🧄Simplifying Levi-Civita and Kronecker Delta Identities

The analysis shows that the tensor identity $\varepsilon_{i j k} \varepsilon_{j k \ell}$ simplifies to $2 \delta_{i \ell}$. This derivation highlights how the epsilon-delta relation is a powerful algebraic tool that connects the Levi-Civita symbol (representing the cross product) and the Kronecker delta (representing the dot product). This relationship allows for complex vector identities, such as the bac-cab rule, to be proven rigorously through a systematic, algebraic process rather than relying on geometric intuition or tedious component expansions. The method involves a proof by cases, which is more efficient than checking all possible index combinations.

🎬Visualizing the Epsilon-Delta Identity

A complex tensor identity can be broken down into a series of simple, step-by-step calculations. By visualizing the summation of the individual products, you can clearly see how the final, simplified result is reached. The animation shows that the non-zero terms in the summation are exactly what's needed to produce the final value.

🧄Mathematical Proof

Simplifying Levi-Civita and Kronecker Delta Identities | Proof and Derivationviadean on Notion