“Tangent Map”的概念、定义、习题解题方法及翻译-数学辞典

单词

Tangent Map

释义

Tangent Map

If , then the tangent map associated to is a Vector Bundle Homeomorphism (i.e., a Map between the Tangent Bundles of and respectively). Thetangent map corresponds to Differentiation by the formula

(1)

where

(i.e.,

is a curve passing through the base point to

at time 0 with velocity

).In this case, if

and

, then the Chain Rule is expressed as

(2)

In other words, with this way of formalizing differentiation, the Chain Rule can be remembered by saying that``the process of taking the tangent map of a map is functorial.'' To a topologist, the form

(3)

for all

, is more intuitive than the usual form of the Chain Rule.

See also Diffeomorphism

References

Gray, A. ``Tangent Maps.'' §9.3 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 168-171, 1993.

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