This course presents the main analytical tools for the mathematical study of equations at partial derivatives and consists of six chapters. Chapter 1 recalls some essential concepts concerning the topology of standard vector spaces. Chapter 2 introduces the integration of Lebesgue and its applications in resolving issues in which integration into the Riemann's sense of smell smacks of some important defects. Chapter 3 outlines the spaces for Lebesgue of a higher order which represent a generalisation of Lebesgue's space introduced in Chapter 2. Chapter 4 introduces the notion of the Fourier transform of functions Lebesgue integrable and its extension has integrable edge functions. Chapter 5 develops the notion of Hilbert's spaces in the case of the innate dimension. Chapter 6 is a brief introduction to the distribution theory that generalizes the notion of function. At the end of this course, the engineering student will know the fundamental notions to assimilate the basis of the variational analysis of the problems of partial derivative equations (PDEs), which opens the door to the method of the nis elements and the methods of discretization
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Mathematics for engineering I
I n my opinion, a mathematician, in so far as he is a mathematician, need not preoccupy himself with philosophy, HENRI LEBESGUES.
Content of the module
1. Direct methods for solving linear systems
(a) Reminder on the Gauss removal method
(b) LU and Cholesky factorization of a matrix: living conditions and unicity,
(c) Algorithm and number of operations: band matrix case
2. Iterative methods for solving linear systems
(a) Matrix standards and packaging
(b) Jacobi and Gauss-Seidel methods: algorithms and convergence for matrices diagonal strictly dominant
(c) Relaxation methods
(d) Cases of symmetrical denies positive matrices
(e) Convergence rates: case of tridiagonal matrices
3. Optimization without constraints
(a) Existence of a minimum
(b) Problems of convex and unicite optimizations
(c) Optimal condition
(d) Problems of lesser square
(e) Downhill methods and algorithms for pitch gradient, pitch gradient and optimal pitch gradient
and the gradient combines Application and convergence for a quadratic problem
4. Optimization with constraints
(a) Existence and uniqueness of an optimization problem with constraints
(b) Euler's optimal condition and inequation
(c) Projection on a convex
(d) Application to a quadratic problem with linear constraints: Conditions of
Kuhn and Tucker
(e) Case of a quadratic problem
(f) Projected gradient method
(g) Uzuwa method