Derivative Rules

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[edit] 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.

D_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.)

[edit] Covariant Derivatives

Covariant derivatives are defined by

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

[edit] 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 D_a S \\ {\mathcal L}_\beta V^c & = & \beta^a D_a V^c - V^a D_a \beta^c \\ {\mathcal L}_\beta W_c & = & \beta^a D_a W_c + W_a D_c \beta^a \\ {\mathcal L}_\beta \rho & = & D_a (\rho\beta^a) \\ {\mathcal L}_\beta \rho^w & = & \beta^a D_a\rho^w + w \rho^w D_a\beta^a \end{array}

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

{\mathcal L}_\beta g_{ab} = D_a\beta_b + D_b \beta_a
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