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

$\complement\cdots$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.

✍️Mathematical Proof

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1. Proving the Cross Product Rules with the Levi-Civita Symbol
2. Proving the Epsilon-Delta Relation and the Bac-Cab Rule
3. Simplifying Levi-Civita and Kronecker Delta Identities
4. Dot Cross and Triple Products
5. Why a Cube's Diagonal Angle Never Changes
6. How the Cross Product Relates to the Sine of an Angle
7. Finding the Shortest Distance and Proving Orthogonality for Skew Lines
8. A Study of Helical Trajectories and Vector Dynamics
9. The Power of Cross Products: A Visual Guide to Precessing Vectors
10. Divergence and Curl Analysis of Vector Fields
11. Unpacking Vector Identities: How to Apply Divergence and Curl Rules
12. Commutativity and Anti-symmetry in Vector Calculus Identities
13. Double Curl Identity Proof using the epsilon-delta Relation
14. The Orthogonality of the Cross Product Proved by the Levi-Civita Symbol and Index Notation
15. Surface Parametrisation and the Verification of the Gradient-Normal Relationship
16. Proof and Implications of a Vector Operator Identity
17. Conditions for a Scalar Field Identity
18. Solution and Proof for a Vector Identity and Divergence Problem
19. Kinematics and Vector Calculus of a Rotating Rigid Body
20. Work Done by a Non-Conservative Force and Conservative Force
21. The Lorentz Force and the Principle of Zero Work Done by a Magnetic Field
22. Calculating the Area of a Half-Sphere Using Cylindrical Coordinates
23. Divergence Theorem Analysis of a Vector Field with Power-Law Components
24. Total Mass in a Cube vs. a Sphere
25. Momentum of a Divergence-Free Fluid in a Cubic Domain
26. Total Mass Flux Through Cylindrical Surfaces
27. Analysis of Forces and Torques on a Current Loop in a Uniform Magnetic Field
28. Computing the Integral of a Static Electromagnetic Field
29. Surface Integral to Volume Integral Conversion Using the Divergence Theorem
30. Circulation Integral vs. Surface Integral
31. Using Stokes' Theorem with a Constant Scalar Field
32. Verification of the Divergence Theorem for a Rotating Fluid Flow
33. Integral of a Curl-Free Vector Field
34. Boundary-Driven Cancellation in Vector Field Integrals
35. The Vanishing Curl Integral
36. Proving the Generalized Curl Theorem
37. Computing the Magnetic Field and its Curl from a Dipole Vector Potential
38. Proving Contravariant Vector Components Using the Dual Basis
39. Verification of Orthogonal Tangent Vector Bases in Cylindrical and Spherical Coordinates
40. Vector Field Analysis in Cylindrical Coordinates
41. Vector Field Singularities and Stokes' Theorem
42. Compute Parabolic coordinates-related properties
43. Analyze Flux and Laplacian of The Yukawa Potential
44. Verification of Vector Calculus Identities in Different Coordinate Systems
45. Analysis of a Divergence-Free Vector Field
46. The Uniqueness Theorem for Vector Fields
47. Analysis of Electric Dipole Force Field

🧄Proof and Derivation-2

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