The tangent map associated with a differentiable map is the bundle map induced by between the tangent bundles of and , respectively. The tangent map corresponds to differentiation by the formula

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

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

for all , is more intuitive than the usual form of the chain rule.

See also

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.

Referenced on Wolfram|Alpha

Tangent Map

Cite this as:

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

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