---
tag: FZBA1HA9ZM
kind: concept
name: Christoffel symbols
msc: [53B05, 53B20]
sources:
  - url: https://en.wikipedia.org/wiki/Christoffel_symbols
same_as:
  wikidata: Q847816
  wikipedia: Christoffel symbols
---

## Summary

The Christoffel symbols $\Gamma^k_{ij}$ are the components of an [[affine connection|JX65KSVHSJ]] in a coordinate chart.
They record how the covariant derivative of one coordinate vector field along another fails to vanish.
For the Levi-Civita connection of a Riemannian metric they are given by an explicit formula in the first derivatives of the metric.
They appear in the geodesic equation and in coordinate formulas for curvature.

## Explanations

### L1 human

In curvilinear coordinates, such as polar coordinates in the plane, the coordinate directions change from point to point.
Even a constant vector field then has components that vary.
The Christoffel symbols are the correction numbers that account for this turning of the coordinate axes.
In Cartesian coordinates on the flat plane they are all zero, while in polar coordinates some of them are not, for example $\Gamma^r_{\theta\theta} = -r$.