Such a system may be written as a single vector equation,
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 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.
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 intervalJ of the real lineR containing t0, such that
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
It is local in the sense that it is defined only for times in J, and not necessarily for all t ≥ t0 (let alone t ≤ t0). 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:
In the special case that M is some open subset of Rn, this is the familiar derivative
where α1, ..., αn 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 maps. Note that the tangent bundle TJ of J is the trivial bundleJ × R and there is a canonical cross-section ι of this bundle such that ι(t) = 1 (or, more precisely, (t, 1) ∈ ι) for all t ∈ J. The curve α induces a bundle mapα∗ : TJ → TM so that the following diagram commutes:
Then the time derivative α′ is the compositionα′ = α∗ o ι, and α′(t) is its value at some point t ∈ J.