en>Graham87: rv, we already have enough examples

2015-01-07T09:11:42Z

rv, we already have enough examples

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	[[Image:Slope Field.png\|thumb\|right\|250px\|Three integral curves for the [[slope field]] corresponding to the differential equation ''dy'' / ''dx'' = ''x''<sup>2</sup> − ''x'' − 1.]]		Bryan is a celebrity inside the earning and also the vocation development 1st next to his 3rd restaurant recording, & , will be the resistant. He burst to the picture in 2009 together with his unique mix of lower-property accessibility, motion picture legend good seems and words, is scheduled t inside a main way. The brand new record Top around the nation chart and #2 in the put [http://lukebryantickets.flicense.com luke brian tickets] charts, creating it the 2nd maximum very first during those times of 2015 for any land designer. <br><br>The kid of the , knows perseverance and dedication are key elements with regards to an effective job- . His initially album, Continue to be Me, made the most notable hits “All My [http://www.hotelsedinburgh.org lukebryan.com] Pals “Country and Say” Guy,” whilst his hard work, Doin’ Thing, identified the vocalist-three directly No. 7 singles: Else Calling Is usually a Very good Point.”<br><br>From the slip of 2001, Tour: Bryan And which had an amazing set of , including Metropolitan. “It’s much like you are getting a approval to go to another level, affirms those performers that have been a part of the Concert touraround in a larger sized degree of performers.” It packaged as among the best trips in its 10-12 months record.<br><br>Review my website the today show luke bryan ([http://minioasis.com simply click the next document])
	~~In [[mathematics]], an '''integral curve'''~~ is a ~~[[parametric curve]] that represents a specific solution to an [[ordinary differential equation]] or system of equations. If~~ the ~~differential equation is represented as a [[vector field]] or [[slope field]], then~~ the ~~corresponding integral curves are [[tangent]]~~ to ~~the field at each point.~~

	~~Integral curves are known by various other names~~, ~~depending on the nature and interpretation of~~ the ~~differential equation or vector field~~. ~~In [[physics]], integral curves for an [[electric field]] or [[magnetic field]] are known as [[field line]]s, and integral curves for~~ the ~~[[velocity field]]~~ of ~~a [[fluid]] are known as [[Streamlines, streaklines~~, and ~~pathlines\|streamlines]]. In [[dynamical systems theory\|dynamical systems]]~~, ~~the integral curves for a differential equation that governs~~ a ~~[[dynamical system\|system]] are referred to as [[trajectory\|trajectories]] or [[orbit (dynamics)\|orbits]]~~.

	~~==Definition==~~
	~~Suppose that '''F''' is a [~~[~~vector field]]~~: ~~that is, a [[vector-valued function]] with [[Cartesian coordinate system\|Cartesian coordinates]] (''F''<sub>1<~~/~~sub>,''F''<sub>2<~~/~~sub>,.~~..~~,''F''<sub>''n''</sub>); and '''x'''(''t'') a [[parametric curve]~~] ~~with Cartesian coordinates (''x''<sub>1</sub>(''t''),''x''<sub>2</sub>(''t''),...~~,~~''x''<sub>''n''</sub>(''t'')). Then '''x'''(''t'') is an '''integral curve''' of '''F''' if~~ it ~~is a solution of~~ the ~~following [[autonomous system (mathematics)\|autonomous system]]~~ of ~~ordinary differential equations:~~
	~~:<math>\begin{align}~~
	~~\frac{dx_1}{dt} &= F_1(x_1,\ldots,x_n) \\~~
	~~&\vdots \\~~
	~~\frac{dx_n}{dt} &= F_n(x_1,\ldots,x_n)~~.
	~~\end{align}~~
	<~~/math~~>
	~~Such a system may be written as a single vector equation~~
	:<~~math~~>~~\mathbf{x}'(t) = \mathbf{F}(\mathbf{x}(t)).\!\,</math>~~
	~~This equation says precisely that the tangent vector to the curve at any point '''x'''(''t'') along the curve is precisely~~ the ~~vector '''F'''('''x'''(''t''))~~, and ~~so the curve '''x'''(''t'') is tangent at each point~~ to ~~the vector field '''F'''~~.

	~~If a given vector field is [[Lipschitz continuous]]~~, ~~then the [[Picard–Lindelöf theorem]] implies that there exists a unique flow for small time.~~

	~~==Generalization~~ to ~~differentiable manifolds==~~
	~~===Definition===~~

	~~Let ''M''~~ be ~~a [[Banach manifold]] of class ''C''<sup>''r''</sup> with ''r'' ≥ 2. As usual~~, ~~T''M'' denotes~~ the [~~[tangent bundle]] of ''M'' with its natural [[projection (mathematics)\|projection]] ''π''<sub>''M''<~~/~~sub> : T''M'' → ''M'' given by~~

	~~:<math>\pi_{M} : (x, v) \mapsto x.<~~/~~math>~~

	~~A vector field on ''M'' is a [[Fiber bundle#Sections\|cross-section]] of the tangent bundle T''M'', i~~.e. ~~an assignment to every point of the manifold ''M'' of a tangent vector to ''M'' at that point~~. ~~Let ''X'' be a vector field on ''M'' of class ''C''<sup>''r''−1</sup>~~ and ~~let ''p'' ∈ ''M''. An '''integral curve''' for ''X'' passing through ''p'' at time ''t''<sub>0</sub> is a curve ''α'' : ''J'' → ''M'' of class ''C''<sup>''r''−1</sup>~~, ~~defined on an [[interval (mathematics)\|open interval]] ''J'' of the [[real line]] '''R''' containing ''t''<sub>0</sub>~~, ~~such that~~

	~~:<math>\alpha (t_{0}) = p;\~~,~~</math>~~
	:~~<math>\alpha' (t) = X (\alpha (t)) \mbox{ for all } t \in J.</math>~~

	~~===Relationship to ordinary differential equations===~~

	~~The above definition of an integral curve ''α'' for a vector field ''X'', passing through ''p'' at time ''t''<sub>0</sub>, is the same as saying that ''α'' is~~ a ~~local solution to the ordinary differential equation/initial value problem~~

	~~:<math>\alpha (t_{0}) = p;\,</math>~~
	~~:<math>\alpha' (t) = X (\alpha (t))~~.\,<~~/math~~>

	~~It is local in the sense that it is defined only for times in ''J'', and not necessarily for all ''t'' ≥ ''t''~~<~~sub~~>~~0</sub> (let alone ''t'' ≤ ''t''<sub>0</sub>). Thus,~~ the ~~problem~~ of ~~proving the existence and uniqueness of integral curves is the same as that of finding solutions to ordinary differential equations/initial value problems and showing that they are unique.~~

	~~===Remarks on the time derivative===~~

	~~In the above~~, ~~''α''′(''t'') denotes the derivative~~ of ~~''α'' at time ''t''~~, ~~the "direction ''α'' is pointing" at time ''t''~~. ~~From~~ a ~~more abstract viewpoint~~, ~~this is the [[Fréchet derivative]]:~~

	~~:<math>(\mathrm{d}_t f) (+1) \in \mathrm{T}_{\alpha (t)} M.</math>~~

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

	~~:<math>\left( \frac{\mathrm{d} \alpha_{1}}{\mathrm{d} t}, \dots, \frac{\mathrm{d} \alpha_{n}}{\mathrm{d} t} \right),</math>~~

	~~where ''α''<sub>1</sub>, ..., ''α''<sub>''n''</sub> are the coordinates for ''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 T''J'' of ''J'' is the [[Fiber bundle#Trivial bundle\|trivial bundle]] ''J'' × '''R''' and there is a [[canonical form\|canonical]] cross~~-~~section ''ι'' of this bundle such that ''ι''(''t'') = 1 (or, more precisely, (''t'', 1)) for all ''t'' ∈ ''J''~~. ~~The curve ''α'' induces a [[bundle map]] ''α''~~<~~sub~~>∗<~~/sub~~> ~~: T''J'' → T''M'' so that~~ the ~~following diagram commutes:~~

	:[[~~Image~~:~~CommDiag TJtoTM~~.~~png]]~~

	~~Then the time derivative ''α''′ is~~ the ~~[[function composition\|composition~~]~~] ''α''′ = ''α''<sub>∗</sub> <small>o</small> ''ι'', and ''α''′(''t''~~) ~~is its value at some point ''t'' ∈ ''J''.~~

	~~==References==~~
	* {{cite book \| authorlink=Serge Lang \| last=Lang \| first=Serge \| title=Differential manifolds \| publisher=Addison-Wesley Publishing Co., Inc. \| location=Reading, Mass.–London–Don Mills, Ont. \| year=1972 }}

	~~[[Category:Differential geometry]]~~
	~~[[Category:Ordinary differential equations]]~~

en>Monkbot: /* Examples */Fix CS1 deprecated date parameter errors

2014-01-29T08:36:48Z

Examples: Fix CS1 deprecated date parameter errors

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en>Jellyfish dave: Examples/Merged auto-mobiles and F1 cars together

2012-08-24T14:15:36Z

Examples/Merged auto-mobiles and F1 cars together

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en>Graham87: rv, we already have enough examples

en>Monkbot: /* Examples */Fix CS1 deprecated date parameter errors

en>Jellyfish dave: Examples/Merged auto-mobiles and F1 cars together