🎬Directional Derivative equals Covariant Derivative for any scalar field

For a scalar field, the covariant derivative is identical to the familiar directional derivative. The reason is that scalar fields, being simple values without directional components, do not change based on the coordinate system. Unlike vectors or tensors, they are unaffected by the curvature of space. The key takeaway is that the more complex machinery of the covariant derivative simplifies to the basic directional derivative when acting on a scalar, highlighting their simple, un-curved nature.

The Metric Tensor Covariant Derivatives and Tensor Densities | Condensed Notesviadean on Notion