Examples
For a scalar field, covariant differentiation is simply partial differentiation:
For a contravariant vector field, we have:
For a covariant vector field, we have:
For a type (2,0) tensor field, we have:
For a type (0,2) tensor field, we have:
For a type (1,1) tensor field, we have:
The notation above is meant in the sense
One must always remember that covariant derivatives do not commute, i.e. . It is actually easy to show that:
where is the Riemann tensor. Similarly,
and
The latter can be shown by taking (without loss of generality) that .
Read more about this topic: Covariant Derivative
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