Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Mathematical Complements: df or δf

In course

PHYS-106(a): General physics : thermodynamics

Le but du cours de Physique générale est de fournir les notions de base nécessaires à la compréhension des phénomènes physiques. L'objectif est atteint lorsqu'il est possible de prévoir quantitativement les conséquences de ces phénomènes avec des outils th ...

Description

This lecture covers mathematical tools for differentials of functions of multiple variables, including partial derivatives, second derivatives, and integrals along paths, surfaces, and volumes. It explores the concept of conservative quantities and exact total differentials.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructors (2)

Cécile Hébert

Cécile Hébert was born in France in 1970. She obtained her Ingeneer degree (physics) and her PhD degree (design of a new energy filter for transmission electron microscopy, under the direction of Prof. B. Jouffrey) at the Ecole Centrale in Paris. As a post doc with Prof. P. Schattschneider, she was working on the calculation of the fine structure of ionisation edges in EELS. As a research assistant at the Vienna University of Technology, she was one of the main participant to the CHIRALTEM project dealing with the measurement of magnetic circular dichroism in the electron microscope.

Romuald Houdré

Curriculum Vitae CV 2011 Appointed as Adjunct Professor 2006 Appointed as Maitre d'Enseignement et de Recherche 2004 Joins the "Laboratory of Quantum Electronics" led by Prof. B. Deveaud-Plédran 2001-2004 Appointed as "Adjoint Scientifique" at the Institute for Quantum Photonics and Electronics (previously Institute for Micro and Optoelectronics led by Prof. M. Ilegems) 1998 Habilitation, University Pierre et Marie Curie, Paris 6 (France) 1997 Invited researcher at NTT, Optoelectronics Department (Atsugi, Japan) 1988-2000 "Collaborateur scientifique" at the Institut for Micro and Optoelectronics with Prof. M.Ilegems at the Swiss Federal Institut of Technology in Lausanne (Switzerland). In charge of the Molecular Beam Epitaxy (1988-1996) and the research on optical microcavities (1996-2000) 1987-1988 Laboratoire de Physique de la Matière Condensée at Ecole Polytechnique (France). 1986-1987 Postdoctoral fellow at the University of Illinois at Urbana-Champaign (U.S.A.) with Prof. H.Morkoç in the molecular beam epitaxy group 1983-1985 Ph.D. thesis on the photoemission from quantum wells and superlattices under negative electron affinity at Laboratoire de Physique de la Matière Condensée, Ecole Polytechnique (France), G.Lampel and C.Hermann as advisors

Official source

Ontological neighbourhood

Mathematics

Analysis: Calculus

Related concepts (32)

Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry. The partial derivative of a function with respect to the variable is variously denoted by It can be thought of as the rate of change of the function in the -direction.

Derivative

In mathematics, the derivative shows the sensitivity of change of a function's output with respect to the input. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.

Integral

In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation. Integration started as a method to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Today integration is used in a wide variety of scientific fields.

Second derivative

In calculus, the second derivative, or the second-order derivative, of a function f is the derivative of the derivative of f. Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the second derivative of the position of an object with respect to time is the instantaneous acceleration of the object, or the rate at which the velocity of the object is changing with respect to time. In Leibniz notation: where a is acceleration, v is velocity, t is time, x is position, and d is the instantaneous "delta" or change.

Symmetry of second derivatives

In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility of interchanging the order of taking partial derivatives of a function of n variables without changing the result under certain conditions (see below). The symmetry is the assertion that the second-order partial derivatives satisfy the identity so that they form an n × n symmetric matrix, known as the function's Hessian matrix.

Related lectures (37)

Taylor Polynomials and Function Properties

Covers Taylor polynomials, function properties, double integrals, partial derivatives, and function boundaries.

Integrals of Functions MOOC: Warm-up for EPFL

Covers integrals as the reciprocal function of the derivative and explains displacement integral of velocity and variation of speed and acceleration in rectilinear motion.

Mathematical Reminders: Derivatives, Primitives, Integrals

Covers mathematical reminders on derivatives, primitives, and integrals, including common functions and Taylor series expansions.

Partial Derivatives: Derivative of an Integral with Parameter-dependent Bounds

Covers the derivative of an integral with parameter-dependent bounds and the gradient.

Partial Derivatives: Introduction