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Informations générales


John H. Maddocks


Cours: lundis de 10h15 à 12h et de 13h15 à 14h, salle CM2
Exercices: vendredis de 13h15 à 15h, salles CE1100 (A-L) et CE1101 (M-Z)

Assistant principale:

Alessandro Patelli



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.


  • 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.


Part I (9-10 weeks) : Vector calculus
Part II (3-4 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 first-order linear constant-coefficient 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 self-adjoint Sturm-Liouville (ODE) operators (two-point boundary condition problems for second-order 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.


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 !!!

Week-by-week 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. 61--67.
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 68--80).
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 95--99).
Week 8 (7.11) Stokes Theorem completed, and complements.
Week 9 (14.11) Divergence Theorem (Stuart Chapter 6, Descloux pp 100--102). Applications of the Divergence Theorem, including intrinsic definitions of div, grad and curl (Descloux pp. 107--109). Figure for lemma 6.4 .
Week 10 (21.11) Convex, star-shaped, 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 1-8), wave and Laplace equations (Stuart Part II, pp 8-11).
Week 13 (12.12) Continuation of separation of variables. Sturm-Liouville operators and generalised Fourier series (Stuart Part II, pp 18-28).
Week 14 (19.12) Convergence results for Fourier series (Descloux, pp 144-148), and connexions between Fourier, sine and cosine series (Descloux, pp 152-162). Integration and differentiation of Fourier series. Complex notation (Descloux, pp 167-171). Afternoon: Mathematica Demo on Fourier Analysis. Download here the Mathematica files prepared by Dr. Thomas Lessinnes.


First year Analysis I & II and Linear Algebra. Specifically:

  • Integration over 1, 2, and 3-dimensional domains
  • Change of variables in 1, 2, and 3-dimensional domains
  • Integration by parts in 1 dimensional integrals
  • Chain rule for partial derivatives
Analyse I (polycopié de J. Stubbe, 2012)
Analyse II (polycopié de J. Stubbe, 2013)


Séries d'exercices Corrigés


Examens précédents: