In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (differentiability class) it has over its domain. [1]
A function of class is a function of smoothness at least k; that is, a function of class is a function that has a kth 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 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.
Consider an open set 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.
The function is continuous, but not differentiable at x = 0, so it is of class C0, but not of class C1.
For each even integer k, the function is continuous and k times differentiable at all x. At x = 0, however, is not (k + 1) times differentiable, so is of class Ck, but not of class Cj where j > k.
The function is differentiable, with derivative
Because oscillates as x → 0, is not continuous at zero. Therefore, is differentiable but not of class C1.
The function is 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.
The exponential function 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 .
The bump function is smooth, so of class C∞, but it is not analytic at x = ±1, and hence is not of class Cω. The function f is an example of a smooth function with compact support.
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 non-negative integers, such that , 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 .
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.
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 (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.
The various order of parametric continuity can be described as follows: [7]
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:
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 if they satisfy equations known as Beta-constraints. For example, the Beta-constraints for continuity are:
where , , and are arbitrary, but is constrained to be positive. [8] : 65 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.[ citation needed ]
A rounded rectangle (with ninety degree circular arcs at the four corners) has continuity, but does not have 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 continuity is required, then cubic splines are typically chosen; these curves are frequently used in industrial design.
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 [ citation needed ].
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 [ citation needed ].
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.
Given a smooth manifold , of dimension and an atlas then a map is smooth on if for all there exists a chart such that and 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 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: 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]
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
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity of change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a derivative is called differentiation.
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.
In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fiber bundles.
In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. There are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial.
In mathematical analysis, Lipschitz continuity, named after German mathematician 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 that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the Lipschitz constant of the function. For instance, every function that is defined on an interval and has a bounded first derivative is Lipschitz continuous.
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In 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 interior point in its domain. A differentiable function is smooth and does not contain any break, angle, or cusp.
In mathematics, a foliation is an equivalence relation on an n-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension p, modeled on the decomposition of the real coordinate space Rn into the cosets x + Rp of the standardly embedded subspace Rp. The equivalence classes are called the leaves of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable, or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class Cr it is usually understood that r ≥ 1. The number p is called the dimension of the foliation and q = n − p is called its codimension.
In mathematics, smooth functions and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The converse is not true, as demonstrated with the counterexample below.
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices.
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In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers , or a subset of that contains an interval of positive length. Most real functions that are considered and studied are differentiable in some interval. The most widely considered such functions are the real functions, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers.
In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.
In mathematics, a real or complex-valued function f on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are real constants C ≥ 0, α > 0, such that for all x and y in the domain of f. More generally, the condition can be formulated for functions between any two metric spaces. The number is called the exponent of the Hölder condition. A function on an interval satisfying the condition with α > 1 is constant. If α = 1, then the function satisfies a Lipschitz condition. For any α > 0, the condition implies the function is uniformly continuous. The condition is named after Otto Hölder.
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In mathematics, and more precisely, in functional Analysis and PDEs, the Schauder estimates are a collection of results due to Juliusz Schauder concerning the regularity of solutions to linear, uniformly elliptic partial differential equations. The estimates say that when the equation has appropriately smooth terms and appropriately smooth solutions, then the Hölder norm of the solution can be controlled in terms of the Hölder norms for the coefficient and source terms. Since these estimates assume by hypothesis the existence of a solution, they are called a priori estimates.
In mathematics, the Kodaira–Spencer map, introduced by Kunihiko Kodaira and Donald C. Spencer, is a map associated to a deformation of a scheme or complex manifold X, taking a tangent space of a point of the deformation space to the first cohomology group of the sheaf of vector fields on X.
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