is the metric, and are the Christoffel symbols.
is the covariant derivative, and is the partial derivative with respect to .
is a scalar.
is a contravariant vector.
is a covariant vector.
is a scalar density of weight 1, and is a scalar density of weight . (Note that is a density of weight 1, where is the determinant of the metric. It follows that is a scalar.)
Covariant derivatives are defined by
The extension to tensors of different type is straightforward. Note the useful result for the divergence of a vector field:
Lie derivatives with respect to a vector field are defined by
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:
Note the useful result for the Lie derivative of the metric tensor: