텐서 표기법(Indicial notation)

 

 

벡터와 텐서의 표기법인 Indicial notation에 대해서 정리하였습니다.

 

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

 

 

Einstein summation convention(아인슈타인의 합규약)
$$ \vec{a} \cdot \vec{b} = {\color{red}{\displaystyle\sum_{i=1}^{3}}}{a_i b_i} = \color{red}{a_i b_ i} $$		$i : \text{dummy index}$
중복된 인덱스는 dummy index 로 $\sum$ 기호가 생략된 것

 

 

Kronecker Delta(크로네커 델타)
$$ {\color{red}{\delta_{ij}}} = \begin{cases} 0 & \quad \text{if } i \neq j, \\ 1 & \quad \text{if } i = j. \end{cases} $$		$$ \delta_{ij} = \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} $$ $$i \in \{1,2,3\}, \quad j \in \{1,2,3\}$$
$$ \delta_{ij} = \delta_{ji} $$		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)
$$ \epsilon_{ijk} = \begin{cases} 0 & \quad ijk \text{ 가 그 외 다른값을 가질 때}, (21개)\\ +1 & \quad ijk \in \{123,231,312\}, (3개)\\ -1 & \quad ijk \in \{132,321,213\}. (3개) \end{cases} $$
$$\epsilon_{ijk} = - \epsilon_{ikj}  $$
$$\epsilon_{ijk} \epsilon_{ilm} = \delta_{jl}\delta_{km} - \delta_{jm}\delta_{kl}$$		$\delta$ 와 $\epsilon$ 의 관계
↓
$$\epsilon_{ijk} \epsilon_{pjk} = 2\delta_{ip}$$
↓
$$ \epsilon_{ijk} \epsilon_{ijk} = 6 $$

 

 

 

Vector operations
Dot product (내적)
$$ \vec{a} \cdot \vec{b} = |a||b|cos\theta$$		$$ = \color{red}{a_i b_i} $$
Cross product (외적)
$$ \vec{a} \times \vec{b} = |a||b|sin\theta = \begin{vmatrix} \hati & \hatj & \hatk \\  a_1 & a_2 & a_3 \\  b_1 & b_2 & b_3 \end{vmatrix} $$		$$ = \color{red}{\epsilon_{ijk}  a_j b_k}$$
Magnitude (크기)
$$ |\vec{a}|^2$$		$$ = \color{red}{a_i a_i} $$
Scalar triple product (스칼라 삼중곱)
$$ \vec{a} \cdot (\vec{b} \times \vec{c}) = \vec{b} \cdot (\vec{c} \times \vec{a}) = \vec{c} \cdot (\vec{a} \times \vec{b}) $$		$$ =\color{red}{\epsilon_{ijk} a_i b_j c_k}$$
Vector triple product (벡터 삼중곱) [BAC-CAB:백캡]
$$ \vec{a} \times (\vec{b} \times \vec{c}) = \vec{b}(\vec{a} \cdot \vec{c}) - \vec{c}(\vec{a} \cdot \vec{b})$$		$$= \color{red}{\epsilon_{ijk} a_j \epsilon_{klm} b_l c_m} $$

 

 

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

 

 

 

Gradient Operation
scalar
$$ \nabla f  = \begin{bmatrix} \dfrac{\partial f}{\partial x_1} \\ \dfrac{\partial f}{\partial x_2} \\ \dfrac{\partial f}{\partial x_3}  \end{bmatrix} $$	$$ = \color{red}{f_{,i}} $$
vector
$$ \nabla \vec{v} = \begin{bmatrix} \pdx[x_1]{v_1} & \pdx[x_2]{v_1} & \pdx[x_3]{v_1} \\ \pdx[x_1]{v_2} & \pdx[x_2]{v_2} & \pdx[x_3]{v_2} \\ \pdx[x_1]{v_3} & \pdx[x_2]{v_3} & \pdx[x_3]{v_3} \end{bmatrix} $$	$$ = \color{red}{v_{i,j}}  $$
divergence
$$ \nabla \cdot \vec{v} = \pdx[x_1]{v_1} + \pdx[x_2]{v_2} + \pdx[x_3]{v_3} $$	$$ = \color{red}{v_{i,i}} $$
curl
$$ \nabla \times \vec{v} = \begin{bmatrix} \pdx[x_2]{v_3} - \pdx[x_3]{v_2} \\ \pdx[x_3]{v_1} - \pdx[x_1]{v_3} \\ \pdx[x_1]{v_2} - \pdx[x_2]{v_1} \end{bmatrix} $$	$$ = \color{red}{\epsilon_{ijk} v_{k,j}} $$
Laplacian
$$ \nabla^2 f  = \dfrac{\partial^2 v_1}{\partial x_1^2} + \dfrac{\partial^2 v_2}{\partial x_2^2} + \dfrac{\partial^2 v_3}{\partial x_3^2} $$	$$ = \color{red}{f_{,ii}} $$

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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