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

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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

Visualizing the Epsilon Delta Identity

Visualizing the Epsilon-Delta Identity

🧄Mathematical Proof

Simplifying Levi-Civita and Kronecker Delta Identities.html

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Proving the Cross Product Rules with the Levi-Civita Symbol

Proving the Epsilon-Delta Relation and the Bac-Cab Rule

Simplifying Levi-Civita and Kronecker Delta Identities

Dot Cross and Triple Products

Why a Cube's Diagonal Angle Never Changes

How the Cross Product Relates to the Sine of an Angle

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