The totally antisymmetric tensor, $\eta^{ a _1 \ldots a _{ N }}=\varepsilon^{ a 1 \ldots a { N }} / \sqrt{ g }$, is a true tensor (weight $w=0$ ) formed by dividing the Levi-Civita symbol by $\sqrt{g}$. Its divergence vanishes identically ( $\nabla { aN } \eta^{ a 1 \ldots a { N }}= 0$ ) because it is covariantly constant ( $\nabla_b \eta^{a_1 \ldots a_N}=0$ ), a fundamental property of the Levi-Civita connection that preserves the volume element. The explicit proof requires recognizing the identity $\sum{i=1}^N \Gamma{a_N c}^{a_i} \eta^{a_1 \ldots c_1 \ldots a_N}=\Gamma{a{N c}}^c \eta^{a_1 \ldots a_N}$, which, combined with the hint $\Gamma{ ab }^{ b }= \delta { a } \ln (\sqrt{ g })$, demonstrates that the two non-vanishing terms in the covariant derivative ( $\partial{a_N} \eta$ and $\Gamma \eta$ ) perfectly cancel each other out.

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✍️Mathematical Proof

$\complement\cdots$Counselor

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1. Nature of the Tensor : The quantity $\eta^{a_1 \ldots a_N}=\varepsilon^{a_1 \ldots a_N} / \sqrt{g}$ is the Levi-Civita tensor (often denoted $\epsilon$ ). It's formed by dividing the Levi-Civita symbol ( $\varepsilon$, a tensor density of weight +1 ) by $\sqrt{g}$ (a scalar density of weight +1 ). This makes $\eta$ a true tensor (a tensor density of weight $w=0$ ).
2. Covariant Constancy : The Levi-Civita tensor $\eta^{\alpha_1 \ldots a_N}$ is covariantly constant in any Riemannian manifold, meaning its covariant derivative vanishes identically: $\nabla _{ b } \eta^{ a _1 \ldots a _{ N }} = 0$. This is a fundamental property related to the metric compatibility of the connection.
3. Divergence is a Contraction: The divergence of $\eta$ is a contraction of its covariant derivative ( $\nabla_{a_N} \eta^{a_1 \ldots a_N}$ ). Since the full covariant derivative is zero, the divergence must also be zero: $\nabla {\text {an }} \eta^{a_1 \ldots a{ N }}= 0$.
4. General Principle : The identity $\nabla_{a_N} \eta^{a_1 \ldots a_N}=0$ is the geometric statement that the volume element is preserved under parallel transport, which is consistent with using a metric-compatible connection (the Levi-Civita connection).

✍️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

🧄Proof and Derivation-1

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