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# Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous. [1]

71 relations: Aesthetics, Affine transformation, Analytic continuation, Analytic function, Architecture, Atlas (topology), Banach space, Brian A. Barsky, Bump function, Circle, Compact space, Conic section, Continuous function, Curve, Derivative, Differentiable function, Differentiable manifold, Differential structure, Domain of a function, Eccentricity (mathematics), Ellipse, Exponential function, Fabius function, Fourier series, Fréchet space, Function (mathematics), Glossary of topology, Gottfried Wilhelm Leibniz, Holomorphic function, Hyperbola, Implicit function, Industrial design, Integer, Jean-Victor Poncelet, Johannes Kepler, Line (geometry), Lipschitz continuity, Manifold, Mathematical analysis, Meagre set, Non-analytic smooth function, Norm (mathematics), Open set, Parabola, Parametric equation, Partial derivative, Partial differential equation, Partition of unity, Projectively extended real line, Quasi-analytic function, ... Expand index (21 more) »

## Aesthetics

Aesthetics (also spelled esthetics) is a branch of philosophy that explores the nature of art, beauty, and taste, with the creation and appreciation of beauty.

## Affine transformation

In geometry, an affine transformation, affine mapBerger, Marcel (1987), p. 38.

## Analytic continuation

In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function.

## Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series.

## Architecture

Architecture is both the process and the product of planning, designing, and constructing buildings or any other structures.

## Atlas (topology)

In mathematics, particularly topology, one describes a manifold using an atlas.

## Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.

## Brian A. Barsky

Brian A. Barsky is a Professor at the University of California, Berkeley, working in computer graphics and geometric modeling as well as in optometry and vision science.

## Bump function

In mathematics, a bump function is a function f:\mathbf^n\rightarrow \mathbf on a Euclidean space \mathbf^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported.

## Circle

A circle is a simple closed shape.

## Compact space

In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).

## Conic section

In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane.

## Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

## Curve

In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight.

## Derivative

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).

## 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.

## Differentiable 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.

## Differential structure

In mathematics, an n-dimensional differential structure (or differentiable structure) on a set M makes M into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows for differential calculus on the manifold.

## Domain of a function

In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.

## Eccentricity (mathematics)

In mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section.

## Ellipse

In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.

## Exponential function

In mathematics, an exponential function is a function of the form in which the argument occurs as an exponent.

## Fabius function

In mathematics, the Fabius function is an example of an infinitely differentiable function that is nowhere analytic, found by.

## Fourier series

In mathematics, a Fourier series is a way to represent a function as the sum of simple sine waves.

## Fréchet space

In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## Glossary of topology

This is a glossary of some terms used in the branch of mathematics known as topology.

## Gottfried Wilhelm Leibniz

Gottfried Wilhelm (von) Leibniz (or; Leibnitz; – 14 November 1716) was a German polymath and philosopher who occupies a prominent place in the history of mathematics and the history of philosophy.

## Holomorphic function

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.

## Hyperbola

In mathematics, a hyperbola (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 is the solution set.

## Implicit function

In mathematics, an implicit equation is a relation of the form R(x_1,\ldots, x_n).

## Industrial design

Industrial design is a process of design applied to products that are to be manufactured through techniques of mass production.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## Jean-Victor Poncelet

Jean-Victor Poncelet (1 July 1788 – 22 December 1867) was a French engineer and mathematician who served most notably as the Commanding General of the École Polytechnique.

## Johannes Kepler

Johannes Kepler (December 27, 1571 – November 15, 1630) was a German mathematician, astronomer, and astrologer.

## Line (geometry)

The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.

## Lipschitz continuity

In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions.

## Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

## Mathematical analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

## Meagre set

In the mathematical fields of general topology and descriptive set theory, a meagre set (also called a meager set or a set of first category) is a set that, considered as a subset of a (usually larger) topological space, is in a precise sense small or negligible.

## Non-analytic smooth function

In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions.

## Norm (mathematics)

In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero.

## Open set

In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.

## Parabola

In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped.

## Parametric equation

In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters.

## Partial derivative

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).

## Partial differential equation

In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.

## Partition of unity

In mathematics, a partition of unity of a topological space X is a set R of continuous functions from X to the unit interval such that for every point, x\in X,.

## Projectively extended real line

In real analysis, the projectively extended real line (also called the one-point compactification of the real line), is the extension of the number line by a point denoted.

## Quasi-analytic function

In mathematics, a quasi-analytic class of functions is a generalization of the class of real analytic functions based upon the following fact.

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.

## Range (mathematics)

In mathematics, and more specifically in naive set theory, the range of a function refers to either the codomain or the image of the function, depending upon usage.

## Real line

In mathematics, the real line, or real number line is the line whose points are the real numbers.

## Recursion

Recursion occurs when a thing is defined in terms of itself or of its type.

## Riemannian manifold

In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.

## Sheaf (mathematics)

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.

## Sinuosity

Sinuosity, sinuosity index, or sinuosity coefficient of a continuously differentiable curve having at least one inflection point is the ratio of the curvilinear length (along the curve) and the Euclidean distance (straight line) between the end points of the curve.

## Smooth number

In number theory, a smooth (or friable) number is an integer which factors completely into small prime numbers.

## 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 itself and its derivatives up to a given order.

## Speed

In everyday use and in kinematics, the speed of an object is the magnitude of its velocity (the rate of change of its position); it is thus a scalar quantity.

## Spline (mathematics)

In mathematics, a spline is a function defined piecewise by polynomials.

## Sports car

A sports car, or sportscar, is a small, usually two-seater, two-door automobile designed for spirited performance and nimble handling.

## Support (mathematics)

In mathematics, the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero.

## Surface (topology)

In topology and differential geometry, a surface is a two-dimensional manifold, and, as such, may be an "abstract surface" not embedded in any Euclidean space.

## 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.

## Tangent bundle

In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M. As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector.

## Tangent space

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 one point from the other.

## Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

## Transcendental number

In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a nonzero polynomial equation with integer (or, equivalently, rational) coefficients.

## Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle.

## Union (set theory)

In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.

## References

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