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.
