The fundamental difference between the two expressions lies in their transformation behavior: the simple partial derivative ($\partial_a v^b$) is not a tensor because its transformation rule includes an extra, "inhomogeneous" term involving the second partial derivative of the coordinate transformation. This term means the derivative's components depend on how the coordinate system is curved or accelerated, violating the principle of physical invariance. The covariant derivative ($\nabla_a v^b$) solves this problem by adding the Christoffel symbol correction ($\Gamma_{a c}^b v^c$). The transformation rule for the Christoffel symbols contains a term that is mathematically designed to exactly cancel the non-tensorial, second-derivative term present in the partial derivative's transformation, ensuring that the resulting expression, $\nabla_a v^b$, obeys the pure, homogeneous tensor transformation rule and represents a physically invariant quantity.

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✍️Mathematical Proof

$\complement\cdots$Counselor

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• Partial Derivative is Not a Tensor: The simple partial derivative, $\partial_a v^b$, does not transform as a tensor because its transformation rule contains an extra, "inhomogeneous" term with a second partial derivative of the coordinate transformation. This term makes the derivative's components dependent on the specific coordinate system used, violating the principle of physical invariance.
• The Covariant Derivative as a Correction: The covariant derivative, $\nabla_a v^b$, is specifically defined to correct this non-tensorial behavior. It adds the Christoffel symbol term ( $\Gamma_{a c}^b v^c$ ), which also has a transformation rule containing a second derivative of the coordinate transformation.
• Invariance Through Cancellation: The key insight is that the non-tensorial term from the Christoffel symbols exactly cancels the problematic term in the partial derivative's transformation rule. This cancellation leaves only the pure, homogeneous tensor transformation, confirming that $\nabla_a v^b$ is a true tensor. This ensures that the physical object it represents is independent of the chosen coordinate system.

✍️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. ‣

🧄Proof and Derivation-1

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