🧄Tensor vs. Non-Tensor Transformation of Derivatives

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.

Tensor vs. Non-Tensor Transformation of Derivatives | Proof and Derivationviadean on Notion