Analysis III (for SV, MT)
MATH-203(a)
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).
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.
Revision of Fourier analysis and applications.