{\partial v_1}{\partial x_1}}& {\displaystyle \frac{\partial v_1}{\partial x_2}}& {\displaystyle \frac{\partial v_1}{\partial x_3}}\\ {\displaystyle \frac{\partial v_2}{\partial x_1}}& {\displaystyle \frac{\partial v_2}{\partial x_2}}& {\displaystyle \frac{\partial v_2}{\partial x_3}}\\ {\displaystyle \frac{\partial v_3}{\partial x_1}}& {\displaystyle \frac{\partial v_3}{\partial x_2}}& {\displaystyle \frac{\partial v_3}{\partial x_3}}\end{array}\right]$$	$$=v_{i,j}$$
divergence
$$\mathrm{\nabla }\cdot \overrightarrow{v}={\displaystyle \frac{\partial v_1}{\partial x_1}}+{\displaystyle \frac{\partial v_2}{\partial x_2}}+{\displaystyle \frac{\partial v_3}{\partial x_3}}$$	$$=v_{i,i}$$
curl
$$\mathrm{\nabla }\times \overrightarrow{v}=\left[\begin{array}{c}{\displaystyle \frac{\partial v_3}{\partial x_2}}-{\displaystyle \frac{\partial v_2}{\partial x_3}}\\ {\displaystyle \frac{\partial v_1}{\partial x_3}}-{\displaystyle \frac{\partial v_3}{\partial x_1}}\\ {\displaystyle \frac{\partial v_2}{\partial x_1}}-{\displaystyle \frac{\partial v_1}{\partial x_2}}\end{array}\right]$$	$$={ϵ}_{ijk}{v}_{k,j}$$
Laplacian
$${\mathrm{\nabla }}^{2}f={\displaystyle \frac{{\partial }^2{v}_1}{\partial x_1^2}}+{\displaystyle \frac{{\partial }^2{v}_2}{\partial x_2^2}}+{\displaystyle \frac{{\partial }^2{v}_3}{\partial x_3^2}}$$	$$=f_{,ii}$$

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[1] https://physastro-msci.tripod.com/webonmediacontents/indicial.pdf

[2] https://en.m.wikipedia.org/wiki/File:Levi-Civita_Symbol_cen.svg