🎬the gradient of a scalar field is a covariant vector

The gradient of a scalar is a covector (1-form): its components change under coordinate transforms as $\alpha^{\prime}_a = \frac{\partial x^b}{\partial y^a}\alpha_b$. Despite the changing components, the underlying geometric gradient (its action on vectors / reconstructed physical vectors) is invariant - components change to preserve the same geometry.

Covariant Nature of the Gradient | Proof and Derivationviadean on Notion