AKA '''Einstein summation convention'''??? 같은건가 sub인가?
''[[Date(2022-02-18T02:54:22)]] summation convention 이 더 나은 pagename?''

Chasnov https://youtu.be/CWIaPrwLyjM?si=T8or3V_5-92PhnLY
Ex.
$\sum_{i=1}^3\delta_{ii}\equiv\delta_{ii}=3$
$\epsilon_{ijk}\epsilon_{ijk}=\epsilon_{123}\epsilon_{123}+\cdots=6$
$\epsilon_{ijk}\epsilon_{lmn}=\begin{vmatrix}\delta_{il}&\delta_{im}&\delta_{in}\\\delta_{jl}&\delta_{jm}&\delta_{jn}\\\delta_{kl}&\delta_{km}&\delta_{kn}\end{vmatrix}$
$\vec{A}\cdot\vec{B}=A_iB_i$
$(\vec{A}\times\vec{B})_i = \epsilon_{ijk} A_j B_k$
$\begin{align}(\vec{A}\times\vec{B})_1 &= \epsilon_{1jk} A_j B_k \\ &= \epsilon_{123}A_2B_3 + \epsilon_{132}A_3B_2 \\ &= A_2B_3-A_3B_2\end{align}$
MKL [[크로네커_델타,Kronecker_delta]] [[레비치비타_기호,Levi-Civita_symbol]]

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MKL
[[합,sum]] (curr goto [[덧셈,addition]])

Google:einstein.notation
Google:einstein.convention

= LCY =

ex.
일단 전제는
 $\vec{A}=iA_x+jA_y+kA_z$
 $\vec{A}=(A_x,A_y,A_z)$
 $\partial_x\equiv\frac{\partial}{\partial x}$
 $\vec{\nabla}=\hat{i}\partial_x+\hat{j}\partial_y+\hat{k}\partial_z$
 $\vec{\nabla}=(\partial_x,\partial_y,\partial_z)$
 $\vec{A}{}_j$ 는 A의 j번째 성분????

이하 파란색이 아인슈타인 표기법으로 표기된 것

[[내적,inner_product]]
$\vec{A}\cdot\vec{B}=A_xB_x+A_yB_y+A_zB_z=\sum_{i=1}^3 A_iB_i$
 $\color{blue}=A_i B_i$

[[기울기,gradient]]관련하여
$\vec{\nabla}\phi(x,y,z)=\hat{i}\partial_x\phi+\hat{j}\partial_y\phi+\hat{k}\partial_z\phi$
 $\blue{=\hat{e_i}\partial_i \phi}$
e-hat 표기법은 모르겠는데... [[레비치비타_기호,Levi-Civita_symbol]]의 "벡터곱,vector_product,cross_product에 사용된다는데" 에서 유추해야

[[발산,divergence]]관련하여
$\vec{\nabla}\cdot\vec{A}=\partial_xA_x+\partial_yA_y+\partial_zA_z$
 $\blue =\partial_i A_i$

[[회전,curl]]
$\vec{\nabla}\times\vec{A}=\begin{vmatrix}i&j&k\\ \partial_x&\partial_y&\partial_z \\ A_x&A_y&A_z \end{vmatrix}$
 $\blue =\hat{e_i}\epsilon_{ijk}\partial_j A_k$

[[삼중곱,triple_product]]
$\nabla\cdot(\vec{A}\times\vec{B})=\vec{B}\cdot(\nabla\times\vec{A})-\vec{A}\cdot(\nabla\times\vec{B})$
 $=\partial_i(\vec{A}\times\vec{B})_i$
 $=\partial_i(\epsilon_{ijk}A_jB_k)$
 $=\epsilon_{ijk}\partial_i(A_jB_k)$
 $=\epsilon_{ijk}\left[(\partial_iA_j)B_k+A_j(\partial_iB_k)\right]$
 $=(\epsilon_{ijk} \partial_i A_j)B_k+A_j(\epsilon_{ijk} \partial_i B_k)$
 $=(\nabla\times\vec{A})_k + A_j (-\epsilon_{jik}\partial_i B_k)$
 $=(\nabla\times\vec{A})_k + A_j(-(\nabla\times\vec{B})_j)$
 $=(\nabla\times\vec{A})\cdot\vec{B}-\vec{A}\cdot(\nabla\times\vec{B})$

$\nabla\times(\vec{A}\times\vec{B})$
 $=\hat{e_i}\epsilon_{ijk}\partial_j(\vec{A}\times\vec{B})_k$
 $=\hat{e_i}\epsilon_{ijk}\partial_j(\epsilon_{kij}A_iB_j)$
i, j, k는 dummy index이다. $($ 의 왼쪽 ijk와 오른쪽 ijk는 다른 것.
 $=\hat{e_i}\epsilon_{ijk}\partial_j(\epsilon_{klm}A_lB_m)$
 $=\hat{e_i}\epsilon_{ijk}\epsilon_{klm}\partial_j(A_lB_m)$
여기서
 $\epsilon_{ijk}\epsilon_{klm}$
 $=\epsilon_{kij}\epsilon_{klm}$
 $=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$

나머지는 H/W라고.

이상 CHK
[[http://www.kocw.net/home/search/kemView.do?kemId=1269801 src 이창영]]1강


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Related:
[[크로네커_델타,Kronecker_delta]]
[[레비치비타_기호,Levi-Civita_symbol]]
[[벡터,vector]]
[[행렬,matrix]] matrix_multiplication [[대각합,trace]]
[[텐서,tensor]]
[[내적,inner_product]] [[스칼라곱,scalar_product,dot_product]]
[[외적,outer_product]] [[벡터곱,vector_product,cross_product]]
첨자
 위첨자superscript
 아래첨자subscript

Twins:
https://mathworld.wolfram.com/EinsteinSummation.html
https://freshrimpsushi.github.io/posts/einstein-notation/
https://planetmath.org/einsteinsummationconvention

[[WpEn:Einstein_notation]]
[[WpKo:아인슈타인_표기법]]

chk
https://everything2.com/title/summation+convention

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Up: [[표기법,notation]]