Tangent Map

If f:M->N, then the tangent map Tf associated to f is a vector bundle homeomorphism Tf:TM->TN (i.e., a map between the tangent bundles of M and N respectively). The tangent map corresponds to differentiation by the formula

 Tf(v)=(f degreesphi)^'(0),

where phi^'(0)=v (i.e., phi is a curve passing through the base point to v in TM at time 0 with velocity v). In this case, if f:M->N and g:N->O, then the chain rule is expressed as

 T(f degreesg)=Tf degreesTg.

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

 (f degreesg)^'(a)=f^'(g(a)) degreesg^'(a),

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

See also


Explore with Wolfram|Alpha


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 Web Resource.

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