# What is a covariant derivative

1. Jul 23, 2014

### Greg Bernhardt

Definition/Summary

Covariant derivative, $D$, is a coordinate-dependent adjustment to ordinary derivative which makes each partial derivative of each coordinate unit vector zero: $D\hat{\mathbf{e}}_i/\partial x_j\ =\ 0$

The adjustment is made by a linear operator known both as the connection, $\Gamma^i_{\ jk}$, and as the Christoffel symbol, $\{^{\ i\ }_{j\ k}\}$.

Covariant derivative of the metric ($g_{ij}$) is zero.

Covariant derivative (unlike ordinary derivative) of any tensor is also a tensor (with an extra covariant index).

A vector $V^i$ is parallel transported along a curve $x^i(s)$ with tangent $T^i(s)$ if its covariant directional derivative in the direction of that tangent is zero: $(T.D)V^i/ds\ =\ 0$

Equations

$$\frac{DV^i}{\partial x^j}\left(\text{also written}\ \frac{DV^i}{Dx^j}\ \text{or}\ \nabla_jV^i\ \text{or}\ V^i_{\ ;\,j}\right)\ =\ \frac{\partial V^i}{\partial x^j}\ -\ \Gamma^i_{\ jk}V^k$$

$$\frac{D\,(a^i\hat{\mathbf{e}}_i)}{\partial x^j}\ =\ 0\ \text{for constant}\ a^i$$

$$\frac{Dg^{ij}}{\partial x^k}\ =\ 0$$

$$\Gamma^i_{\ jk}\ =\ \Gamma^i_{\ kj}$$

$$\Gamma^i_{\ jk}\ =\ \{^{\ i\ }_{j\ k}\}\ =\ \frac{1}{2}\,g^{il}\left(\frac{\partial g_{jl}}{\partial x^k}\ +\ \frac{\partial g_{kl}}{\partial x^j}\ -\ \frac{\partial g_{jk}}{\partial x^l}\right)$$

Parallel transport of a vector $V^i$ along a curve $x^i(s)$:

$$\frac{dx^j}{ds}\frac{DV^i}{\partial x^j}\ =\ 0\ \ \text{or}\ \ dV^i = -\Gamma^i_{\ jk}V^kdx^j$$

Geodesic deviation equation:

$$\frac{D^2\,\delta x^{\alpha}}{d\tau^2}\ =\ -\,R^{\alpha}_{\ \mu\beta\sigma}\,V^{\mu}\,V^{\sigma}\,\delta x^{\beta}$$

Extended explanation

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