Editing Integral curve (section)

===Remarks on the time derivative===

In the above, {{math|''α''&prime;(''t'')}} denotes the derivative of {{math|''α''}} at time {{math|''t''}}, the "direction {{math|''α''}} is pointing" at time {{math|''t''}}. From a more abstract viewpoint, this is the [[Fréchet derivative]]:

<math display="block">(\mathrm{d}_t\alpha) (+1) \in \mathrm{T}_{\alpha (t)} M.</math>

In the special case that {{math|''M''}} is some [[open subset]] of {{math|'''R'''<sup>''n''</sup>}}, this is the familiar derivative

<math display="block">\left( \frac{\mathrm{d} \alpha_1}{\mathrm{d} t}, \dots, \frac{\mathrm{d} \alpha_n}{\mathrm{d} t} \right),</math>

where {{math|''α''<sub>1</sub>, ..., ''α''<sub>''n''</sub>}} are the coordinates for {{math|''α''}} with respect to the usual coordinate directions.

The same thing may be phrased even more abstractly in terms of [[induced homomorphism|induced maps]]. Note that the tangent bundle {{math|T''J''}} of {{math|''J''}} is the [[Fiber bundle#Trivial bundle|trivial bundle]] {{math|''J'' × '''R'''}} and there is a [[canonical form|canonical]] cross-section {{math|''ι''}} of this bundle such that {{math|1=''ι''(''t'') = 1}} (or, more precisely, {{math|(''t'', 1) ∈ ''ι''}}) for all {{math|''t'' ∈ ''J''}}. The curve {{math|''α''}} induces a [[bundle map]] {{math|''α''<sub>∗</sub> : T''J'' → T''M''}} so that the following diagram commutes:

:[[Image:CommDiag TJtoTM.png]]

Then the time derivative {{math|''α''&prime;}} is the [[function composition|composition]] {{math|1=''α''&prime; = ''α''<sub>∗</sub> <small>o</small> ''ι'', and ''α''&prime;(''t'')}} is its value at some point&nbsp;{{math|''t'' ∈ ''J''}}.