Category: Smooth functions

In mathematics, mollifiers (also known as approximations to the identity) are smooth functions with special properties, used for example in distribution theory to create sequences of smooth functions
Sard's theorem
In mathematics, Sard's theorem, also known as Sard's lemma or the Morse–Sard theorem, is a result in mathematical analysis that asserts that the set of critical values (that is, the image of the set o
Fundamental lemma of calculus of variations
In mathematics, specifically in the calculus of variations, a variation δf of a function f can be concentrated on an arbitrarily small interval, but not a single point.Accordingly, the necessary condi
Jet (mathematics)
In mathematics, the jet is an operation that takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain. Although this is the defini
Spaces of test functions and distributions
In mathematical analysis, the spaces of test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application of distributions. Test functions are usual
Differentiable function
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-verti
Submersion (mathematics)
In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a sub
Flat function
In mathematics, especially real analysis, a flat function is a smooth function all of whose derivatives vanish at a given point . The flat functions are, in some sense, the antitheses of the analytic
Non-analytic smooth function
In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of
Connection (principal bundle)
In mathematics, and especially differential geometry and gauge theory, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fiber
Morse theory
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insi
Smooth approximation
No description available.
Colombeau algebra
In mathematics, a Colombeau algebra is an algebra of a certain kind containing the space of Schwartz distributions. While in classical distribution theory a general multiplication of distributions is
Immersion (mathematics)
In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential (or pushforward) is everywhere injective. Explicitly, f : M → N is an immersion if is an i
Gevrey class
In mathematics, the Gevrey classes on a domain , introduced by Maurice Gevrey, are spaces of functions 'between' the space of analytic functions and the space of smooth (infinitely differentiable) fun
Distribution (mathematics)
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to
Pushforward (differential)
In differential geometry, pushforward is a linear approximation of smooth maps on tangent spaces. Suppose that φ : M → N is a smooth map between smooth manifolds; then the differential of φ, , at a po
Morse–Bott function
No description available.
Connection (fibred manifold)
In differential geometry, a fibered manifold is surjective submersion of smooth manifolds Y → X. Locally trivial fibered manifolds are fiber bundles. Therefore, a notion of connection on fibered manif
Critical point (mathematics)
Critical point is a wide term used in many branches of mathematics. When dealing with functions of a real variable, a critical point is a point in the domain of the function where the function is eith
Constructive function theory
In mathematical analysis, constructive function theory is a field which studies the connection between the smoothness of a function and its degree of approximation. It is closely related to approximat
Cartan connection
In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general c
Connection form
In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms. Historically, c
Schwartz space
In mathematics, Schwartz space is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on t
Rank (differential topology)
In mathematics, the rank of a differentiable map between differentiable manifolds at a point is the rank of the derivative of at . Recall that the derivative of at is a linear map from the tangent spa
Ridge detection
In image processing, ridge detection is the attempt, via software, to locate ridges in an image, defined as curves whose points are local maxima of the function, akin to geographical ridges. For a fun
Bump function
In mathematics, a bump function (also called a test function) is a function on a Euclidean space which is both smooth (in the sense of having continuous derivatives of all orders) and compactly suppor
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: If f is an analytic function on an interval [a,b] ⊂ R, an
Frölicher space
In mathematics, Frölicher spaces extend the notions of calculus and smooth manifolds. They were introduced in 1982 by the mathematician Alfred Frölicher.
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called differentiability class. At the very minimum, a fu
Affine connection
In differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were