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