Inverse function theorem, the Glossary

Table of Contents

1. 63 relations: Adjugate matrix, Éditions Hermann, Édouard Goursat, Émile Picard, Banach fixed-point theorem, Banach manifold, Banach space, Bounded operator, Cauchy sequence, Chain rule, Compact space, Continuous function, Contraction mapping, Derivative, Determinant, Diffeomorphism, Differentiable function, Differentiable manifold, Differential calculus, Domain of a function, Effective method, Embedding, Exhaustion by compact sets, Extreme value theorem, Fixed-point theorem, Formula, Function (mathematics), Geometric series, Henri Cartan, Holomorphic function, Implicit function theorem, Injective function, Inverse function, Jacobian conjecture, Jacobian matrix and determinant, Jean Dieudonné, Lars Hörmander, Linear map, Local diffeomorphism, Manifold, Mathematics, Mean value theorem, Moore–Penrose inverse, Multivariable calculus, Nash–Moser theorem, Necessity and sufficiency, Neighbourhood (mathematics), Newton's method, O-minimal theory, Ordinary differential equation, ... Expand index (13 more) »

2. Inverse functions
3. Theorems in calculus
4. Theorems in real analysis

Adjugate matrix

In linear algebra, the adjugate of a square matrix is the transpose of its cofactor matrix and is denoted by.

See Inverse function theorem and Adjugate matrix

Éditions Hermann

Éditions Hermann is a French publishing house founded in 1876, by the French professor of mathematics Arthur Hermann.

See Inverse function theorem and Éditions Hermann

Édouard Goursat

Édouard Jean-Baptiste Goursat (21 May 1858 – 25 November 1936) was a French mathematician, now remembered principally as an expositor for his Cours d'analyse mathématique, which appeared in the first decade of the twentieth century.

See Inverse function theorem and Édouard Goursat

Émile Picard

Charles Émile Picard (24 July 1856 – 11 December 1941) was a French mathematician.

See Inverse function theorem and Émile Picard

Banach fixed-point theorem

In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points.

See Inverse function theorem and Banach fixed-point theorem

Banach manifold

In mathematics, a Banach manifold is a manifold modeled on Banach spaces.

See Inverse function theorem and Banach manifold

Banach space

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

See Inverse function theorem and Banach space

Bounded operator

In functional analysis and operator theory, a bounded linear operator is a linear transformation L: X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector spaces (a special type of TVS), then L is bounded if and only if there exists some M > 0 such that for all x \in X, \|Lx\|_Y \leq M \|x\|_X.

See Inverse function theorem and Bounded operator

Cauchy sequence

In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses.

See Inverse function theorem and Cauchy sequence

Chain rule

In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and. Inverse function theorem and chain rule are theorems in calculus.

See Inverse function theorem and Chain rule

Compact space

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space.

See Inverse function theorem and Compact space

Continuous function

In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function.

See Inverse function theorem and Continuous function

Contraction mapping

In mathematics, a contraction mapping, or contraction or contractor, on a metric space (M, d) is a function f from M to itself, with the property that there is some real number 0 \leq k such that for all x and y in M, The smallest such value of k is called the Lipschitz constant of f. Contractive maps are sometimes called Lipschitzian maps.

See Inverse function theorem and Contraction mapping

Derivative

The derivative is a fundamental tool of calculus that quantifies the sensitivity of change of a function's output with respect to its input.

See Inverse function theorem and Derivative

Determinant

In mathematics, the determinant is a scalar-valued function of the entries of a square matrix.

See Inverse function theorem and Determinant

Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds.

See Inverse function theorem and Diffeomorphism

Differentiable function

In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. Inverse function theorem and differentiable function are multivariable calculus.

See Inverse function theorem and Differentiable function

Differentiable 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 apply calculus.

See Inverse function theorem and Differentiable manifold

Differential calculus

In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change.

See Inverse function theorem and Differential calculus

Domain of a function

In mathematics, the domain of a function is the set of inputs accepted by the function.

See Inverse function theorem and Domain of a function

Effective method

In logic, mathematics and computer science, especially metalogic and computability theory, an effective methodHunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Press, 1971 or effective procedure is a procedure for solving a problem by any intuitively 'effective' means from a specific class.

See Inverse function theorem and Effective method

Embedding

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. Inverse function theorem and embedding are differential topology.

See Inverse function theorem and Embedding

Exhaustion by compact sets

In mathematics, especially general topology and analysis, an exhaustion by compact sets of a topological space X is a nested sequence of compact subsets K_i of X (i.e. K_1\subseteq K_2\subseteq K_3\subseteq\cdots), such that K_i is contained in the interior of K_, i.e. K_i\subseteq\text(K_) for each i and X.

See Inverse function theorem and Exhaustion by compact sets

Extreme value theorem

In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed and bounded interval, then f must attain a maximum and a minimum, each at least once. Inverse function theorem and extreme value theorem are theorems in calculus and theorems in real analysis.

See Inverse function theorem and Extreme value theorem

Fixed-point theorem

In mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x).

See Inverse function theorem and Fixed-point theorem

Formula

In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a chemical formula.

See Inverse function theorem and Formula

Function (mathematics)

In mathematics, a function from a set to a set assigns to each element of exactly one element of.

See Inverse function theorem and Function (mathematics)

Geometric series

In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms.

See Inverse function theorem and Geometric series

Henri Cartan

Henri Paul Cartan (8 July 1904 – 13 August 2008) was a French mathematician who made substantial contributions to algebraic topology.

See Inverse function theorem and Henri Cartan

Holomorphic function

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space.

See Inverse function theorem and Holomorphic function

Implicit function theorem

In multivariable calculus, the implicit function theorem is a tool that allows relations to be converted to functions of several real variables. Inverse function theorem and implicit function theorem are theorems in calculus and theorems in real analysis.

See Inverse function theorem and Implicit function theorem

Injective function

In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies.

See Inverse function theorem and Injective function

Inverse function

In mathematics, the inverse function of a function (also called the inverse of) is a function that undoes the operation of. Inverse function theorem and inverse function are inverse functions.

See Inverse function theorem and Inverse function

Jacobian conjecture

In mathematics, the Jacobian conjecture is a famous unsolved problem concerning polynomials in several variables.

See Inverse function theorem and Jacobian conjecture

Jacobian matrix and determinant

In vector calculus, the Jacobian matrix of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. Inverse function theorem and Jacobian matrix and determinant are multivariable calculus.

See Inverse function theorem and Jacobian matrix and determinant

Jean Dieudonné

Jean Alexandre Eugène Dieudonné (1 July 1906 – 29 November 1992) was a French mathematician, notable for research in abstract algebra, algebraic geometry, and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the Éléments de géométrie algébrique project of Alexander Grothendieck, and as a historian of mathematics, particularly in the fields of functional analysis and algebraic topology.

See Inverse function theorem and Jean Dieudonné

Lars Hörmander

Lars Valter Hörmander (24 January 1931 – 25 November 2012) was a Swedish mathematician who has been called "the foremost contributor to the modern theory of linear partial differential equations".

See Inverse function theorem and Lars Hörmander

Linear map

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication.

See Inverse function theorem and Linear map

Local diffeomorphism

In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between smooth manifolds that preserves the local differentiable structure. Inverse function theorem and local diffeomorphism are inverse functions.

See Inverse function theorem and Local diffeomorphism

Manifold

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

See Inverse function theorem and Manifold

Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

See Inverse function theorem and Mathematics

Mean value theorem

In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. Inverse function theorem and mean value theorem are theorems in calculus and theorems in real analysis.

See Inverse function theorem and Mean value theorem

Moore–Penrose inverse

In mathematics, and in particular linear algebra, the Moore–Penrose inverse of a matrix, often called the pseudoinverse, is the most widely known generalization of the inverse matrix.

See Inverse function theorem and Moore–Penrose inverse

Multivariable calculus

Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables (multivariate), rather than just one.

See Inverse function theorem and Multivariable calculus

Nash–Moser theorem

In the mathematical field of analysis, the Nash–Moser theorem, discovered by mathematician John Forbes Nash and named for him and Jürgen Moser, is a generalization of the inverse function theorem on Banach spaces to settings when the required solution mapping for the linearized problem is not bounded. Inverse function theorem and Nash–Moser theorem are inverse functions.

See Inverse function theorem and Nash–Moser theorem

Necessity and sufficiency

In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements.

See Inverse function theorem and Necessity and sufficiency

Neighbourhood (mathematics)

In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.

See Inverse function theorem and Neighbourhood (mathematics)

Newton's method

In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function.

See Inverse function theorem and Newton's method

O-minimal theory

In mathematical logic, and more specifically in model theory, an infinite structure (M,<,...) that is totally ordered by Knight, Pillay and Steinhorn (1986), Pillay and Steinhorn (1988).

See Inverse function theorem and O-minimal theory

Ordinary differential equation

In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable.

See Inverse function theorem and Ordinary differential equation

Picard–Lindelöf theorem

In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution.

See Inverse function theorem and Picard–Lindelöf theorem

Pushforward (differential)

In differential geometry, pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces.

See Inverse function theorem and Pushforward (differential)

Rank (differential topology)

In mathematics, the rank of a differentiable map f:M\to N between differentiable manifolds at a point p\in M is the rank of the derivative of f at p. Recall that the derivative of f at p is a linear map from the tangent space at p to the tangent space at f(p).

See Inverse function theorem and Rank (differential topology)

Rank (linear algebra)

In linear algebra, the rank of a matrix is the dimension of the vector space generated (or spanned) by its columns.

See Inverse function theorem and Rank (linear algebra)

Real closed field

In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers.

See Inverse function theorem and Real closed field

Roger Godement

Roger Godement (1 October 1921 – 21 July 2016) was a French mathematician, known for his work in functional analysis as well as his expository books.

See Inverse function theorem and Roger Godement

Semi-continuity

In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity.

See Inverse function theorem and Semi-continuity

Serge Lang

Serge Lang (May 19, 1927 – September 12, 2005) was a French-American mathematician and activist who taught at Yale University for most of his career.

See Inverse function theorem and Serge Lang

Submersion (mathematics)

In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective.

See Inverse function theorem and Submersion (mathematics)

The American Mathematical Monthly

The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894.

See Inverse function theorem and The American Mathematical Monthly

Total derivative

In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Inverse function theorem and total derivative are multivariable calculus.

See Inverse function theorem and Total derivative

Variable (mathematics)

In mathematics, a variable (from Latin variabilis, "changeable") is a symbol that represents a mathematical object.

See Inverse function theorem and Variable (mathematics)

Vector-valued function

A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors.

See Inverse function theorem and Vector-valued function

See also

Inverse functions

• Arg max
• Branch point
• Equation solving
• Fatou–Bieberbach domain
• Inverse function
• Inverse function rule
• Inverse function theorem
• Inverse hyperbolic functions
• Inverse problems
• Inverse trigonometric functions
• Jankov–von Neumann uniformization theorem
• Lagrange inversion theorem
• Lagrange reversion theorem
• Local diffeomorphism
• Local inverse
• Logarithm of a matrix
• Nash–Moser theorem
• Self number

Theorems in calculus

• Bioche's rules
• Cantor's intersection theorem
• Chain rule
• Darboux's theorem (analysis)
• Differentiation of integrals
• Differentiation rules
• Divergence theorem
• Extreme value theorem
• Fermat's theorem (stationary points)
• Fubini's theorem
• Fundamental theorem of calculus
• General Leibniz rule
• Generalized Stokes theorem
• Gradient theorem
• Green's theorem
• Helmholtz decomposition
• Implicit function theorem
• Increment theorem
• Integral of inverse functions
• Integration by parts
• Integration using Euler's formula
• Intermediate value theorem
• Inverse function rule
• Inverse function theorem
• L'Hôpital's rule
• Linearity of differentiation
• Mean value theorem
• Monotone convergence theorem
• Pappus's centroid theorem
• Power rule
• Product rule
• Quotient rule
• Reciprocal rule
• Rolle's theorem
• Stokes' theorem
• Taylor's theorem
• Triple product rule
• Uniqueness theorem for Poisson's equation

Theorems in real analysis

• Abel's theorem
• Anderson's theorem
• Arzelà–Ascoli theorem
• Bernstein's theorem on monotone functions
• Blumberg theorem
• Bohr–Favard inequality
• Caristi fixed-point theorem
• Darboux's theorem (analysis)
• Dini's theorem
• Discontinuities of monotone functions
• Dominated convergence theorem
• Extreme value theorem
• Fatou–Lebesgue theorem
• Fermat's theorem (stationary points)
• Fubini's theorem on differentiation
• Fundamental theorem of calculus
• Glaeser's composition theorem
• Hardy's inequality
• Heine–Borel theorem
• Identity theorem
• Implicit function theorem
• Intermediate value theorem
• Inverse function theorem
• Kirszbraun theorem
• Kolmogorov–Arnold representation theorem
• L'Hôpital's rule
• Lagrange inversion theorem
• Lebesgue differentiation theorem
• Lusin's theorem
• Mean value theorem
• Monotone convergence theorem
• Nested intervals
• Riemann series theorem
• Riesz–Fischer theorem
• Rolle's theorem
• Routh–Hurwitz theorem
• Steinhaus theorem
• Sturm's theorem
• Taylor's theorem
• Titchmarsh convolution theorem
• Uniform limit theorem
• Vitali–Carathéodory theorem
• Watson's lemma
• Zahorski theorem
• Śleszyński–Pringsheim theorem

References

[1] https://en.wikipedia.org/wiki/Inverse_function_theorem

Also known as Constant rank theorem, Derivative rule for inverses, Inverse transformation theorem, Inversion theorem.

, Picard–Lindelöf theorem, Pushforward (differential), Rank (differential topology), Rank (linear algebra), Real closed field, Roger Godement, Semi-continuity, Serge Lang, Submersion (mathematics), The American Mathematical Monthly, Total derivative, Variable (mathematics), Vector-valued function.