Tangent Map
المؤلف:
Gray, A.
المصدر:
"Tangent Maps." §11.3 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press
الجزء والصفحة:
pp. 250-255
29-5-2021
2107
Tangent Map
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.
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