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Concept# Implicit function

Summary

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.
The implicit function theorem provides conditions under which some kinds of implicit equations define implicit functions, namely those that are obtained by equating to zero multivariable functions that are continuously differentiable.
A common type of implicit function is an inverse function. Not all functions have a unique inverse function. If g is a function of x that has a unique inverse, then the inverse function of g, called g−1, is the unique function giving a solution of the equation
for x in terms of y. This solution can then be written as
Defining g−1 as the inverse of g is an implicit definition. For some functions g, g−1(y) can be written out explicitly as a closed-form expression — for instance, if g(x) = 2x − 1, then g−1(y) = 1/2(y + 1). However, this is often not possible, or only by introducing a new notation (as in the product log example below).
Intuitively, an inverse function is obtained from g by interchanging the roles of the dependent and independent variables.
Example: The product log is an implicit function giving the solution for x of the equation y − xex = 0.
Algebraic function
An algebraic function is a function that satisfies a polynomial equation whose coefficients are themselves polynomials. For example, an algebraic function in one variable x gives a solution for y of an equation
where the coefficients ai(x) are polynomial functions of x. This algebraic function can be written as the right side of the solution equation y = f(x). Written like this, f is a multi-valued implicit function.

Official source

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MATH-106(a): Analysis II

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

MATH-106(c): Analysis II

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

MATH-106(b): Analysis II

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

Level set

In mathematics, a level set of a real-valued function f of n real variables is a set where the function takes on a given constant value c, that is: When the number of independent variables is two, a level set is called a level curve, also known as contour line or isoline; so a level curve is the set of all real-valued solutions of an equation in two variables x_1 and x_2. When n = 3, a level set is called a level surface (or isosurface); so a level surface is the set of all real-valued roots of an equation in three variables x_1, x_2 and x_3.

Function of several real variables

In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of a real variable to several variables. The "input" variables take real values, while the "output", also called the "value of the function", may be real or complex.

Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.

Implicit Functions Theorem

Explains the Implicit Functions Theorem and conditions for defining functions implicitly.

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Explores the tangent plane to implicit functions, emphasizing equations and graphical representations.

Implicit Functions

Covers directional derivatives, implicit functions, and finding equations of tangents.

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