Differential geometry MOC

Differential form

Let be a -manifold. A differential -form is a totally contravariant, totally antisymmetric tensor field. and a general differential form is a direct sum of these.

As a section

The above is equivalent to a -section of the th exterior power of the cotangent bundle, i.e.

A general (non-homogenous) differential form is a -section of the exterior algebra bundle

Exterior product

The exterior algebra bundle induces the exterior product of differential forms, so that

which acts on vector fields as

Local coördinates

If are a chart then locally a tensor field is given by

and if is antisymmetric,

where

and is the Exterior derivative.

Related concepts

• Exterior derivative
• Volume form
• Hodge dual

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