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爱因斯坦求和约定 - Wikipedia

爱因斯坦求和约定

维基百科，自由的百科全书

一般力學及工程會用互相垂直單位向量i, j 及 k來描述3維空間的向量。

\mathbf{u} = u_x \mathbf{i} + u_y \mathbf{j} + u_z \mathbf{k}

把 i, j及 k 寫成 e1, e2 及e3,所以一個向量可以寫成:

\mathbf{u} = u_1 \mathbf{e}_1 + u_2 \mathbf{e}_2 + u_3 \mathbf{e}_3 = \sum_{i = 1}^3 u_i \mathbf{e}_i

根據愛因斯坦求和约定（Einstein summation convention）或愛因斯坦標記法(Einstein notation)，若單項中有標號出現兩次且分別位於上標及下標，則此項代表著所有之和。

u^j \mathbf{e}_j = \sum_{i = 1}^3 u_i = \mathbf{u}
\langle \mathbf{u},\mathbf{v} \rangle = \langle u^i \mathbf{e}_i,v^j \mathbf{e}_j \rangle

或者:

\mathbf{u} \cdot \mathbf{v} = u_i v_j ( \mathbf{e}_i \cdot \mathbf{e}_j )

這裡

\mathbf{e}_i \cdot \mathbf{e}_j = \delta_{ij}

\ \delta_{ij} 就是克罗内克尔δ。當 i = j,\ \delta_{ij}=1,否則就是0。邏輯上，可以把方程式中的i轉為j或者把j轉為i。

\mathbf{u} \cdot \mathbf{v} = u_i v_j\delta_{ij}= u_i v_i = u_j v_j

叉積

\mathbf{u} \times \mathbf{v}= \sum_{j = 1}^3 u_j \mathbf{e}_j \times \sum_{k = 1}^3 v_k \mathbf{e}_k = u_j \mathbf{e}_j \times v_k \mathbf{e}_k = u_j v_k (\mathbf{e}_j \times \mathbf{e}_k ) = \epsilon_{ijk} \mathbf{e}_i u_j v_k

這裡的 \mathbf{e}_j \times \mathbf{e}_k = \epsilon_{ijk} \mathbf{e}_i。\ \epsilon_{ijk} 是Levi-Civita 符號。它的定義：

\epsilon_{ijk} = \left\{ \begin{matrix} +1 & \mbox{if } (i,j,k) \mbox{ is } (1,2,3), (2,3,1) \mbox{ or } (3,1,2)\\ -1 & \mbox{if } (i,j,k) \mbox{ is } (3,2,1), (1,3,2) \mbox{ or } (2,1,3)\\ 0 & \mbox{otherwise: }i=j \mbox{ or } j=k \mbox{ or } k=i \end{matrix} \right.


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