Inverse function theorem

In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its derivative is continuous and non-zero at the point. The theorem also gives a formula for the derivative of the inverse function. In multivariable calculus, this theorem can be generalized to any continuously differentiable, vector-valued function whose Jacobian determinant is nonzero at a point in its domain, giving a formula for the Jacobian matrix of the inverse. There are also versions of the inverse function theorem for complex holomorphic functions, for differentiable maps between manifolds, for differentiable functions between Banach spaces, and so forth. The theorem was first established by Picard and Goursat using an iterative scheme: the basic idea is to prove a fixed point theorem using the contraction mapping theorem. Statements For functions of a single variable,

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications

Related people

Related units

Related concepts

Related courses

Related lectures

Summary

Official source

About this result

Related publications (2)

Related publications

Related people

Related units

Related concepts (15)

Related concepts

Related courses (20)

Étudier les concepts fondamentaux d'analyse, et le calcul différentiel et intégral des fonctions réelles de plusieurs variables.

Etudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles de plusieurs variables.

The purpose of the course is to introduce the basic notions of linear algebra and its applications.

Related courses

Related lectures (38)

Related lectures

Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or n-manifold for sho

Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are

Tangent space

In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics

Equidistribution of Heegner points and ternary quadratic forms

Dimitar Petkov Jetchev

We prove new equidistribution results for Galois orbits of Heegner points with respect to single reduction maps at inert primes. The arguments are based on two different techniques: primitive representations of integers by quadratic forms and distribution relations for Heegner points. Our results generalize an equidistribution result with respect to a single reduction map established by Cornut and Vatsal in the sense that we allow both the fundamental discriminant and the conductor to grow. Moreover, for fixed fundamental discriminant and variable conductor, we deduce an effective surjectivity theorem for the reduction map from Heegner points to supersingular points at a fixed inert prime. Our results are applicable to the setting considered by Kolyvagin in the construction of the Heegner points Euler system.

Springer Verlag2011

We study the Gowers uniformity norms of functions over Z/pZ which are trace functions of l-adic sheaves. On the one hand, we establish a strong inverse theorem for these functions, and on the other hand this gives many explicit examples of functions with Gowers norms of size comparable to that of "random" functions.

Cambridge University Press2013

Étudier les concepts fondamentaux d'analyse, et le calcul différentiel et intégral des fonctions réelles de plusieurs variables.

Etudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles de plusieurs variables.

The purpose of the course is to introduce the basic notions of linear algebra and its applications.