4.3: Exercises

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Apr 30, 2021
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- 4.2: Linear Equations
- 5: Gaussian Elimination

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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

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

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