Analyse III pour physiciens
Informations générales
Enseignant:
Horaires:
Cours: lundis de 10h15 à 12h et de 13h15 à 14h, salle CM2
Exercices: vendredis de 13h15 à 15h, salles CE1100 (AL) et CE1101 (MZ)
Assistant principale:
Cours
Polycopiés
 Notes of the course by prof. J. Descloux
 Notes of the course by prof. C. A. Stuart: part I and part II
In addition to learning the material, you need to start to learn how to read descriptions of the same material from slightly different point of view. The two polycopiés are deliberately two different perspectives. The course is a third viewpoint, in between.
Book
 Cours d'Analyse 1, Analyse vectorielle, Srishti D. Chatterji, Presses polytechniques et universitaires romandes.
In particular the chapter 6, Analyse vectorielle, covers all the material of Part I of this lecture. This book must be considered as an OPTIONAL complement to the lecture.
Contents
Part I (910 weeks) : Vector calculus
Part II (34 weeks) : Basics of Fourier series
(not so much connexion between the two parts)
Contents of part I
 Integrals over curves in $\mathbb{R}^N$, $N \geq 2$
 Definitions
 Change of parametrisation
 Oriented or not
 Integrals over surfaces in $\mathbb{R}^3$
 Vector operators on scalars and vector fields (div, grad, curl, and product rules)
 Integration by parts on:
 Domains in $\mathbb{R}^2$  Green's theorem
 Domains in $\mathbb{R}^N$  divergence theorem
 Surfaces in $\mathbb{R}^3$  Stokes theorem
 Intrinsic definitions of div, grad, curl
 Scalar & vector potentials, Helmholtz decomposition
Contents of part II
 Eigenvector solution of the initial value problem for systems of firstorder linear constantcoefficient ordinary differential equations (ODE), as an introduction to the separation of variables method for the solution of linear, constant coefficient partial differential equations (PDE). (not in a polycopie, in principle material of first year)
 Introduction to Fourier series via application of the separation of variables method to the heat, wave and Laplace equations.
 Eigenfunctions of selfadjoint SturmLiouville (ODE) operators (twopoint boundary condition problems for secondorder ODE), and generalised Fourier series.
 Statement of convergence results for Fourier series, and connexions between full, sine and cosine Fourier series. If time, statement of results on the integration and differentiation of Fourier series. Complex notation.
Announcement
We announce that there will not be exercise session the following days:
 Friday 4.11.2016 (Class cancelled by EPFL's direction). We will upload the exercice sheet with already the solutions and you can ask question during the exercise session of the week after.
 Friday 23.12.2016. We will replace the last exercice session with a RAQ session before the exam.
!!! A revision session is organised Friday 13th January 2017 from 10h to 12h in CE 5 !!!
Weekbyweek correspondence
Week 1 (19.9)  Jeûne Fédéral, pas de cours lundi 19.9. La première séance d'exercice aura lieu comme prévu vendredi 23.9. 
Week 2 (26.9)  Introductory part and Grammian, just in class notes. Arcs and line integrals, definitions. Class follows Stuart polycopié Chapter 1 quite closely. Same material treated much more quickly and in less depth in Descloux pp. 6167. 
Week 3 (3.10)  Arcs and line integrals continued, motivation of definition, independence of definition to choice of arc parametrisation. Oriented arcs, and curvilinear integrals. (Stuart Sections 1.2 and 1.3). 
Week 4 (10.10)  Paths, and closed paths, closed paths in the plane, orientation and exterior normals (Stuart 1.4 and 1.5). Start surfaces and surface integrals. (Stuart Chapter 3, Descloux pp 6880). 
Week 5 (17.10)  Motivation of surface integral formulas and independence of parametrisation. Examples of Nappe with different parametrisations here. Complement to surface integration here. 
Week 6 (24.10)  Complements regarding surface integrals, including oriented nappes, and solid angle (only in Descloux p. 77). Integration by parts in the plane (Stuart Chapter 4). 
Week 7 (31.10)  Green's theorem in the plane (Stuart 4.3). Stokes' theorem, ie integration by parts on surfaces (Stuart Chapter 5, Descloux pp 9599). 
Week 8 (7.11)  Stokes Theorem completed, and complements. 
Week 9 (14.11)  Divergence Theorem (Stuart Chapter 6, Descloux pp 100102). Applications of the Divergence Theorem, including intrinsic definitions of div, grad and curl (Descloux pp. 107109). Figure for lemma 6.4 . 
Week 10 (21.11)  Convex, starshaped, simply connected, and (path) connected domains. Scalar potentials for vector fields (Stuart 7.1). 
Week 11 (28.11)  Morning: vector potentials (Stuart 7.2, 7.3) and Helmholtz decomposition. Afternoon: Start of Fourier series and method of separation of variables (Stuart Part II, Descloux p. 129). 
Week 12 (5.12)  Introduction to separation of variables for heat (Stuart Part II, pp 18), wave and Laplace equations (Stuart Part II, pp 811). 
Week 13 (12.12)  Continuation of separation of variables. SturmLiouville operators and generalised Fourier series (Stuart Part II, pp 1828). 
Week 14 (19.12)  Convergence results for Fourier series (Descloux, pp 144148), and connexions between Fourier, sine and cosine series (Descloux, pp 152162). Integration and differentiation of Fourier series. Complex notation (Descloux, pp 167171). Afternoon: Mathematica Demo on Fourier Analysis. Download here the Mathematica files prepared by Dr. Thomas Lessinnes. 
Requirements
First year Analysis I & II and Linear Algebra. Specifically:
 Integration over 1, 2, and 3dimensional domains
 Change of variables in 1, 2, and 3dimensional domains
 Integration by parts in 1 dimensional integrals
 Chain rule for partial derivatives
Analyse II (polycopié de J. Stubbe, 2013)
Exercices
Séries d'exercices  Corrigés 

Examen
Examens précédents:

Examen 2017 et solution 2017
 Examen 2016 et solution 2016
 Examen 2014 et solution 2014
 Examen 2013 et solution 2013
 Examen 2012 et solution 2012
 Examen 2011 et solution 2011
 Examen 2010 et solution 2010
 Examen 2009 (sans solution)