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.
See also Diffeomorphism
References
Gray, A. ``Tangent Maps.'' §9.3 in Modern Differential Geometry of Curves and Surfaces.
Boca Raton, FL: CRC Press, pp. 168171, 1993.
© 19969 Eric W. Weisstein
19990526