 # 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

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...