Math for BE

But du cours

• Double integrals: definition, Fubini’s theorem, change of variables, Jacobian, polar coordinates; applications: center of mass, moment of inertia, area calculation, volume under a surface

• Triple integrals: definition and applications, cylindrical and spherical coordinates, change of variables with the Jacobian, practical applications

• Curves and line integrals: curves and parametric equations, polar coordinates, definition and calculation of line integrals, applications to vector fields and curve length calculations

• Fundamental theorems: Green–Riemann theorem, Ostrogradski’s and Stokes’ theorems, gradient and curl theorems, calculation of the flux of a vector field

Acquis d'apprentissage visés

• Formulate and evaluate double and triple integrals in Cartesian coordinates

• Formulate and evaluate line integrals in various geometric configurations

• Perform change of variables in multiple integrals (polar, cylindrical, and spherical coordinates)

• Apply multiple integrals to physics and engineering problems (volume, center of mass, moment of inertia calculations, etc.)

• Apply line integrals to physics problems (curve length, work done by a force, flux of vector fields, etc.)

• Use the Green–Riemann theorem and other fundamental theorems to solve physical and geometric problems

Prérequis

• Differentiation and integration of single-variable functions

• Multivariable functions and partial derivatives

• Vector calculus

• Calculation of the determinant of a $3 \times 3$ matrix

Programme

1. Double and triple integrals in Cartesian coordinates: definition, geometric interpretation, calculation over simple domains.

1. Change of variables in multiple integrals: introduction to polar, cylindrical, and spherical coordinates, practical applications.

1. Line integrals: definition, orientation, calculation over parameterized curves in various geometric contexts.

1. Applications in physics and engineering: volume, center of mass, moment of inertia, work of a force, flux of fields calculations.

1. Fundamental theorems: Green–Riemann theorem and introduction to Stokes and Gauss for solving concrete problems.

Modalités d'évaluation

2 Written exams (2h/2h)