2018-10-05

2018-10-05 本文已影响0人 carpediemmlf

Vector calculus

Cartesian Unit Vectors

linearly independent: $\vec{a} = \hat{e_i}a_i$ is uniquely represented
orthonormal: $\hat{e_i}\cdot \hat{e_i} = 1$ $\hat{e_i}\cdot\hat{e_j} = 0$
right handed: $[e_1, e_2, e_3] = +1$

suffix notation：
$\delta_{ij} = \left \{ \begin{array}{lr} +1 & i = j\\ 0 & i \neq j \end{array} \right .$

$\epsilon_{ijk} = \left \{ \begin{array}{lr} +1 & even\ perm\\ -1 & odd\ perm\\ 0 & otherwise \end{array} \right .$

Einstein's summation convention
$\vec{a}\cdot\vec{b} = a_ib_i$

free suffix: appearing once; free variables used should be agreed on by LHS and RHS
dummy suffix: appearing twice and twice only

Contraction: set one free suffix equal to another

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