🎬how the partial and covariant derivatives behave in a polar coordinate system

The green vector in the demo is a truly constant physical object (its Cartesian components are fixed), its representation in the polar coordinate system is constantly changing. The covariant derivative correctly captures this change, demonstrating that a derivative must account for both the change in a vector's components and the change in the basis vectors themselves. This distinction is fundamental in fields like general relativity, where coordinates are often not fixed. The visualization proves that if a vector field is constant in one coordinate system, its covariant derivative is not necessarily zero in another.

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