Smoothness

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