lepl1105  2021-2022  Louvain-la-Neuve

5 credits
30.0 h + 30.0 h
Q2
Language
French
Prerequisites
This course supposes acquired the notions of mathematics developed in the courses LEPL1101 and LEPL1102.
Main themes
Functions of several real variables. Continuity and differentiability. Optimization problems, vector analysis and integral theorems. Linear differential equations. Modelling of simple problems.
Aims

At the end of this learning unit, the student is able to :

1 At the end of the course the students will be able to 
  • Express metric notions in Rn using the language of general topology.
  •  Study limits, continuity, directional derivatives and differentiability for functions of several variables.
  • Apply Taylor polynomial in order to approximate a function.
  • Locate and identify free extrema of a function; locate extrema under constraints of a function using the technique of Lagrange multipliers.
  • Calculating multiple integrals possibly using a change of variables.
  • Calculate line integrals, surface integrals, the flow of a vector field along a curve and the flow of a vector field through a surface possibly using Stokes type theorems.
  • Apply the resolving method for linear differential equations with constant coefficients of order n.
  • Analyse and write rigorously statements and demonstrations on the mathematical content specified below, and illustrate them with examples and counter-examples.
 
Content
  • Linear constant-coefficient ordinary differential equations of any order, Cauchy problem
  • Scalar and vector-valued real functions of several variables, topology, continuity
  • Differentiability, partial and directional derivatives, chain rule, tangent plane, gradient and Jacobian matrix
  • Higher order partial derivatives and Taylor polynomial
  • Unconstrained and constrained extrema, Lagrange multipliers
  • Multiple integrals and changes of variables
  • Line and surface integrals, circulation and flux of a vector field
  • Notion of boundary and Stokes-type theorems
Teaching methods
Lectures in a large auditorium, supervised exercise (APE) and problem (APP) sessions in small groups, possibly online exercises.
Evaluation methods
Students will be evaluated with an individual written exam, based on the above-mentioned learning outcomes. 
Bibliography
Teaching materials
  • Multivariable Calculus with Applications par Peter D. Lax et Maria Shea Terrell, Springer, 2017.
Faculty or entity


Programmes / formations proposant cette unité d'enseignement (UE)

Title of the programme
Sigle
Credits
Prerequisites
Aims
Bachelor in Engineering : Architecture

Bachelor in Engineering