 TOPICS # 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). The tangent map corresponds to differentiation by the formula (1)

where (i.e., is a curve passing through the base point to in 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.

Diffeomorphism

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## References

Gray, A. "Tangent Maps." §11.3 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 250-255, 1997.

Tangent Map

## Cite this as:

Weisstein, Eric W. "Tangent Map." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TangentMap.html