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,
The latter can be shown by taking (without loss of generality) that .
Read more about this topic: Covariant Derivative
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Famous quotes containing the word examples:
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (15331592)
“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 (16701733)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (18961966)