The three verified identities confirm that the covariant derivative ( $\nabla_a$ ) is the mathematically robust replacement for the simple partial derivative in general coordinate systems. Identity (a) confirms that $\nabla _{ a }$ rigorously obeys the Leibniz Product Rule for all tensor products, ensuring algebraic consistency. Identity (b) shows that when differentiating a scalar quantity, the complex Christoffel corrections naturally cancel out, meaning the simple partial derivative ( $\partial_a$ ) is only equivalent to $\nabla _{ a }$ in this one specific, contracted scenario. Finally, Identity (c) verifies the principle of Metric Compatibility ( $\nabla_a g^{a b}=0$ ), proving that the metric tensor is parallel transported and guarantees that raising or lowering a vector's indices using the metric can be done either before or after differentiation without changing the physical result.

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✍️Mathematical Proof

$\complement\cdots$Counselor

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• The Covariant Derivative Obeys the Product Rule (Leibniz Rule): Identity (a), $\nabla_a\left(v^b T^{c d}\right)=v^b \nabla_a T^{c d}+T^{c d} \nabla_a v^b$, explicitly confirms that the covariant derivative acts on products of tensor fields just like the standard derivative in calculus. The verification involved expanding the definition of the covariant derivative for the product and regrouping the resulting terms based on the definitions of $\nabla_a v^b$ and $\nabla_a T^{c d}$.
• The Partial Derivative of a Scalar is Trivial (Identity b): Identity (b), $\partial_a\left(v^b w_b\right)= v^b \nabla_a w_b+w_b \nabla_a v^b$, shows that when taking the derivative of a scalar quantity (a quantity with all indices contracted, like ($v^b w_b$), the $\partial_a$ operator can be freely replaced by the $\nabla_a$ operator. Crucially, the non-tensorial Christoffel terms cancel each other out on the righthand side, leaving the partial derivative equal to the covariant product rule, thereby highlighting that the simple partial derivative is equivalent to the covariant derivative only when operating on scalars.
• Metric Compatibility is Essential (Identity c): Identity (c), $\nabla_a v^a=g^{a b} \nabla_a v_b$, demonstrates the direct consequence of metric compatibility, the defining property that the covariant derivative of the metric tensor is zero ($\nabla_a g^{a b}=0$). By starting with the tensor identity $v^a=g^{a b} v_b$ and applying $\nabla_a$ to both sides, the identity is verified because the term containing $\nabla_a g^{a b}$ vanishes, proving that raising or lowering indices using the metric commutes with the covariant derivative.

✍️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

🧄Proof and Derivation-1

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