Suppose we have a vector field \\(\alpha\\) on manifold \\(M\\), which is a function that assigns to each \\(s \in M\\) a tangent vector in \\(\mathit{Tangent}(s)\\).

Let \\(\beta: I \subseteq \mathbb{R} \rightarrow M\\) be a differentiable function that maps an open interval \\(I\\) of the reals into \\(M\\). So, \\(\beta\\) is a path in the manifold, with a well-defined tangent vector at every point along the curve. For \\(t \in I\\), we have \\(\beta(t) \in M\\), and \\(\beta'(t) \in \mathit{Tangent}(\beta(t))\\).

Next, \\(\beta(t)\\) is called an _integral curve_, or _flow line_, of the vector field \\(\alpha\\) if the tangent vector at every point in the curve equals the value of the vector field at that point on the manifold. That is, if \\(\beta'(t) = \alpha(\beta(t))\\), for all \\(t \in I\\).

Let \\(\beta: I \subseteq \mathbb{R} \rightarrow M\\) be a differentiable function that maps an open interval \\(I\\) of the reals into \\(M\\). So, \\(\beta\\) is a path in the manifold, with a well-defined tangent vector at every point along the curve. For \\(t \in I\\), we have \\(\beta(t) \in M\\), and \\(\beta'(t) \in \mathit{Tangent}(\beta(t))\\).

Next, \\(\beta(t)\\) is called an _integral curve_, or _flow line_, of the vector field \\(\alpha\\) if the tangent vector at every point in the curve equals the value of the vector field at that point on the manifold. That is, if \\(\beta'(t) = \alpha(\beta(t))\\), for all \\(t \in I\\).