TheInfoList

In
mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
, the smoothness of a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
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 ga ...
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 In some cultures, a surname, family name, or last name is the portion of one's personal name that indicates their family, tribe or community. Practices vary by culture. The family name may be placed at either the start of ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

s. It is a measure of the highest order of derivative that exists for a function. Consider an
open set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
on the
real line In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
and a function ''f'' defined on that set with real values. Let ''k'' be a non-negative
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
. 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 ga ...
. 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 , the Taylor series of a is an of terms that are expressed in terms of the function's s at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor's series are named after ...
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 (mathematics), function f: \mathbf^n \to \mathbf on a Euclidean space \mathbf^n which is both smooth function, smooth (in the sense of having Continuous function, continuo ...

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 mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

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 function $f(x) = \beginx & \mbox x \geq 0, \\ 0 &\text x < 0\end$ is continuous, but not differentiable at , so it is of class ''C''0, but not of class ''C''1. The function $g(x) = \beginx^2\sin & \textx \neq 0, \\ 0 &\textx = 0\end$ is differentiable, with derivative $g'(x) = \begin-\mathord + 2x\sin(\tfrac) & \textx \neq 0, \\ 0 &\textx = 0.\end$ 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 set, closed (i.e., containing all its limit points) and bounded set, bounded (i.e., having all ...
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 function (mathematics), functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there ex ...
. The functions $f(x)=, x, ^$ 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 The exponential function is a mathematical function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of a ...

is analytic, and hence falls into the class ''C''ω. The
trigonometric function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s are also analytic wherever they are defined. The
bump function In mathematics, a bump function (also called a test function) is a Function (mathematics), function f: \mathbf^n \to \mathbf on a Euclidean space \mathbf^n which is both smooth function, smooth (in the sense of having Continuous function, continuo ...

$f(x) = \begine^ & \text , x, < 1, \\ 0 &\text\end$ 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 Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations, pacts are usually between two or more sovereign state A sovereign state is a po ...
.

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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
$\frac(y_1,y_2,\ldots,y_n)$ 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 $f_i(x_1,x_2,\ldots,x_n)=(\pi_i\circ f)(x_1,x_2,\ldots,x_n)=\pi_i(f(x_1,x_2,\ldots,x_n)) \text i=1,2,3,\ldots,m$ 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
seminorm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...
s $p_=\sup_\left, f^(x)\$ 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 set, closed (i.e., containing all its limit points) and bounded set, bounded (i.e., having all ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
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 normed space, norm that is a combination of Lp norm, ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a s ...
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 Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, bodies (objects), and systems of bodies (groups of objects) without considerin ...

, with which the parameter traces out the curve. Parametric continuity is a concept applied to
parametric curve In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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: zeroth derivative is continuous (curves are continuous) * ''C''1: zeroth and first derivatives are continuous * ''C''2: zeroth, first and second derivatives are continuous * ''Cn'': 0-th through ''n''-th derivatives are continuous

Geometric continuity

The concept of geometrical continuity 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 Conical surface, surface of a cone (geometry), cone with a plane (mathematics), plane. The three types of conic section are the hyperbola, the par ...

(and related shapes) by mathematicians such as
Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the "#1666–1676, 1666–1676" section. ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist, and diplomat. He is a promin ...
,
Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German , , , and writer on music. He is a key figure in the 17th-century , best known for his , and his books ', ', and '. These works also provided one of the foundations for ...

, and
Poncelet The poncelet (symbol p) is an obsolete unit of power, once used in France and replaced by ''cheval vapeur'' (cv, metric horsepower Horsepower (hp) is a unit of measurement A unit of measurement is a definite magnitude (mathematics), mag ...

. The concept was an early attempt at describing, through geometry rather than algebra, the concept of continuity 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 , an ellipse is a surrounding two , 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 , which is the special type of ellipse in which the two focal points are t ...

tends to a
circle A circle is a shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

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 ...
approaches zero, or to a
parabola In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

as it approaches one; and a
hyperbola File:Hyperbel-def-ass-e.svg, 300px, Hyperbola (red): features In mathematics, a hyperbola () (adjective form hyperbolic, ) (plural ''hyperbolas'', or ''hyperbolae'' ()) is a type of smooth function, smooth plane curve, curve lying in a plane, defi ...

tends to a
parabola In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

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 Lite ...

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 Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative ...

. 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 of a function and of $\infty$ (see
projectively extended real line Image:Real projective line.svg, 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 projectiv ...
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 (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

or
surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
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 Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

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 Philosophy (from , ) is the study of general and fundamental questions, such as those about Metaphysics, existence, reason, Epistemology, knowledge, Ethics, values, Philosophy of m ...

, 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 ''archi ...

and
sports car A sports car is a car designed with an emphasis on dynamic performance, such as Automobile handling, handling, acceleration, top speed, or thrill of driving. Sports cars originated in Europe in the early 1900s and are currently produced by man ...
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 (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 , 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 Image:Parametic Cubic Spline.svg, Single knots at 1/3 and 2/3 establish a spline of three cubic polynomials meeting with ''C''2 continuity. Triple knots at both ends of the interval ensure that the curve interpolates the end points In mathematics, ...
are typically chosen; these curves are frequently used in
industrial design Industrial design is a process of design A design is a plan or specification for the construction of an object or system or for the implementation of an activity or process, or the result of that plan or specification in the form of a prototyp ...

.

Other concepts

Relation to analyticity

While all
analytic function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 (mathematics), function f: \mathbf^n \to \mathbf on a Euclidean space \mathbf^n which is both smooth function, smooth (in the sense of having Continuous function, continuo ...

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 Sine wave, sinusoids combined by a weighted summation. With appropriate weights, one cycle (or ''period'') of the summation can be made to approximate an ...
; 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
'' and
topology glossary This is a glossary of some terms used in the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and c ...
); 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 vector space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an ...
s, for example to show that
Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ''g'p'' on the tangent space ''T'p'M'' at each poin ...
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 (mathematics), function f: \mathbf^n \to \mathbf on a Euclidean space \mathbf^n which is both smooth function, smooth (in the sense of having Continuous function, continuo ...

on the real line, that is, a smooth function ''f'' that takes the value 0 outside an interval 'a'',''b''and such that $f(x) > 0 \quad \text \quad a < x < b.\,$ Given a number of overlapping intervals on the line, bump functions can be constructed on each of them, and on semi-infinite intervals $\left(-\infty, c\right]$ 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 functions; their different behavior relative to existence and analytic continuation is one of the roots of Sheaf (mathematics), sheaf 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 vector space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an ...
$M$, of dimension $m,$ and an
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$\mathfrak = \_\alpha,$ then a map $f:M\to \R$ is smooth on $M$ if for all $p \in M$ there exists a chart $\left(U, \phi\right) \in \mathfrak,$ such that $p \in U,$ and $f \circ \phi^ : \phi\left(U\right) \to \R$ is a smooth function from a neighborhood of $\phi\left(p\right)$ in $\R^m$ to $\R$ (all partial derivatives up to a given order are continuous). Smoothness can be checked with respect to any
chart A chart is a graphical representation Graphic communication as the name suggests is communication using graphic elements. These elements include symbols such as glyphs and icon (computing), icons, images such as drawings and photographs, and ca ...
of the atlas that contains $p,$ since the smoothness requirements on the transition functions between charts ensure that if $f$is smooth near $p$ in one chart it will be smooth near $p$ in any other chart. If $F : M \to N$ is a map from $M$ to an $n$-dimensional manifold $N$, then $F$ is smooth if, for every $p \in M,$ there is a chart $\left(U,\phi\right)$ containing $p,$ and a chart $\left(V, \psi\right)$ containing $F\left(p\right)$ such that $F\left(U\right) \subset V,$ and $\psi \circ F \circ \phi^ : \phi\left(U\right) \to \psi\left(V\right)$ is a smooth function from $\R^n.$ Smooth maps between manifolds induce linear maps between
tangent spaces In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
: 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 $F\left(p\right)$: $F_ : T_p M \to T_N,$ 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 differen ...

, the pushforward is a vector bundle homomorphism: $F_* : TM \to TN.$ 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 (disambiguation), pushforward. Precomposition Precomposition with a Function (mathematics), function probabl ...
, 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 (geometry), vector to each point in a subset of Space (mathematics), space. For instance, a vector field in the plane can be visualised as a collection of arrows with a gi ...

and
differential forms In the mathematical fields of differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems i ...
, 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.

Smooth functions between subsets of manifolds

There is a corresponding notion of smooth map for arbitrary subsets of manifolds. If $f : X \to Y$ is a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
whose
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 ...
and
range Range may refer to: Geography * Range (geographic)A range, in geography, is a chain of hill A hill is a landform A landform is a natural or artificial feature of the solid surface of the Earth or other planetary body. Landforms together ...
are subsets of manifolds $X \subseteq M$ and $Y \subseteq N$ respectively. $f$ is said to be smooth if for all $x \in X$ there is an open set $U \subseteq M$ with $x \in U$ and a smooth function $F : U \to N$ such that $F\left(p\right) = f\left(p\right)$ for all $p \in U \cap X.$