Covariant derivation

D(expr,$-$I,...,$-$N) $\Rightarrow$

The operator of covariant derivation is denoted by D. The same rules are valid for this as for the partial derivative, however the covariant derivatives with respect to upper indices are not converted to lower indices automatically and the covariant derivative of the metric tensor and that of the the Levi-Civita tensor are equal to zero.

The non-operator notation for the partial and covariant derivatives, which is frequently used in the literature, is applied in the output as shown by the examples above. In this convention the partial derivative is denoted by a comma, the covariant derivative by a semicolon.

_i  T^&ldots;_\>\>&ldots; = T^&ldots;_{\>\>&ldots;  ,  i} d_i  T^&ldots;_\>\>&ldots; = T^&ldots;_{\>\>&ldots;  ;  i}

The notation is applied in the output of the calculations if the switch NAT of REDUCE is ON (default setting).