🧄Verification of Covariant Derivative Identities

The verification of these identities confirms three core operational aspects of the covariant derivative: first, the a\nabla_a operator rigorously obeys the Leibniz Product Rule for all tensor products, as shown in identity (a). Second, identity (b) demonstrates that when the derivative is applied to a scalar quantity (where all indices are contracted, like vbwbv^b w_b ), the simple partial derivative ( a\partial_a ) becomes equivalent to the covariant derivative ( a\nabla_a ) because the Christoffel symbol corrections naturally cancel out. Finally, identity (c) verifies the principle of Metric Compatibility ( agab=0\nabla_a g^{a b}=0 ), proving that the divergence of a contravariant vector ( ava\nabla_a v^a ) is equal to the divergence of its dual form ( gabavbg^{a b} \nabla_a v_b ), confirming that the metric tensor is parallel transported and can be used to raise or lower indices before or after differentiation without affecting the result.

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