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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.]]
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| 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.
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| 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]].
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| ==Definition==
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| 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:
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| :<math>\begin{align}
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| \frac{dx_1}{dt} &= F_1(x_1,\ldots,x_n) \\
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| &\vdots \\
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| \frac{dx_n}{dt} &= F_n(x_1,\ldots,x_n).
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| \end{align}
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| </math> | |
| Such a system may be written as a single vector equation
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| :<math>\mathbf{x}'(t) = \mathbf{F}(\mathbf{x}(t)).\!\,</math>
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| 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'''.
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| If a given vector field is [[Lipschitz continuous]], then the [[Picard–Lindelöf theorem]] implies that there exists a unique flow for small time.
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| ==Generalization to differentiable manifolds==
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| ===Definition===
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| 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
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| :<math>\pi_{M} : (x, v) \mapsto x.</math>
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| 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
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| :<math>\alpha (t_{0}) = p;\,</math>
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| :<math>\alpha' (t) = X (\alpha (t)) \mbox{ for all } t \in J.</math> | |
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| ===Relationship to ordinary differential equations===
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| 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
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| :<math>\alpha (t_{0}) = p;\,</math>
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| :<math>\alpha' (t) = X (\alpha (t)).\,</math>
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| 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.
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| ===Remarks on the time derivative===
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| 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]]:
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| :<math>(\mathrm{d}_t f) (+1) \in \mathrm{T}_{\alpha (t)} M.</math>
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| In the special case that ''M'' is some [[open subset]] of '''R'''<sup>''n''</sup>, this is the familiar derivative
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| :<math>\left( \frac{\mathrm{d} \alpha_{1}}{\mathrm{d} t}, \dots, \frac{\mathrm{d} \alpha_{n}}{\mathrm{d} t} \right),</math>
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| where ''α''<sub>1</sub>, ..., ''α''<sub>''n''</sub> are the coordinates for ''α'' with respect to the usual coordinate directions.
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| 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:
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| :[[Image:CommDiag TJtoTM.png]]
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| 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''.
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| ==References==
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| * {{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 }}
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| [[Category:Differential geometry]]
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| [[Category:Ordinary differential equations]]
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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 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.”
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