(adsbygoogle = window.adsbygoogle || []).push({}); 1. Variables

Given a generalized basis in three dimensions: [tex]e_{1},e_{2},e_{3}[/tex] and the standard Kronecker delta [tex]\delta_{ij}[/tex], and using Einstein summation.

With the vector [tex]\textbf{x},\textbf{y},\textbf{z}[/tex] i'm trying to simplify this problem:

2. Problem

[tex]\delta_{il} . \delta_{jm} . x_{j}[/tex]

3. My attempt

[tex]\delta_{il}.\delta_{jm} . \textbf{x} . e_{j}

= (e_{i}. e_{l}) . (e_{j} . e_{m}) . \textbf{x} . e_{j}

= (e_{j}. e_{j}) . (e_{l} . e_{m} . e_{j}) . \textbf{x}

= 1 . (e_{l} . e_{m} . e_{j}) . \textbf{x} [/tex]

Surely this leads to [tex]\delta_{il} . \delta_{jm} . x_{j} = 0[/tex] as [tex]e_{l} , e_{m} , e_{j}[/tex] are all orthagonal ?

Ultimately I'm trying to prove that

[tex](\delta_{il} . \delta_{jm} - \delta_{jl} . \delta_{im}).x_{j}.y_{l}.z_{m}

= y_{i}.x_{j}.z_{j} - z_{i}.x_{j}.y_{j}[/tex]

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# Homework Help: 3D Vectors

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