Definition:Piecewise Smooth Vector Field along Family of Curves
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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 $\Gamma$.
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 6$: Geodesics and Distance. Geodesics and Minimizing Curves