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Date: 26-9-2016
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Date: 28-7-2021
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Date: 5-5-2021
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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
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(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
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(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
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(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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دراسة: عدم ترتيب الغرفة قد يدل على مشاكل نفسية
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علماء: تغير المناخ تسبب في ارتفاع الحرارة خلال موسم الحج
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شعبة فاطمة بنت أسد للدراسات القرآنية تختتم دورة ينابيع الرحمة
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