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.

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✍️Mathematical Proof

$\gg$Mathematical Structures Underlying Physical Laws

$\complement\cdots$Counselor

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Result of the Derivation: The identity $\varepsilon_{i j k} \varepsilon_{j k \ell}=2 \delta_{i \ell}$ is derived by applying a systematic process of index reordering and summation. The process correctly shows how to use the single-index contraction identity, which is $\sum_p \varepsilon_{p q r} \varepsilon_{p s t}=\delta_{q s} \delta_{r t}-\delta_{q t} \delta_{r s}$.
The Epsilon-Delta Relation is a powerful tool: This relation is a fundamental algebraic identity in vector analysis. It connects the Levi-Civita symbol ( $\varepsilon_{i j k}$ ), which represents the cross product and geometric concepts like orientation and perpendicularity, with the Kronecker delta ( $\delta_{i j}$ ), which represents the dot product and algebraic concepts like orthogonality. This connection allows for the manipulation of complex vector identities without relying on geometric intuition or lengthy component expansions.
Proof by Cases: Instead of exhaustively checking all possible index combinations, the analysis shows that the identity can be proven rigorously by considering a few key cases based on the properties of the tensors involved. This approach is more efficient and elegant.
Derivation of the Bac-Cab Rule: The analysis demonstrates that the well-known bac-cab rule $( A \times( B \times C )= B ( A \cdot C )- C ( A \cdot B ))$ is a direct consequence of the epsilon-delta relation. The proof starts by expressing the vector identity in index notation, applying the epsilon-delta relation, and simplifying it back into the familiar vector form.

🎬Demonstration

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.

Visualizing the Epsilon-Delta Identity