---
tag: 5FZDQQHQKM
kind: concept
name: Covariant derivative
msc: [53B05, 53C05]
sources:
  - url: https://en.wikipedia.org/wiki/Covariant_derivative
same_as:
  wikidata: Q2287715
  wikipedia: Covariant derivative
---

## Summary

A covariant derivative differentiates a vector field in the direction of a tangent vector, and the result is again a tangent vector.
It applies more generally to sections of a vector bundle and to tensor fields.
It is the operator $\nabla_X$ that a [[connection|5ZYRRVE8BW]] provides.
Unlike partial derivatives of components, it does not depend on the choice of coordinates.
It also differentiates vector fields along curves, which leads to parallel transport and geodesics.

## Explanations

### L1 human

The partial derivatives of the components of a vector field change when you change coordinates in a non-linear way.
The covariant derivative corrects them by a term that accounts for how the coordinate axes themselves bend.
For a car driving on a sphere, the covariant derivative of its velocity is the part of the acceleration that is tangent to the sphere.
A curve with zero covariant acceleration is as straight as possible on the surface; on a sphere these are the great circles.