#### An integral curve

If a vector field is given, then a velocity vector is defined at each point using (8.10). Imagine a point that starts at some at time and then moves according to the velocities expressed in . Where should it travel? Its trajectory starting from can be expressed as a function , in which the domain is a time interval, . A trajectory represents an integral curve (or solution trajectory) of the differential equations with initial condition if (8.14)

for every time . This is sometimes expressed in integral form as (8.15)

and is called a solution to the differential equations in the sense of Caratheodory. Intuitively, the integral curve starts at and flows along the directions indicated by the velocity vectors. This can be considered as the continuous-space analog of following the arrows in the discrete case, as depicted in Figure 8.2b.

Example 8..9 (Integral Curve for a Constant Velocity Field)   The simplest case is a constant vector field. Suppose that a constant field and is defined on . The integral curve from is . It can be easily seen that (8.14) holds for all . Example 8..10 (Integral Curve for a Linear Velocity Field)   Consider a velocity field on . Let and . The function represents the integral curve from . At , , which is the initial state. If can be verified that for all , (8.14) holds. This is a simple example of a linear velocity field. In general, if each is a linear function of the coordinate variables , , , then a linear velocity field is obtained. The integral curve is generally found by determining the eigenvalues of the matrix when the velocity field is expressed as . See  for numerous examples. A basic result from differential equations is that a unique integral curve exists to if is smooth. An alternative condition is that a unique solution exists if satisfies a Lipschitz condition. This means that there exists some constant such that (8.16)

for all , and denotes the Euclidean norm (vector magnitude). The constant is often called a Lipschitz constant. Note that if satisfies the Lipschitz condition, then it is continuous. Also, if all partial derivatives of over all of can be bounded by a constant, then is Lipschitz. The expression in (8.16) is preferred, however, because it is more general (it does not even imply that is differentiable everywhere).

Steven M LaValle 2020-08-14