Covariant Derivative - Examples

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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