4.3: Exercises

Exercise \(\PageIndex{1}\)

Write Python code to construct a 3D array of size \(3\times 3\times 3\) corresponding to the Levi-Civita tensor,

\[\varepsilon_{ijk} = \begin{cases} +1 & \text{if } (i,j,k) \text{ is } (1,2,3), (2,3,1) \text{ or } (3,1,2), \\ -1 & \text{if } (i,j,k) \text{ is } (3,2,1), (1,3,2) \text{ or } (2,1,3), \\ \;\;\,0 & \text{if }i=j \text{ or } j=k \text{ or } k=i \end{cases}\]
Then, using the tensordot function, verify the identity \(\sum_i \varepsilon_{ijk} \varepsilon_{imn}=\delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km}\).