Definition:Piecewise Smooth Vector Field along Family of Curves

Definition
Let $M$ be a smooth manifold.

Let $X$ be a continuous vector field along $\Gamma$.

Let $I = \closedint a b$ is a closed real interval.

Let $J$ is an open real interval.

Let $\Gamma : J \times I \to M$ be an admissible family of curves, where $\times$ denotes the cartesian product.

Suppose there is an admissible subdivision $\tuple {a_0, a_1, a_2, \ldots, a_{n - 1}, a_n}$ of $I$.

Suppose for all $i \in \N_{>0} : i \le n$ the restriction of $X$ to a rectangle $J \times \closedint {a_{i - 1}} {a_i}$ is smooth.

Then $X$ is said to be a piecewise smooth vector field along $\Gamma$.