Piecewise smooth path
Let be a differential manifold and be two (not necessarily distinct) points. A piecewise smooth path from to is a map such that:
- There exists a partition such that restricted to is smooth.
Set of all piecewise smooth paths between two points
Further information: Path space of a manifold
The set of all piecewise smooth paths in a differential manifold between two points is denoted as , sometimes simply as or simply .
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From the infinite-dimensional manifold definition, the following definition emerges for the tangent space to a piecewise smooth curve with respect to the path space.
The tangent space of a path is defined as the vector space of all piecewise smooth vector fields along for which . In other words, it is an infinitesimal change in the path, with no change occurring at the endpoints.