Smoothness Explained
In mathematical analysis, the smoothness of a function is a property measured by the number, called differentiability class, of continuous derivatives it has over its domain.[1]
A function of class
is a function of smoothness at least ; that is, a function of class
is a function that has a th derivative that is continuous in its domain.
A function of class
or
-function (pronounced
C-infinity function) is an
infinitely differentiable function, that is, a function that has derivatives of all orders (this implies that all these derivatives are continuous).
Generally, the term smooth function refers to a
-function. However, it may also mean "sufficiently differentiable" for the problem under consideration.
Differentiability classes
Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function.
on the
real line and a function
defined on
with real values. Let
k be a non-negative
integer. The function
is said to be of differentiability
class
if the derivatives
exist and are
continuous on
If
is
-differentiable on
then it is at least in the class
since
are continuous on
The function
is said to be
infinitely differentiable,
smooth, or of
class
if it has derivatives of all orders on
(So all these derivatives are continuous functions over
)
[2] The function
is said to be of
class
or
analytic, if
is smooth (i.e.,
is in the class
) and its
Taylor series expansion around any point in its domain converges to the function in some
neighborhood of the point. There exist functions that are smooth but not analytic;
is thus strictly contained in
Bump functions are examples of functions with this property.
To put it differently, the class
consists of all continuous functions. The class
consists of all
differentiable functions whose derivative is continuous; such functions are called
continuously differentiable. Thus, a
function is exactly a function whose derivative exists and is of class
In general, the classes
can be defined
recursively by declaring
to be the set of all continuous functions, and declaring
for any positive integer
to be the set of all differentiable functions whose derivative is in
In particular,
is contained in
for every
and there are examples to show that this containment is strict (
). The class
of infinitely differentiable functions, is the intersection of the classes
as
varies over the non-negative integers.
Examples
Example: Continuous (C0) But Not Differentiable
The functionis continuous, but not differentiable at, so it is of class C0, but not of class C1.
Example: Finitely-times Differentiable (C)
For each even integer, the functionis continuous and times differentiable at all . At, however,
is not times differentiable, so
is of class
C, but not of class
C where .
Example: Differentiable But Not Continuously Differentiable (not C1)
The functionis differentiable, with derivative
Because
oscillates as → 0,
is not continuous at zero. Therefore,
is differentiable but not of class
C1.
Example: Differentiable But Not Lipschitz Continuous
The functionis differentiable but its derivative is unbounded on a compact set. Therefore,
is an example of a function that is differentiable but not locally
Lipschitz continuous.
Example: Analytic (C)
is analytic, and hence falls into the class
Cω. The
trigonometric functions are also analytic wherever they are defined, because they are linear combinations of complex exponential functions
and
.
Example: Smooth (C) but not Analytic (C)
The bump functionis smooth, so of class C∞, but it is not analytic at, and hence is not of class Cω. The function is an example of a smooth function with compact support.
Multivariate differentiability classes
A function
defined on an open set
of
is said
[3] to be of class
on
, for a positive integer
, if all
partial derivatives exist and are continuous, for every
\alpha1,\alpha2,\ldots,\alphan
non-negative integers, such that
\alpha=\alpha1+\alpha2+ … +\alphan\leqk
, and every
. Equivalently,
is of class
on
if the
-th order
Fréchet derivative of
exists and is continuous at every point of
. The function
is said to be of class
or
if it is continuous on
. Functions of class
are also said to be
continuously differentiable.
A function
, defined on an open set
of
, is said to be of class
on
, for a positive integer
, if all of its components
are of class
, where
are the natural
projections
defined by
. It is said to be of class
or
if it is continuous, or equivalently, if all components
are continuous, on
.
The space of Ck functions
Let
be an open subset of the real line. The set of all
real-valued functions defined on
is a
Fréchet vector space, with the countable family of
seminorms
where
varies over an increasing sequence of
compact sets whose
union is
, and
.
The set of
functions over
also forms a Fréchet space. One uses the same seminorms as above, except that
is allowed to range over all non-negative integer values.
The above spaces occur naturally in applications where functions having derivatives of certain orders are necessary; however, particularly in the study of partial differential equations, it can sometimes be more fruitful to work instead with the Sobolev spaces.
Continuity
The terms parametric continuity (Ck) and geometric continuity (Gn) were introduced by Brian Barsky, to show that the smoothness of a curve could be measured by removing restrictions on the speed, with which the parameter traces out the curve.[4] [5] [6]
Parametric continuity
Parametric continuity (Ck) is a concept applied to parametric curves, which describes the smoothness of the parameter's value with distance along the curve. A (parametric) curve
is said to be of class
Ck, if
exists and is continuous on
, where derivatives at the end-points
and
are taken to be
one sided derivatives (from the right at
and from the left at
).
As a practical application of this concept, a curve describing the motion of an object with a parameter of time must have C1 continuity and its first derivative is differentiable—for the object to have finite acceleration. For smoother motion, such as that of a camera's path while making a film, higher orders of parametric continuity are required.
Order of parametric continuity
The various order of parametric continuity can be described as follows:[7]
: zeroth derivative is continuous (curves are continuous)
: zeroth and first derivatives are continuous
: zeroth, first and second derivatives are continuous
: 0-th through
-th derivatives are continuous
Geometric continuity
A curve or surface can be described as having
continuity, with
being the increasing measure of smoothness. Consider the segments either side of a point on a curve:
The curves touch at the join point.
The curves also share a common tangent direction at the join point.
The curves also share a common center of curvature at the join point.
In general,
continuity exists if the curves can be reparameterized to have
(parametric) continuity.
[8] [9] A reparametrization of the curve is geometrically identical to the original; only the parameter is affected.
Equivalently, two vector functions
and
such that
have
continuity at the point where they meet ifthey satisfy equations known as Beta-constraints. For example, the Beta-constraints for
continuity are:
\begin{align}
g(1)(0)&=\beta1f(1)(1)\\
g(2)(0)&=
f(2)(1)+\beta2f(1)(1)\\
g(3)(0)&=
f(3)(1)+3\beta1\beta2f(2)(1)+\beta3f(1)(1)\\
g(4)(0)&=
f(4)(1)+
f(3)(1)+(4\beta1\beta3+3\beta
f(2)(1)+\beta4f(1)(1)\\
\end{align}
where
,
, and
are arbitrary, but
is constrained to be positive.In the case
, this reduces to
and
, for a scalar
(i.e., the direction, but not necessarily the magnitude, of the two vectors is equal).
While it may be obvious that a curve would require
continuity to appear smooth, for good aesthetics, such as those aspired to in
architecture and
sports car design, higher levels of geometric continuity are required. For example, reflections in a car body will not appear smooth unless the body has
continuity.
A (with ninety degree circular arcs at the four corners) has
continuity, but does not have
continuity. The same is true for a, with octants of a sphere at its corners and quarter-cylinders along its edges. If an editable curve with
continuity is required, then
cubic splines are typically chosen; these curves are frequently used in
industrial design.
Other concepts
Relation to analyticity
While all analytic functions are "smooth" (i.e. have all derivatives continuous) on the set on which they are analytic, examples such as bump functions (mentioned above) show that the converse is not true for functions on the reals: there exist smooth real functions that are not analytic. Simple examples of functions that are smooth but not analytic at any point can be made by means of Fourier series; another example is the Fabius function. Although it might seem that such functions are the exception rather than the rule, it turns out that the analytic functions are scattered very thinly among the smooth ones; more rigorously, the analytic functions form a meagre subset of the smooth functions. Furthermore, for every open subset A of the real line, there exist smooth functions that are analytic on A and nowhere else .
It is useful to compare the situation to that of the ubiquity of transcendental numbers on the real line. Both on the real line and the set of smooth functions, the examples we come up with at first thought (algebraic/rational numbers and analytic functions) are far better behaved than the majority of cases: the transcendental numbers and nowhere analytic functions have full measure (their complements are meagre).
The situation thus described is in marked contrast to complex differentiable functions. If a complex function is differentiable just once on an open set, it is both infinitely differentiable and analytic on that set .
Smooth partitions of unity
Smooth functions with given closed support are used in the construction of smooth partitions of unity (see partition of unity and topology glossary); these are essential in the study of smooth manifolds, for example to show that Riemannian metrics can be defined globally starting from their local existence. A simple case is that of a bump function on the real line, that is, a smooth function f that takes the value 0 outside an interval [''a'',''b''] and such that
Given a number of overlapping intervals on the line, bump functions can be constructed on each of them, and on semi-infinite intervals
and
to cover the whole line, such that the sum of the functions is always 1.
From what has just been said, partitions of unity do not apply to holomorphic functions; their different behavior relative to existence and analytic continuation is one of the roots of sheaf theory. In contrast, sheaves of smooth functions tend not to carry much topological information.
Smooth functions on and between manifolds
, of dimension
and an
atlas ak{U}=\{(U\alpha,\phi\alpha)\}\alpha,
then a map
is
smooth on
if for all
there exists a chart
such that
and
f\circ\phi-1:\phi(U)\to\R
is a smooth function from a neighborhood of
in
to
(all partial derivatives up to a given order are continuous). Smoothness can be checked with respect to any chart of the atlas that contains
since the smoothness requirements on the transition functions between charts ensure that if
is smooth near
in one chart it will be smooth near
in any other chart.
If
is a map from
to an
-dimensional manifold
, then
is smooth if, for every
there is a chart
containing
and a chart
containing
such that
and
\psi\circF\circ\phi-1:\phi(U)\to\psi(V)
is a smooth function from
Smooth maps between manifolds induce linear maps between tangent spaces: for
, at each point the
pushforward (or differential) maps tangent vectors at
to tangent vectors at
:
and on the level of the
tangent bundle, the pushforward is a vector bundle homomorphism:
The dual to the pushforward is the
pullback, which "pulls" covectors on
back to covectors on
and
-forms to
-forms:
F*:\Omegak(N)\to\Omegak(M).
In this way smooth functions between manifolds can transport
local data, like
vector fields and
differential forms, from one manifold to another, or down to Euclidean space where computations like integration are well understood.
Preimages and pushforwards along smooth functions are, in general, not manifolds without additional assumptions. Preimages of regular points (that is, if the differential does not vanish on the preimage) are manifolds; this is the preimage theorem. Similarly, pushforwards along embeddings are manifolds.[10]
Smooth functions between subsets of manifolds
There is a corresponding notion of smooth map for arbitrary subsets of manifolds. If
is a
function whose
domain and
range are subsets of manifolds
and
respectively.
is said to be
smooth if for all
there is an open set
with
and a smooth function
such that
for all
See also
Notes and References
- Web site: Smooth Function. Weisstein. Eric W.. mathworld.wolfram.com. en. 2019-12-13. 2019-12-16. https://web.archive.org/web/20191216043114/http://mathworld.wolfram.com/SmoothFunction.html. live.
- Book: Warner, Frank W.. Frank Wilson Warner. 1983. Foundations of Differentiable Manifolds and Lie Groups. Springer. 978-0-387-90894-6. 5 [Definition 1.2]. 2014-11-28. 2015-10-01. https://web.archive.org/web/20151001012659/https://books.google.com/books?id=t6PNrjnfhuIC&pg=PA5&dq=%22f+is+differentiable+of+class+Ck%22. live.
- Book: Henri Cartan. Cours de calcul différentiel. 1977. Paris: Hermann. Henri Cartan.
- Ph.D. . Barsky . Brian A. . 1981 . The Beta-spline: A Local Representation Based on Shape Parameters and Fundamental Geometric Measures . University of Utah, Salt Lake City, Utah.
- Book: Brian A. Barsky. Computer Graphics and Geometric Modeling Using Beta-splines. 1988. Springer-Verlag, Heidelberg. 978-3-642-72294-3.
- Book: Richard H. Bartels. John C. Beatty. Brian A. Barsky. An Introduction to Splines for Use in Computer Graphics and Geometric Modeling. 1987. Morgan Kaufmann. 978-1-55860-400-1. Chapter 13. Parametric vs. Geometric Continuity.
- Web site: Michiel . van de Panne . Parametric Curves . Fall 1996 Online Notes . 1996 . University of Toronto, Canada . 2019-09-01 . 2020-11-26 . https://web.archive.org/web/20201126212511/https://www.cs.helsinki.fi/group/goa/mallinnus/curves/curves.html . live .
- Brian A. . Barsky . Tony D. . DeRose . Geometric Continuity of Parametric Curves: Three Equivalent Characterizations . IEEE Computer Graphics and Applications . 9 . 6 . 1989 . 60–68 . 10.1109/38.41470 . 17893586 .
- Web site: Geometry and Algorithms for Computer Aided Design . Erich . Hartmann . 55 . 2003 . . 2019-08-31 . 2020-10-23 . https://web.archive.org/web/20201023054532/http://www2.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf#page=55 . live .
- Book: Guillemin . Victor . Pollack . Alan . Differential Topology . Englewood Cliffs . Prentice-Hall . 1974 . 0-13-212605-2 .