Index picking function

Table of contents
- Kronecker delta Index picking function
- Levi-Civita symbol Index picking function
  - Levi-Civita symbol as a tensor Levi-Civita symbol

Kronecker delta ( $δ_{i j}$ )

 1  0 

Levi-Civita symbol ( $ε$ )

 1  0 

Denoted by the Greek letter epsilon with \varepsilon encoding in LaTeX.

Definition:

odd permutation: -1
even permutation: 1
not a permutation: 0. This happens iff two more more indices are repeated

Levi-Civita symbol as a tensor

 0  0 

An Introduction to Tensors and Group Theory for Physicists by Nadir Jeevanjee (2011) shows that this is a tensor that represents the volume of a parallelepiped.

It takes as input three vectors, and outputs one real number, the volume. And it is linear on each vector. This perfectly satisfied the definition of a tensor of order (3,0).

Given a basis

(e_{i}, e_{j}, e_{k})

and a function that return the volume of a parallelepiped given by three vectors

V (v_{1}, v_{2}, v_{3})

ε_{i k j} = V (e_{i}, e_{j}, e_{k})

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