In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations.
Name 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 lines, and integral curves for the velocity field of a fluid are known as streamlines. In dynamical systems, the integral curves for a differential equation that governs a system are referred to as trajectories or orbits.
Definition Suppose that F is a static vector field, that is, a vector-valued function with components (F1,F2,...,Fn) in a Cartesian coordinate system, and that x(t) is a parametric curve with Cartesian coordinates (x1(t),x2(t),...,xn(t)). Then x(t) is an integral curve of F if it is a solution of the autonomous system of ordinary differential equations,
d x 1 d t = F 1 ( x 1 , … , x n ) ⋮ d x n d t = F n ( x 1 , … , x n ) . {\displaystyle {\begin{aligned}{\frac {dx_{1}}{dt}}&=F_{1}(x_{1},\ldots ,x_{n})\\&\;\,\vdots \\{\frac {dx_{n}}{dt}}&=F_{n}(x_{1},\ldots ,x_{n}).\end{aligned}}}
Such a system may be written as a single vector equation,
x ′ ( t ) = F ( x ( t ) ) . {\displaystyle \mathbf {x} '(t)=\mathbf {F} (\mathbf {x} (t)).}
This equation says that the vector tangent 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.
Examples
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.
Generalization to differentiable manifolds
Definition Let M be a Banach manifold of class Cr with r ≥ 2. As usual, TM denotes the tangent bundle of M with its natural projection πM : TM → M given by
π M : ( x , v ) ↦ x . {\displaystyle \pi _{M}:(x,v)\mapsto x.}
A vector field on M is a cross-section of the tangent bundle TM, 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 Cr−1 and let p ∈ M. An integral curve for X passing through p at time t0 is a curve α : J → M of class Cr−1, defined on an open interval J of the real line R containing t0, such that
α ( t 0 ) = p ; α ′ ( t ) = X ( α ( t ) ) for all t ∈ J . {\displaystyle {\begin{aligned}\alpha (t_{0})&=p;\\\alpha '(t)&=X(\alpha (t)){\text{ for all }}t\in J.\end{aligned}}}
Relationship to ordinary differential equations The above definition of an integral curve α for a vector field X, passing through p at time t0, is the same as saying that α is a local solution to the ordinary differential equation/initial value problem
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