## Informations générales

##### Horaires:

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)

## Cours

#### Polycopiés

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 (9-10 weeks) : Vector calculus
Part II (3-4 weeks) : Basics of Fourier series
(not so much connection 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.

#### Announcements

A revision session will take place Thursday 11th January from 10h to 12h in CE 5

#### Week-by-week correspondence

 Week 1 (18.9) Jeûne Fédéral, pas de cours lundi 18.9. La première séance d'exercice aura lieu comme prévu vendredi 22.9. Week 2 (25.9) 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 (2.10) 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 (9.10) 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 (16.10) Motivation of surface integral formulas and independence of parametrisation. Examples of Nappe with different parametrisations here. Complement to surface integration here. Week 6 (23.10) 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 (30.10) Green's theorem in the plane (Stuart 4.3). Stokes' theorem, ie integration by parts on surfaces (Stuart Chapter 5, Descloux pp 95--99). The text of the proof of the basic integration formula of derivatives on a nappe that was presented in the afternoon lecture (rather fast in order to be able to complete it in one lecture) appears almost verbatim (but without the explanatory figures) in the Stuart polycopie pp 32--34, Theorem 5.2 and its proof. One small typo in the polycopie is that in two integrals toward the top of p. 34 the same symbol u appears twice, once as an unknown finction and once as the integration variable. The function and the integration variable should be different symbols (x was used in class for the integration variable). Week 8 (6.11) Stokes Theorem completed, and complements. Before next Monday please look at the figure for lemma 6.4 (link below). Week 9 (13.11) 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 (20.11) Convex, star-shaped, simply connected, and (path) connected domains. Scalar potentials for vector fields (Stuart 7.1). Week 11 (27.11) 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). There have been some small changes made to question 5 in série 11, the new version is online now. There are also some small but significant changes to the corrigé compared to previous years which is also online now. Week 12 (4.12) 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 (11.12) Continuation of separation of variables. Sturm-Liouville operators and generalised Fourier series (Stuart Part II, pp 18-28). Some animations demonstrate the convergence of sine and cosine series calculated in problem sheet 13. There is also a document which gives some examples of Fourier series taken from the book Analyse de Fourier by Murray R. Spiegel. Week 14 (18.12) 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.

#### Requirements

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)