Hello, you have come here looking for the meaning of the word
tangent space. In DICTIOUS you will not only get to know all the dictionary meanings for the word
tangent space, but we will also tell you about its etymology, its characteristics and you will know how to say
tangent space in singular and plural. Everything you need to know about the word
tangent space you have here. The definition of the word
tangent space will help you to be more precise and correct when speaking or writing your texts. Knowing the definition of
tangent space, as well as those of other words, enriches your vocabulary and provides you with more and better linguistic resources.
English
Noun
tangent space (plural tangent spaces)
- (topology, differential topology) An n-dimensional vector space that represents the set of all vectors tangent to given n-dimensional differentiable manifold M at point x.
2001, Stephen Semmes, Some Novel Types of Fractal Geometry, Oxford University Press, page 124:This gives a degenerate norm on the tangent space of Hn at the origin, and one can use left translations to define similar norms at the tangent spaces of Hn at all other points.
2011, Loring W. Tu, An Introduction to Manifolds, 2nd edition, Springer, page 178:In a Lie group G, because left translation by an element g G is a diffeomorphism that maps a neighborhood of the identity to a neighborhood of g, all the local information about the group is concentrated in a neighborhood of the identity, and the tangent space of the identity assumes a special importance.
Moreover, one can give the tangent space TeG a Lie bracket [ , ], so that in addition to being a vector space, it becomes a Lie algebra, called the Lie algebra of the Lie group.
2014, Peter K. Friz, Martin Hairer, A Course on Rough Paths: With an Introduction to Regularity Structures, Springer, page 7:More dramatically, there are situations where one has a sequence of smooth manifolds such that the limit is again a smooth manifold, but with a limiting “tangent space” which has nothing to do with the actual tangent space of the limit!
Usage notes
Typically denoted TxM, although the manifold may be omitted if understood.
More strictly, for any smooth curve in M passing through x, its derivative at x is a vector in TxM.
Holonyms
Related terms
Translations
vector space tangent to a manifold
Further reading