TheInfoList

is a smooth function with
compact support In mathematics, the support of a real-valued function ''f'' is the subset of the domain containing the elements which are not mapped to zero. If the domain of ''f'' is a topological space, the support of ''f'' is instead defined as the smallest c ...
. In
mathematical analysis#REDIRECT Mathematical analysis#REDIRECT Mathematical analysis {{R from other capitalisation ...
{{R from other capitalisation ...
, the smoothness of a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriented ...
is a property measured by the number of
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous gam ...
derivatives Derivative may refer to: In mathematics and economics *Brzozowski derivative in the theory of formal languages *Derivative in calculus, a quantity indicating how a function changes when the values of its inputs change. *Formal derivative, an opera ...
it has over some domain. At the very minimum, a function could be considered "smooth" if it is differentiable everywhere (hence continuous). At the other end, it might also possess derivatives of all
orders Orders is a surname. Etymology The etymology of 'Orders' is unclear, although there are records extant dating the surname in its current spelling back to the 17th century in the Warminster region of Wiltshire and the 16th century in Cambridgeshire, ...
in its
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Doma ...
, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or $C^$ function).

Differentiability classes

Differentiability class is a classification of functions according to the properties of their
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For ...
s. It is a measure of the highest order of derivative that exists for a function. Consider an
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space—that is, when a distance is defined—open sets are the sets that, with every point , contain all points that are sufficiently near to (that ...
on the
real line In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one. It can be though ...
and a function ''f'' defined on that set with real values. Let ''k'' be a non-negative
integer An integer (from the Latin ''integer'' meaning "whole") is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and  are not. The set of intege ...
. The function ''f'' is said to be of (differentiability) class ''Ck'' if the derivatives ''f''′, ''f''″, ..., ''f''(''k'') exist and are
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous gam ...
. The function ''f'' is said to be infinitely differentiable, smooth, or of class ''C'', if it has derivatives of all orders. The function ''f'' is said to be of class ''C''ω, or analytic, if ''f'' is smooth ''and'' if its
Taylor series In mathematics, the Taylor series of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this ...
expansion around any point in its domain converges to the function in some neighborhood of the point. ''C''ω is thus strictly contained in ''C''.
Bump function In mathematics, a bump function (also called a test function) is a function f: \mathbf^n \to \mathbf on a Euclidean space \mathbf^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set ...
s are examples of functions in ''C'' but ''not'' in ''C''ω. To put it differently, the class ''C''0 consists of all continuous functions. The class ''C''1 consists of all
differentiable function In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each ...
s whose derivative is continuous; such functions are called continuously differentiable. Thus, a ''C''1 function is exactly a function whose derivative exists and is of class ''C''0. In general, the classes ''Ck'' can be defined
recursively Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
by declaring ''C''0 to be the set of all continuous functions, and declaring ''Ck'' for any positive integer ''k'' to be the set of all differentiable functions whose derivative is in ''C''''k''−1. In particular, ''Ck'' is contained in ''C''''k''−1 for every ''k'' > 0, and there are examples to show that this containment is strict (''Ck'' ⊊ ''C''''k''−1). The class ''C'' of infinitely differentiable functions, is the intersection of the classes ''Ck'' as ''k'' varies over the non-negative integers.

Examples

The ''C''0 function for and 0 otherwise. 300px, A smooth function that is not analytic. The function : $f\left(x\right) = \beginx & \mboxx \ge 0, \\ 0 &\textx < 0\end$ is continuous, but not differentiable at , so it is of class ''C''0, but not of class ''C''1. The function :$g\left(x\right) = \beginx^2\sin & \textx \neq 0, \\ 0 &\textx = 0\end$ is differentiable, with derivative :$g\text{'}\left(x\right) = \begin-\mathord + 2x\sin\left(\tfrac\right) & \textx \neq 0, \\ 0 &\textx = 0.\end$ Because $\cos\left(1/x\right)$ oscillates as → 0, $g\text{'}\left(x\right)$ is not continuous at zero. Therefore, $g\left(x\right)$ is differentiable but not of class ''C''1. Moreover, if one takes $g\left(x\right) = x^\sin\left(1/x\right)$ ( ≠ 0) in this example, it can be used to show that the derivative function of a differentiable function can be unbounded on a
compact set In mathematics, more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (i.e., containing all its limit points) and bounded (i.e., having all its points lie within so ...
and, therefore, that a differentiable function on a compact set may not be locally
Lipschitz continuous In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such ...
. The functions : $f\left(x\right)=, x, ^$ where is even, are continuous and times differentiable at all . But at they are not times differentiable, so they are of class ''C'', but not of class ''C'' where . The
exponential function In mathematics, an exponential function is a function of the form where is a positive real number, and the argument occurs as an exponent. For real numbers and , a function of the form f(x)=ab^ is also an exponential function, since it can ...
is analytic, and hence falls into the class ''C''ω. The
trigonometric function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...
s are also analytic wherever they are defined. The
bump function In mathematics, a bump function (also called a test function) is a function f: \mathbf^n \to \mathbf on a Euclidean space \mathbf^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set ...
:$f\left(x\right) = \begine^ & \text , x, < 1, \\ 0 &\text\end$ is smooth, so of class ''C'', but it is not analytic at , and hence is not of class ''C''ω. The function is an example of a smooth function with
compact support In mathematics, the support of a real-valued function ''f'' is the subset of the domain containing the elements which are not mapped to zero. If the domain of ''f'' is a topological space, the support of ''f'' is instead defined as the smallest c ...
.

Multivariate differentiability classes

A function $f:U\subset\mathbb^n\to\mathbb$ defined on an open set $U$ of $\mathbb^n$ is said to be of class $C^k$ on $U$, for a positive integer $k$, if all
partial derivatives In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Parti ...
:$\frac\left(y_1,y_2,\ldots,y_n\right)$ exist and are continuous, for every $\alpha_1,\alpha_2,\ldots,\alpha_n$ non-negative integers, such that $\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n\leq k$, and every $\left(y_1,y_2,\ldots,y_n\right)\in U$. Equivalently, $f$ is of class $C^k$ on $U$ if the $k$-th order
Fréchet derivative In mathematics, the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued func ...
of $f$ exists and is continuous at every point of $U$. The function $f$ is said to be of class $C$ or $C^0$ if it is continuous on $U$. A function $f:U\subset\mathbb^n\to\mathbb^m$, defined on an open set $U$ of $\mathbb^n$, is said to be of class $C^k$ on $U$, for a positive integer $k$, if all of its components :$f_i\left(x_1,x_2,\ldots,x_n\right)=\left(\pi_i\circ f\right)\left(x_1,x_2,\ldots,x_n\right)=\pi_i\left(f\left(x_1,x_2,\ldots,x_n\right)\right) \text i=1,2,3,\ldots,m$ are of class $C^k$, where $\pi_i$ are the natural projections $\pi_i:\mathbb^m\to\mathbb$ defined by $\pi_i\left(x_1,x_2,\ldots,x_m\right)=x_i$. It is said to be of class $C$ or $C^0$ if it is continuous, or equivalently, if all components $f_i$ are continuous, on $U$.

The space of ''C''''k'' functions

Let ''D'' be an open subset of the real line. The set of all ''Ck'' real-valued functions defined on ''D'' is a Fréchet vector space, with the countable family of
seminormIn mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and, co ...
s : $p_=\sup_\left, f^\left(x\right)\$ where ''K'' varies over an increasing sequence of
compact set In mathematics, more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (i.e., containing all its limit points) and bounded (i.e., having all its points lie within so ...
s whose union is ''D'', and ''m'' = 0, 1, ..., ''k''. The set of ''C'' functions over ''D'' also forms a Fréchet space. One uses the same seminorms as above, except that ''m'' 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 equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how ...
s, it can sometimes be more fruitful to work instead with the
Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to m ...
s.

Parametric continuity

The terms ''parametric continuity'' 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 In everyday use and in kinematics, the speed (commonly referred to as v) of an object is the magnitude of the change of its position; it is thus a scalar quantity. The average speed of an object in an interval of time is the distance travelled ...
, with which the parameter traces out the curve. Parametric continuity is a concept applied to
parametric curve In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric objec ...
s, which describes the smoothness of the parameter's value with distance along the curve.

Definition

A (parametric) curve is said to be of class ''C''''k'', if $\textstyle \frac$ exists and is continuous on

Order of continuity

The various order of parametric continuity can be described as follows: * ''C''0: ''0''–th derivatives are continuous (curves are continuous) * ''C''1: ''0''–th and first derivatives are continuous * ''C''2: ''0''–th, first and second derivatives are continuous * ''Cn'': ''0''–th through ''n''–th derivatives are continuous

Geometric continuity

The concept of geometrical or geometric continuity was primarily applied to the
conic sections In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of ...

(and related shapes) by mathematicians such as
Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the "1666–1676" section. (; or ; – 14 November 1716) was a prominent German polymath and one of the most important logicians, mathematicians and natural philoso ...
,
Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, and astrologer. He is a key figure in the 17th-century scientific revolution, best known for his laws of planetary motion, and his books ''Astro ...

, and . The concept was an early attempt at describing, through geometry rather than algebra, the concept of
continuity Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous gam ...
as expressed through a parametric function. The basic idea behind geometric continuity was that the five conic sections were really five different versions of the same shape. An
ellipse In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a circle, which is the special type of elli ...
tends to a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. T ...
as the
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-center, in geometry * Eccentricity (graph theory) of a ve ...
approaches zero, or to a
parabola The parabola is a member of the family of conic sections. In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be p ...

as it approaches one; and a
hyperbola 300px, Hyperbola (red): features In mathematics, a hyperbola () (adjective form hyperbolic, ) (plural ''hyperbolas'', or ''hyperbolae'' ()) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it ...

tends to a
parabola The parabola is a member of the family of conic sections. In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be p ...

as the eccentricity drops toward one; it can also tend to intersecting
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Literatu ...
s. Thus, there was ''continuity'' between the conic sections. These ideas led to other concepts of continuity. For instance, if a circle and a straight line were two expressions of the same shape, perhaps a line could be thought of as a circle of infinite
radius In classical geometry, a radius of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the Latin ''radius'', meaning ray but also the spoke of a c ...
. For such to be the case, one would have to make the line closed by allowing the point $x =\infty$ to be a point on the circle, and for $x =+\infty$ and $x =\neg\infty$ to be identical. Such ideas were useful in crafting the modern, algebraically defined, idea of the
continuity Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous gam ...
of a function and of $\infty$ (see
projectively extended real line The projectively extended real line can be visualized as the real number line wrapped around a circle (by some form of stereographic projection) with an additional point at infinity. In real analysis, the projectively extended real line (also cal ...
for more).

Smoothness of curves and surfaces

A
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that app ...
or
surface Water droplet lying on a damask. Surface tension">damask.html" style="text-decoration: none;"class="mw-redirect" title="Water droplet lying on a damask">Water droplet lying on a damask. Surface tension is high enough to prevent floating ...
can be described as having ''Gn'' continuity, with ''n'' being the increasing measure of smoothness. Consider the segments either side of a point on a curve: *''G''0: The curves touch at the join point. *''G''1: The curves also share a common
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More pre ...

direction at the join point. *''G''2: The curves also share a common center of curvature at the join point. In general, ''Gn'' continuity exists if the curves can be reparameterized to have ''C''''n'' (parametric) continuity. A reparametrization of the curve is geometrically identical to the original; only the parameter is affected. Equivalently, two vector functions ''f''(''t'') and ''g''(''t'') have ''Gn'' continuity if and , for a scalar (i.e., if the direction, but not necessarily the magnitude, of the two vectors is equal). While it may be obvious that a curve would require ''G''1 continuity to appear smooth, for good
aesthetics Aesthetics, or esthetics (), is a branch of philosophy that deals with the nature of beauty and taste, as well as the philosophy of art (its own area of philosophy that comes out of aesthetics). It examines subjective and sensori-emotional va ...
, such as those aspired to in
architecture upright=1.45, alt=Plan d'exécution du second étage de l'hôtel de Brionne (dessin) De Cotte 2503c – Gallica 2011 (adjusted), Plan of the second floor (attic storey) of the Hôtel de Brionne in Paris – 1734. Architecture (Latin ''architect ...
and
sports car#REDIRECT Sports car#REDIRECT Sports car {{R from other capitalisation ...
{{R from other capitalisation ...
design, higher levels of geometric continuity are required. For example, reflections in a car body will not appear smooth unless the body has ''G''2 continuity. A ''rounded rectangle'' (with ninety degree circular arcs at the four corners) has ''G''1 continuity, but does not have ''G''2 continuity. The same is true for a ''rounded cube'', with octants of a sphere at its corners and quarter-cylinders along its edges. If an editable curve with ''G''2 continuity is required, then cubic splines are typically chosen; these curves are frequently used in
industrial design Industrial design is a process of design applied to products that are to be manufactured through techniques of mass production. A key characteristic is that design precedes manufacture: the creative act of determining and defining a product's form ...
.

Other concepts

Relation to analyticity

While all
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex anal ...
s are "smooth" (i.e. have all derivatives continuous) on the set on which they are analytic, examples such as
bump function In mathematics, a bump function (also called a test function) is a function f: \mathbf^n \to \mathbf on a Euclidean space \mathbf^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set ...
s (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 In mathematics, a Fourier series () is a periodic function composed of harmonically related sinusoids, combined by a weighted summation. With appropriate weights, one cycle (or ''period'') of the summation can be made to approximate an arbitrary f ...
; 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 number In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial with rational coefficients. The best known transcendental numbers are and . Though only a few classes of transcendental num ...
s 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 Support may refer to: Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a type of structural support to help prevent sideways movement * ...
are used in the construction of smooth partitions of unity (see ''
partition of unityIn mathematics, a partition of unity of a topological space ''X'' is a set ''R'' of continuous functions from ''X'' to the unit interval ,1such that for every point, x\in X, * there is a neighbourhood of ''x'' where all but a finite number of the fun ...
'' and
topology glossary This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundame ...
); these are essential in the study of
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an at ...
s, for example to show that
Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T'p'M'' at each point ''p''. A common convention is to ta ...
s can be defined globally starting from their local existence. A simple case is that of a
bump function In mathematics, a bump function (also called a test function) is a function f: \mathbf^n \to \mathbf on a Euclidean space \mathbf^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set ...
on the real line, that is, a smooth function ''f'' that takes the value 0 outside an interval 'a'',''b''and such that :$f\left(x\right) > 0 \quad \text \quad a < x < b.\,$ 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 don't apply to
holomorphic function A rectangular grid (top) and its image under a conformal map ''f'' (bottom). In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is, at every point of its domain, complex differentiable in a ...
s; their different behavior relative to existence and
analytic continuation Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles". Analytic can also have the following meanings: Natural sciences Chemistry * ...
is one of the roots of
sheaf Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics), a mathematical tool * Sheaf toss, a Scottish sport * River Sheaf, a tributary of River Don in England * ''The Sheaf'', a student-run newspaper serv ...
theory. In contrast, sheaves of smooth functions tend not to carry much topological information.

Smooth functions on and between manifolds

Given a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an at ...
$M$, dimension ''m'', with atlas $\mathfrak=\_\alpha$, then a map $f:M\to \R$ is smooth on ''M'' if for all $p \in M$ there exists a chart $\exists \left(U,\phi\right)\in \mathfrak: p \in U$, such that the pullback of $f$by $\phi^$, denoted $\left(\phi^\right)^* f=f \circ \phi^: \phi\left(U\right) \to \R$is smooth as a function from $\R^m$to $\R$in a neighborhood of $\phi\left(p\right)$(all partial derivatives up to a given order are continuous). Note that smoothness can be checked with respect to any preferred
chart A chart is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent tabular numeric data, functions or s ...
atlas Blaeu's world map, originally prepared by Joan Blaeu for his ''Atlas Maior'', published in the first book of the ''Atlas Van Loon'' (1664). An atlas is a collection of maps; it is typically a bundle of maps of Earth or a region of E ...
, since the smoothness requirements on the transition functions between charts ensure that if $f$is smooth about p in one chart it will be smooth about ''p'' in any other chart of the atlas. If instead $F:M\to N$ is a map from ''$M$'' to an ''n''-dimensional manifold ''$N$'', then ''F'' is smooth if, for every , there is a chart $\left(U,\phi\right)$about ''p'' in ''$M$,'' and a chart $\left(V,\psi\right)$about $F\left(p\right)$in ''$N$'' with $F\left(U\right) \subset V$, such that $\psi \circ F \circ \phi^:\phi\left(U\right)\to\psi\left(V\right)$ is smooth as a function from R''m'' to R''n''. Smooth maps between manifolds induce linear maps between
tangent spaces In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector that gives the displacement of the ...
: for $F:M\to N$, at each point the
pushforward The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things. * Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" ope ...
(or differential) maps tangent vectors at ''p'' to tangent vectors at ''F(p)'': $F_:T_p M \to T_N$, and on the level of the
tangent bundle Image:Tangent bundle.svg, Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom). In diffe ...
, the pushforward is a
vector bundle homomorphism In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space ''X'' (for example ''X'' could be a topological space, a manifold, or an algebraic variety): to e ...
: $F_*:TM\to TN$. The dual to the pushforward is the
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: in ...
, which "pulls" covectors on ''$N$'' back to covectors on ''$M$'', and ''k''-forms to ''k-''forms: $F^*:\Omega^k\left(N\right) \to \Omega^k\left(M\right)$. In this way smooth functions between manifolds can transport local data, like
vector fields In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each attach ...

and
differential forms In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to define integrands over curve ...
, from one manifold to another, or down to Euclidean space where computations like
integration Integration may refer to: Biology *Modular integration, where different parts in a module have a tendency to vary together *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome ...
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.

Smooth functions between subsets of manifolds

There is a corresponding notion of smooth map for arbitrary subsets of manifolds. If is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriented ...
whose domain of a function, domain and range of a function, range are subsets of manifolds and respectively. ''f'' is said to be smooth if for all there is an open set with and a smooth function such that for all .