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

Famous quotes containing the word examples:

    There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.
    Bernard Mandeville (1670–1733)

    Histories are more full of examples of the fidelity of dogs than of friends.
    Alexander Pope (1688–1744)

    No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.
    André Breton (1896–1966)