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Lecture

Functions on R^n: Limits and Partial Differentiation

In course

MATH-431: Theory of stochastic calculus

Introduction to the mathematical theory of stochastic calculus: construction of the Ito stochastic integral, proof of the Ito formula, introduction to stochastic differential equations, Girsanov's theorem and the Feynman-Kac formula, the martingale represe ...

Description

This lecture explores functions f: R^n → R, focusing on the general theory, limits, and partial differentiation. The concept of limits is discussed, emphasizing the definition and visualization of functions. Additionally, partial differentiation is introduced, highlighting the existence and properties of partial derivatives. Examples are provided to illustrate the concepts.

Instructors (2)

Thomas Mountford

D'origine britannique, né en 1961. Il a reçu la bourse présidentielle des jeunes investisseurs (Presidential Young Investigator Grant) en 1990 et le prix de la Fraternité Sloan (Sloan Fellowship) en 1991. Son travail montre que les valeurs critiques d'une classe large de de systèmes de particules proches sont égales à 1, et que, suite à plusieurs travaux sur la trajectoire du mouvement brownien, incluant une simulation numérique, le nombre d'îlots browniens en 2 dimensions tend vers l'infini quand leurs tailles tendent vers 0. Il reçoit le prix Rosenbaum en 1993 lui ouvrant ainsi les portes du Isaac Newton Institute à Cambridge. Il est également décoré par le prix Rollo Davidson en 1995 et nommé membre honoraire de l'Institute of Mathematical Statistics en 2001. Assistant Profeseur de Mathematique a UCLA 1987- 1991 Associate Professeur de Mathematique a UCLA 1991-1993 Professeur de Mathematique a UCLA 1993-2001 Professeur Departement de mathématiques, EPFL dès 2001

Robert Dalang

Robert Dalang, né en 1961, a reçu le diplôme de Mathématicien-EPFL en 1983 et est lauréat du Prix Dommer. Il passe l'année 1985-86 à Cornell University (USA) comme chercheur invité. Il obtient le doctorat au Département de mathématiques de l'EPFL en 1987. Son domaine de spécialisation est la théorie des processeurs stochastiques. En 1987, Robert Dalang est nommé professeur assistant au Département de statistiques de l'Université de Californie à Berkeley (USA). En 1988, il reçoit une bourse post-doctorale du Fonds national scientifique américain et effectue des recherches sur les propriétés markoviennes de processus stochastiques à plusieurs paramètres. En 1990, il est nommé à Tufts University (Boston, USA). Il est promu professeur associé en mai 1993. Une partie importante de ses recherches se font dans le cadre de contrats avec le Fonds national scientifique américain et l'Office de la recherche de l'armée américaine. En collaboration avec le Prof. R. Cairoli du Département de mathématiques de l'EPFL, il a écrit un livre sur l'optimisation stochastique séquentielle publié en 1996 aux éditions John Wiley. M. Dalang est nommé professeur extraordinaire de probabilités au Département de mathématiques en 1995. Il y poursuit des travaux de recherche en processus stochastiques et probabilité appliquée et participe à l'enseignement des processus stochastiques, de la théorie des probabilités et des cours de mathématiques aux sections d'ingénieurs. Il dirige régulièrement des thèses de doctorats, est éditeur de plusieurs journaux de recherche mathématique et travail en collaboration avec des chercheurs de plusieurs universités européennes et américaines.

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Related concepts (32)

Jacobi elliptic functions

In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation for .

Theta function

In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theory. The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called z), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function.

Definition

A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitions (which try to list the objects that a term describes). Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions.

Lemniscate elliptic functions

In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied by Giulio Fagnano in 1718 and later by Leonhard Euler and Carl Friedrich Gauss, among others. The lemniscate sine and lemniscate cosine functions, usually written with the symbols sl and cl (sometimes the symbols sinlem and coslem or sin lemn and cos lemn are used instead), are analogous to the trigonometric functions sine and cosine.

Hyperbolic functions

In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola. Also, similarly to how the derivatives of sin(t) and cos(t) are cos(t) and –sin(t) respectively, the derivatives of sinh(t) and cosh(t) are cosh(t) and +sinh(t) respectively. Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry.

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