# Variation vector field

## Definition

Let be a differential manifold.

Let be a piecewise smooth variation of a curve . The **variation vector field** of is defined as the following element of the tangent space :

## Facts

Given any element of the tangent space of a piecewise smooth path in the path space, there exists a piecewise smooth variation thereof for which it is the variation vector field. This is obtained by simply exponentiating the vector field.