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Lecture

Change of Coordinates: Analysis Week 8

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 concept of changing coordinates, focusing on bijective functions, Jacobian matrices, and reciprocal functions. It delves into the inverse function theorem, spherical coordinates, and examples illustrating the application of these concepts.

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Instructors (3)

Boris Buffoni

David Valentin Strütt

Adélie Eliane Garin

Official source

Related concepts (32)

Injective function

In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct elements; that is, x1 ≠ x2 implies f(x1) f(x2). (Equivalently, f(x1) = f(x2) implies x1 = x2 in the equivalent contrapositive statement.) In other words, every element of the function's codomain is the of one element of its domain. The term must not be confused with that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain.

Inverse function

In mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective, and if it exists, is denoted by For a function , its inverse admits an explicit description: it sends each element to the unique element such that f(x) = y. As an example, consider the real-valued function of a real variable given by f(x) = 5x − 7. One can think of f as the function which multiplies its input by 5 then subtracts 7 from the result.

Bijection

In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set; there are no unpaired elements between the two sets. In mathematical terms, a bijective function f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y.

Partial function

In mathematics, a partial function f from a set X to a set Y is a function from a subset S of X (possibly the whole X itself) to Y. The subset S, that is, the domain of f viewed as a function, is called the domain of definition or natural domain of f. If S equals X, that is, if f is defined on every element in X, then f is said to be a total function. More technically, a partial function is a binary relation over two sets that associates every element of the first set to at most one element of the second set; it is thus a functional binary relation.

Surjective function

In mathematics, a surjective function (also known as surjection, or onto function ˈɒn.tuː) is a function f such that every element y can be mapped from some element x such that f(x) = y. In other words, every element of the function's codomain is the of one element of its domain. It is not required that x be unique; the function f may map one or more elements of X to the same element of Y.

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