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\$$.