$$\require{cancel}$$

# 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 $$\sum_i \varepsilon_{ijk} \varepsilon_{imn}=\delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km}$$.

4.3: Exercises is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Y. D. Chong via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.