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Vipul at 09:01, 5 August 2007

2007-08-05T09:01:24Z

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==Definition==

Let <math>M</math> be a [[differential manifold]] and <math>p,q \in M</math> be two (not necessarily distinct) points. A '''piecewise smooth path''' <math>\omega</math> from <math>p</math> to <math>q</math> is a map <math>\omega:[0,1] \to M</math> such that:

* There exists a partition <math>0 = t_0 < t_1 < \ldots < t_k = 1</math> such that <math>\omega</math> restricted to <math>[t_{i-1},t_i]</math> is smooth.
* <math>\omega(0) = p</math> and <math>\omega(1) = q</math>

==Notions==
===Set of all piecewise smooth paths between two points===

{{further|[[Path space of a manifold]]}}

The set of all piecewise smooth paths in a differential manifold <math>M</math> between two points <math>p,q \in M</math> is denoted as <math>\Omega(M;p,q)</math>, sometimes simply as <math>\Omega(M)</math> or simply <math>\Omega</math>.

{{fillin}}
===Tangent space===

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 <math>\omega \in \Omega</math> is defined as the vector space of all piecewise smooth vector fields <math>W</math> along <math>\omega</math> for which <math>W(0) = W(1) = 0</math>. In other words, it is an infinitesimal change in the path, with no change occurring at the endpoints.

Piecewise smooth path - Revision history

Vipul: 1 revision

Vipul at 09:01, 5 August 2007