Mathematics (06POBMAT)

• Coefficient : 2.5
• Hourly Volume: 66.0h (including 36.0h supervised)
  CTD : 36h supervised (and 6h unsupervised)
  Out-of-schedule personal work : 24h

AATs Lists

Description

1. algebra: linear applications, reduction of endomorphisms
2. analysis
   • Functions of several variables
     • level lines and surfaces
     • limits, continuity, partial derivatives
     • Taylor formulas, differential
     • extrema
   • Line and surface integrals

Learning Outcomes (AAv)

• AAv1 [hours: 18, B1, B2, B3] (diagonalization): By the end of this course, students know how to diagonalize a square matrix and geometrically interpret characteristic elements (eigenvalues, eigenvectors, eigenspaces). They can apply this to situations reducing to solving a system of recurrent sequences, calculating matrix power, solving linear differential systems with constant coefficients.

  • Students know how to determine the characteristic polynomial, eigenvalues, and eigenvectors of a matrix or endomorphism.
  • Students know how to give the transition matrix and diagonal matrix of a diagonalizable endomorphism.
  • Students know how to calculate the nth power of a diagonalizable matrix.
  • Students know how to determine the general term of a sequence $U_{n+1}=AU_n$ with A being a matrix.
  • Students know how to solve a linear differential system that can be written using a diagonalizable matrix.

• AAv2 [hours: 15, B2, B3]: By the end of this course, each student is able to prove the continuity or discontinuity of a function of 2 or 3 variables, calculate first and second partial derivatives particularly in situations of function composition and variable changes.

  • Specifically:
    • Students can calculate partial derivatives without error;
    • Students can interpret differentiability in terms of tangent plane, and relate gradient properties to level curves of a function;
    • Students can calculate a second-order operator in a variable system (Laplacian in polar coordinates);
    • Students can prove a function is continuous in several ways.

• AAv3 [hours: 18, B2, B3]: By the end of this course, each student is able to autonomously conduct research and classification of critical points (and possible extrema) based on the sign of the Hessian determinant. They will be able to propose approaches in cases not determined by classification.

  • Specifically:
    • Students know how to calculate coordinates of critical points;
    • Students know how to conduct local study by reducing second derivative calculation for non-zero $s²-rt$;
    • Students know how to construct a path strategy to justify absence of extremum in indeterminate cases;
    • Students know how to roughly represent level curves in the vicinity of a saddle point.

• AAv4 [hours: 15, B2, B3]: By the end of this course, in relation to physics situations such as vibrating string study or temperature evolution of a bar, each student is able to solve elementary differential equations (1st and 2nd order linear), apply or propose a variable change to reduce to the simple case, particularize solutions verifying boundary conditions.

  • Specifically:
    • Students know how to solve first and second order hyperbolic homogeneous equations with constant coefficients through a well-constructed variable change;
    • Students know how to solve a first-order equation without second member with non-constant coefficients, using level curves method;
    • Students know how to determine a solution verifying boundary or initial conditions in simple cases.

Assessment Methods

One long continuous assessment test (coefficient 1) and the average of several short continuous assessment tests (coefficient 3)

Keywords

Functions of several variables, partial derivatives, extrema, simple integrals, multiple, line and surface integrals.

Prerequisites

Content of Mathematics S5O

Resources

Any analysis book for first and second year preparatory classes.