# Derivative Rules

## Notation

$g_{ab}$ is the metric, and $\Gamma^a_{bc} = \frac{1}{2} g^{ad} ( \partial_b g_{dc} + \partial_c g_{bd} - \partial_d g_{bc})$ are the Christoffel symbols.

$\nabla_a$ is the covariant derivative, and $\partial_a$ is the partial derivative with respect to $x^a$.

$S$ is a scalar.

$V^c$ is a contravariant vector.

$W_c$ is a covariant vector.

$\rho$ is a scalar density of weight 1, and $\rho^w$ is a scalar density of weight $w$. (Note that $\sqrt{|g|}$ is a density of weight 1, where $g$ is the determinant of the metric. It follows that $\rho/\sqrt{|g|}$ is a scalar.)

## Covariant Derivatives

Covariant derivatives are defined by

$\begin{array}{rcl} \nabla_a S & = & \partial_a S \\ \nabla_a V^c & = & \partial_a V^c + \Gamma^c_{ab} V^b \\ \nabla_a W_c & = & \partial_a W_c - \Gamma^b_{ac} W_b \\ \nabla_a \rho & = & \partial_a \rho - \rho \Gamma^c_{ac} \\ & = & \sqrt{|g|} \,\partial_a (\rho/\sqrt{|g|}) \\ \nabla_a \rho^w & = & \partial_a \rho^w - w \rho^w \Gamma^c_{ac} \\ & = & \sqrt{|g|}^w \,\partial_a (\rho^w/\sqrt{|g|}^w) \end{array}$

The extension to tensors of different type is straightforward. Note the useful result for the divergence of a vector field:

$\sqrt{|g|} \nabla_a V^a = \partial_a (\sqrt{|g|} V^a)$

## Lie Derivatives

Lie derivatives with respect to a vector field $\beta^a$ are defined by

$\begin{array}{rcl} {\mathcal L}_\beta S & = & \beta^a \partial_a S \\ {\mathcal L}_\beta V^c & = & \beta^a \partial_a V^c - V^a \partial_a \beta^c \\ {\mathcal L}_\beta W_c & = & \beta^a \partial_a W_c + W_a \partial_c \beta^a \\ {\mathcal L}_\beta \rho & = & \partial_a (\rho \beta^a) \\ {\mathcal L}_\beta \rho^w & = & \beta^a \partial_a\rho^w + w \rho^w \partial_a\beta^a \\ \end{array}$

The extension to tensors of different type is straightforward. Lie derivatives do not rely on the presence of a metric for their definitions. When a metric is present, they can be written in terms of covariant derivatives as follows:

$\begin{array}{rcl} {\mathcal L}_\beta S & = & \beta^a \nabla_a S \\ {\mathcal L}_\beta V^c & = & \beta^a \nabla_a V^c - V^a \nabla_a \beta^c \\ {\mathcal L}_\beta W_c & = & \beta^a \nabla_a W_c + W_a \nabla_c \beta^a \\ {\mathcal L}_\beta \rho & = & \nabla_a (\rho\beta^a) \\ {\mathcal L}_\beta \rho^w & = & \beta^a \nabla_a\rho^w + w \rho^w \nabla_a\beta^a \end{array}$

Note the useful result for the Lie derivative of the metric tensor:

${\mathcal L}_\beta g_{ab} = \nabla_a\beta_b + \nabla_b \beta_a$