🧄Verification of Covariant Derivative Identities

The verification of these identities confirms three core operational aspects of the covariant derivative: first, the $\nabla_a$ 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 $v^b w_b$ ), the simple partial derivative ( $\partial_a$ ) becomes equivalent to the covariant derivative ( $\nabla_a$ ) because the Christoffel symbol corrections naturally cancel out. Finally, identity (c) verifies the principle of Metric Compatibility ( $\nabla_a g^{a b}=0$ ), proving that the divergence of a contravariant vector ( $\nabla_a v^a$ ) is equal to the divergence of its dual form ( $g^{a b} \nabla_a v_b$ ), 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.

Verification of Covariant Derivative Identities | Proof and Derivationviadean on Notion