Covariant derivative motivation quick Q Concept

As we can not meaningfully compare a vector at 2 points acted upon by this operator , because it does not take into account the change due to the coordinate system constantly changing, I conclude that the elementary differential operator must describe a change with respect to space-time,

How do we know that the elementary differential operator does this, instead of describing a change wrt a coordinate system?

Matterwave
Gold Member
As we can not meaningfully compare a vector at 2 points acted upon by this operator , because it does not take into account the change due to the coordinate system constantly changing, I conclude that the elementary differential operator must describe a change with respect to space-time,

What do you mean by "elementary differential operator"? I do not think this is standard terminology. Also, how do you draw this conclusion? I don't see how "must describe a change with respect to space-time" follows from "can not compare a vector at 2 points acted upon by this operator". In general, there is no way to compare a vector at 2 different points on a manifold without defining either a connection plus a path, or a congruence.

oh, sorry,it's the other way around. so the elementary partial derivative gives the change with respect to coordinate system, and the aim of the covariant derivative is to give the change with respect to space-time?

Matterwave