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...