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.

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.