Mathematical Theory applied to Mechanics

Presentation

Programme (detailed contents):

The part on Multivariable Calculus is organised as

I Norms and Scalar products on IR^n

II Functions of two variables: graphs, level set, contour plots, partial derivatives, differential, approximation formula, optimization problems.

III Function of n variables: gradient, jacobian, partial derivatives and chain rule, approximation formula, optimization problems

IV Examples of double integrals computation: exchanging order of integration, change of variable, integration by parts. Examples.

The part on Numerical Analysis is organised as

I Numerical Errors

II Numerical Computation of Integrals (one variable)

III Solving non linear equations

IV Direct resolution of linear systems

V Norms and Condition Number of a Matrix

VI Iterative methods to solve linear systems

VII Interpolation, Splines

VIII Descent Methods, Least Squares

IX Compute eigenvalues and eigenvectors

Organisation: the theoretical part is standard: courses and exercise classes. Concerning the numerical part, the student have access to 9 videos for each item completed with exercises sheet and lecture notes. At the beginning of each labwork session a brief review of the course and exercises is proposed.

   During labwork, we focus on the use of Python and the librairies dedicated to numerical analysis: SciPy, NumPy and MatPlotLib which are free.

Main difficulties for students: Calculus and Programmation.

Objectives

At the end of this module, the student will have understood and be able to explain (main concepts):

Theoretical Part (Multivariable Calculus)

Norms and scalar product on IR^n

Partial derivatives

Notion of differential: gradient, jacobian

Approximation formula

Critical points of a function of several variables

Numerical Part (Numerical Analysis)

Numerical errors, condition numbers

Methods to compute numerically integrals

Methods to solve linear and nonlinear systems

Least square method.

Needed prerequisite

1st year Mathematics

Form of assessment

The evaluation of outcome prior learning is made as a continuous training during the semester. According ot the teaching, the assessment will be different: as a written exam, an oral exam, a record, a written report, peers review...