Analysis III (for SV, MT)

MATH-203(a)

This file is part of the content downloaded from Analysis III (for SV, MT).

General information

Lecture title: Analysis III - MATH-203(a)


  • Instructor: Leonid Monin
  • Main assistant: Veronica Montanaro

Course content:

We welcome you to "Analysis III". This lecture covers important concepts and topics related to vector analysis, Fourier analysis, and their applications to ordinary and partial differential equations. These constitute a crucial part of your mathematical curriculum. In particular:

  • Vector Analysis: Gradient, curl, divergence, and Laplacian operators. Integrals on curves and surfaces. Vector fields and potentials. Green's, divergence, and Stokes' theorems.
  • Fourier Analysis: Fourier series. Parceval identity. Fourier transforms. Plancherel identity. Use and applications.

Bibliography:

The main reference for this course is the book (in french) Analyse avancée pour ingénieurs by B. Dacorogna and C. Tanteri, 4th edition (2018). 
You can find it either at EPFL library or freely available online using this link.

You can also use lecture notes from last year which I will follow (but not perfectly).

Exam information

Instructions for the exam:
  • Before the exam, make sure you know which is your assigned room and seat by looking at the maps and at the student list in the folder below.
  • All students will write the exam in STCC Garden.
  • You can bring a cheat sheet to the exam: A single A4 written on both sides. Fourier transforms table will be provided in the exam booklet.
  • Do not unstaple the booklet.
  • Scratch paper will be provided during the exam (it will be collected at the end of the exam). 
  • Using a calculator or any electronic device is not permitted during the exam.
  • The scoring system of each question will be indicated in the exam booklet.
  • Use a black or dark blue ball-pen and clearly erase with correction fluid if necessary.



September 9 - September 15

Differential operators: Gradient, Curl, Divergence, Laplacian

September 16 - September 22

Formulas with differential operators, Curves, Line integral

September 23 - September 29

Line integral (cont.); Fields that come from Potentials.

Fields that come from Potentials (cont.); Orientation of the boundary in R^2 and Green's theorem

Exterior normal in R^2, Divergence theorem in R^2

 Area formulas, Green's identities, Surface, Surface integral

Regular domains in R^3, Divergence theorem in R^3, Volume formulas, Green's identities in R^3

Boundary of surfaces in R^3 and compatible orientations. Stokes theorem. Recap of vector calculus.

Orthogonality of sin(nx), cos(nx), Fourier Series, Dirichlet theorem,  properties of Fourier series

Dirichlet therorem (continuation), complex Fourier coefficients, properties of Fourier series, Parceval identity


Sin and Cos Fourier series. Fourier transform, properties of Fourier transform, Plancherel identity

Properties of Fourier transform (continuation), a fixed point of Fourier transform, Plancherel identity, sine and cosine Fourier transform, convolutions and their Fourier transform.


Fourier Transform of convolutions (continuation), inverse Fourier transform. Applications of Fourier methods to ODE.

Revision of Fourier analysis and applications.