This is the demonstration of the dual anti-symmetry of the generalised Kronecker delta $\delta_{b_1 . . b_n}^{a_1 . . a_n}$. Defined as a determinant, it is completely anti-symmetric in both its contravariant (upper) indices and its covariant (lower) indices. The derived relation, $\delta_{b_1 \ldots b_n}^{a_1 \ldots a_n}= n!\delta_{\left[b_1\right.}^{a_1} \ldots \delta_{b_n}^{a_n}$, confirms that anti-symmetrising the covariant indices of the simple Kronecker delta product $\delta_{b_1}^{a_1} \ldots \delta_{b_n}^{a_n}$ yields the generalised delta, mirroring the given result for contravariant index anti-symmetrisation. Crucially, the factor of $n!$ appears because it is needed to cancel the $\frac{1}{n!}$ factor inherently present in the definition of the anti-symmetrisation operator, resulting in the determinant definition.

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✍️Mathematical Proof

$\complement\cdots$Counselor

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• Generalised Kronecker Delta Definition: The generalised Kronecker delta, $\delta_{b_1 \ldots b_n}^{a_1 \ldots a_n}$, is defined as the determinant of the matrix formed by the simple Kronecker deltas $\delta_{b_i}^{a_j}$.
• Symmetry Property: The generalised Kronecker delta is completely anti-symmetric in both its contravariant indices ( $\left\{a_1, \ldots, a_n\right\}$ ) and its covariant indices ( $\left\{b_1, \ldots, b_n\right\}$ ).
• Relationship to Anti-symmetrisation: The relation shown, $\delta_{b_1 \ldots b_n}^{a_1 \ldots a_n}=n!\delta_{\left[b_1\right.}^{a_1} \ldots \delta_{\left.b_n\right]^{\prime}}^{a_n}$ confirms that the anti-symmetrisation of the covariant indices of the product of simple deltas $\left(\delta_{b_1}^{a_1} \ldots \delta_{b_n}^{a_n}\right)$ yields the generalised delta, multiplied by a factor of $n!$.
• Equivalence to Contravariant Anti-symmetrisation: This result is formally identical to the given relation where the anti-symmetrisation is applied to the contravariant indices $\left(\delta_{b_1 \ldots b_n}^{a_1 \ldots a_n}=n!\delta_{b_1}^{\left[a_1\right.} \ldots \delta_{b_n}^{\left.a_n\right]}\right)$, demonstrating the dual anti-symmetry property.
• Role of $n!$ Factor: The factor $n!$ in the result exactly cancels the $\frac{1}{n!}$ factor present in the definition of the anti-symmetrisation operator, $\delta_{\left[b_1\right.}^{a_1} \ldots \delta_{\left.b_n\right]}^{a_n}$.

✍️Mathematical Proof

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1. Derivation of Tensor Transformation Properties for Mixed Tensors
2. The Polar Tensor Basis in Cartesian Form
3. Verifying the Rank Two Zero Tensor
4. Tensor Analysis of Electric Susceptibility in Anisotropic Media
5. Analysis of Ohm's Law in an Anisotropic Medium
6. Verifying Tensor Transformations
7. Proof of Coordinate Independence of Tensor Contraction
8. Proof of a Tensor's Invariance Property
9. Proving Symmetry of a Rank-2 Tensor
10. Tensor Symmetrization and Anti-Symmetrization Properties
11. Symmetric and Antisymmetric Tensor Contractions
12. The Uniqueness of the Zero Tensor under Specific Symmetry Constraints
13. Counting Independent Tensor Components Based on Symmetry
14. Transformation of the Inverse Metric Tensor
15. Finding the Covariant Components of a Magnetic Field
16. Covariant Nature of the Gradient
17. Christoffel Symbol Transformation Rule Derivation
18. Contraction of the Christoffel Symbols and the Metric Determinant
19. Divergence of an Antisymmetric Tensor in Terms of the Metric Determinant
20. Calculation of the Metric Tensor and Christoffel Symbols in Spherical Coordinates
21. Christoffel Symbols for Cylindrical Coordinates
22. Finding Arc Length and Curve Length in Spherical Coordinates
23. Solving for Metric Tensors and Christoffel Symbols
24. Metric Tensor and Line Element in Non-Orthogonal Coordinates
25. Tensor vs. Non-Tensor Transformation of Derivatives
26. Verification of Covariant Derivative Identities
27. Divergence in Spherical Coordinates Derivation and Verification
28. Laplace Operator Derivation and Verification in Cylindrical Coordinates
29. Divergence of Tangent Basis Vectors in Curvilinear Coordinates
30. Derivation of the Laplacian Operator in General Curvilinear Coordinates
31. Verification of Tensor Density Operations
32. Verification of the Product Rule for Jacobian Determinants and Tensor Density Transformation
33. Metric Determinant and Cross Product in Scaled Coordinates
34. Vanishing Divergence of the Levi-Civita Tensor
35. Curl and Vector Cross-Product Identity in General Coordinates
36. Curl of the Dual Basis in Cylindrical and Spherical Coordinates
37. Proof of Covariant Index Anti-Symmetrisation

🧄Proof and Derivation-1

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