Differential form

Exterior derivative

The exterior derivative generalizes the concept of differential to general differential forms on a -manifold . For , we have a map

Note that to any 0-form, i.e. continuous function , we can naturally associate a 1-form by the

The general exterior derivative is then the unique extension of this operation to a graded derivation such that , i.e. if and we have

Proof of existence and uniqueness

proof

Local coördinates

Let be a chart. Then for

we have

This can be seen as a special case of From a covariant derivative.

From a covariant derivative

Let denote a torsion-free affine connexion on . Then for , the covariant derivative gives

which is independent of the choice of .¹

Proof

proof

See also

• Covariant derivative

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Footnotes

1. 2009. General relativity, §B.1, pp. 428–429. ↩