Smoooth vector field: Difference between revisions
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==Definition== | ==Definition== | ||
A '''vector field''' on a [[differential manifold]] can be defined in any of the following equivalent ways: | A '''smooth vector field''' on a [[differential manifold]] can be defined in any of the following equivalent ways: | ||
* It is a [[derivation]] from the algebra of <math>C^\infty</math> functions on the manifold, to itself | * It is a [[derivation]] from the algebra of <math>C^\infty</math> functions on the manifold, to itself | ||
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==Related notions== | ==Related notions== | ||
* [[Continuous vector field]] | |||
* [[Tensor field]] | * [[Tensor field]] | ||
* [[Differential form]] | * [[Differential form]] | ||
* [[Tensor of mixed type]] | * [[Tensor of mixed type]] | ||
Revision as of 19:26, 28 February 2008
Definition
A smooth vector field on a differential manifold can be defined in any of the following equivalent ways:
- It is a derivation from the algebra of functions on the manifold, to itself
- It is a section of the tangent bundle
- it is a rule that associates (smoothly) to every point in the manifold a tangent vector