📢Core Properties of the Covariant Derivative

The three verified identities confirm that the covariant derivative is the mathematically robust replacement for the simple partial derivative in general coordinate systems. Identity (a) confirms that the covariant derivative 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 is only equivalent to the covariant derivative in this one specific, contracted scenario. Finally, Identity (c) verifies the principle of Metric Compatibility, 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.

Verification of Covariant Derivative Identities | Proof and Derivationviadean on Notion