Lecture

Implicit Functions: Theory and Examples

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

In course

MATH-106(f): Analysis II

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

Description

This lecture covers the theory of implicit functions, focusing on the relationship between explicit and implicit definitions. Starting with the definition of implicit functions, the instructor explains the concept of separability and differentiability. The lecture delves into the neighborhood of a function and the classification of functions based on their properties. The theorem of implicit functions is illustrated through various examples, emphasizing the conditions for the existence and differentiation of implicit functions. The lecture concludes with a detailed explanation of the derivative of implicit functions and the proof of theorems related to implicit functions.

Instructors (3)

Boris Buffoni

Research activityResearch ResearchCurrent Research Fields https://sma.epfl.ch/~buffoni/home.html Teaching & PhD Courses Analysis II MATH-106(b) The course studies fundamental concepts of analysis and the calculus of functions of several variables. Evolutional partial differential equations MATH-421 Fundamental techniques and theories for partial differential equations of evolution. Application to basic examples: first-order equations, wave equation, heat equation. The Cauchy

David Valentin Strütt

Assistanat Vous visitez cette page dans l'optique de rejoindre mon équipe d'asssistant·es pour un de mes cours d'analyse ? Vous trouverez plus d'informations ici. Contributions auteur de polycopiés d'analysecréateur matheminecraftrecord suisse du plus grand nombre de participations à la construction d'un additionneur binaire en dominos (autoproclamé) Je suis né pile le jour de mon anniversaire. Expériences professionnelles Chargé de communication pour MATH et chargé de cours pour la SMA 2019–2024 EPFLCharg

Adélie Eliane Garin

Teaching & PhD Courses Sustainability ENV-101 This course presents global sustainability issues. It introduces planetary boundaries, their systemic interconnections and social justice issues. The interdisciplinary approach introduces group work and systems thinking (interdependencies in a complex system). Sustainability in my research ENG-650 This course is designed to develop the necessary transversal skills to foster sustainability integration in the present and future rese

Official source

Related concepts (25)

Implicit function

In mathematics, an implicit equation is a relation of the form where R is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is An implicit function is a function that is defined by an implicit equation, that relates one of the variables, considered as the value of the function, with the others considered as the arguments. For example, the equation of the unit circle defines y as an implicit function of x if −1 ≤ x ≤ 1, and y is restricted to nonnegative values.

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. It does so by representing the relation as the graph of a function. There may not be a single function whose graph can represent the entire relation, but there may be such a function on a restriction of the domain of the relation. The implicit function theorem gives a sufficient condition to ensure that there is such a function. More precisely, given a system of m equations fi (x1, .

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-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If x0 is an interior point in the domain of a function f, then f is said to be differentiable at x0 if the derivative exists.

Semi-differentiability

In calculus, a branch of mathematics, the notions of one-sided differentiability and semi-differentiability of a real-valued function f of a real variable are weaker than differentiability. Specifically, the function f is said to be right differentiable at a point a if, roughly speaking, a derivative can be defined as the function's argument x moves to a from the right, and left differentiable at a if the derivative can be defined as x moves to a from the left.

Theory

A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be scientific, belong to a non-scientific discipline, or no discipline at all. Depending on the context, a theory's assertions might, for example, include generalized explanations of how nature works. The word has its roots in ancient Greek, but in modern use it has taken on several related meanings.

Related lectures (64)

Optimal Transport: Theory and Applications

Explores the theory of optimal transport, focusing on Lipschitz functions and uniqueness of solutions.

The Role of Symmetries

Delves into symmetries in physics, covering group theory, perturbation, and quantification.

Implicit Functions: Theory and Applications

Explores implicit functions theory, derivatives, solutions existence, and applications.

Linear Applications and Simple Objects

Covers the bijection between linear applications from L(X) to V and applications from X to U(V).

Natural Transformations: Functors and Categories