# Definition Every [[curve]] has a [[Vector]] at every point, defining a [[vector field]], which can be thought of as as cross-section on the [[tangent bundle]]. The question now arises, given a [[vector field]], is it possible to start at a point $P$ and find a [[curve]] whose [[Vector]] is always the [[vector field]] at whatever point the [[curve]] passes through? This turns out to be true for $C^1$ [[Differentiable Vector Field|differentiable vector fields]]. These curves are known as *integral curves* of the [[vector field]]. Let the components of the [[vector field]] be $V^i(P)$, in some coordinate system $\{x^i\}$, we have $V^i(P) = v^i(x^j)$. We can write the components to the [[Vector]] to a [[curve]] with parameter $\lambda$ as: $ \frac{d x^i}{d\lambda} = v^i(x^j) $ which is a first order ODE with a unique solution which always exists in the [[neighborhood]] of the initial point $P$ by the existence and uniqueness theorems. The paths of different integral curves can never cross except possibly at points where $V^i = 0$ due to the uniqueness of solutions. By solving the equation above with initial conditions for every point $P \in M$, for some [[manifold]] $M$, we see that the integral curves "fill" $M$. Such a [[manifold]] filling set of curves is called a *congruence*. Integral curves can be thought of as "joining the arrows" of a [[vector field]]. Integral curves are invariant sets, i.e. they [[map]] into themselves under time evolution. Take a point along the [[curve]], it will [[map]] into the next point under time evolution and so on, giving the whole integral [[curve]]. The unions and intersections of invariant sets are invariant sets.