A function from a subset of
into
is called a
*smooth function* if derivatives of any order can be taken with
respect to any variables, at any point in the domain of . A
vector field is said to be *smooth* if every one of its
defining functions, , , , is smooth. An alternative
name for a smooth function is a * function*. The
superscript represents the order of differentiation that can be taken.
For a * function*, its derivatives can be taken at least up
to order . A * function* is an alternative name for a
continuous function. The notion of a
homeomorphism can be extended to a
*diffeomorphism*, which is a homeomorphism that is a smooth
function. Two topological spaces are called *diffeomorphic* if there exists a
diffeomorphism between them.

Steven M LaValle 2012-04-20