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Lecture

Change of Coordinates: Notation and Examples

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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 introduces the notation for change of coordinates, covering the book, exercises, and exams. It explains the concept using examples and discusses the notation used in exercises and exams. The lecture progresses to illustrate the concept with examples of plain coordinates and discusses the verification of formulas. The instructor demonstrates the application of the notation in various scenarios, emphasizing the importance of understanding the notation introduced. The lecture concludes with a detailed explanation of the notation's application in different coordinate systems and the verification of formulas.

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Instructors (3)

David Valentin Strütt

Adélie Eliane Garin

Boris Buffoni

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Related concepts (17)

Coordinate system

In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring.

Homogeneous coordinates

In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts.

Curvilinear coordinates

In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible (a one-to-one map) at each point. This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back. The name curvilinear coordinates, coined by the French mathematician Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved.

Notation for differentiation

In differential calculus, there is no single uniform notation for differentiation. Instead, various notations for the derivative of a function or variable have been proposed by various mathematicians. The usefulness of each notation varies with the context, and it is sometimes advantageous to use more than one notation in a given context. The most common notations for differentiation (and its opposite operation, the antidifferentiation or indefinite integration) are listed below.

Leibniz's notation

In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent infinitely small (or infinitesimal) increments of x and y, respectively, just as Δx and Δy represent finite increments of x and y, respectively. Consider y as a function of a variable x, or y = f(x).

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