3D mathematical tools

The aim of this course is to give students the mathematical tools needed to set up and use modeling tools in mechanics.
By the end of this course, they will have mastered the use of functions of several variables, as well as elementary notions of linear algebra.

Basic mathematics (functions, derivatives, basic geometry).

1. Reminder of complex numbers

Notation a + ib. Modulus, argument. Geometric interpretation.
Calculation rules, exponential, log. Trigonometry.

2. Plane and space geometry

Plane geometry: Cartesian and parametric equation of a line. Intersection of two lines.
Geometry in space: equations of lines. Cartesian and parametric equations of a plane. Intersection of two planes, a plane and a line.
Scalar product, vector product. Orthogonality.

3. Linear algebra

Vector space, the Rn case.
Free families, generatrices, bases. Finite dimension. Generated subspaces.
Linear applications. Kernel, image.
Some function spaces.

4. Matrix calculation

Definition. Operations on matrices (sum, matrix-vector product, product of matrices).
Matrix inversion, solving linear systems, rank of a matrix. Gauss pivot.
Eigenvalues, eigenvectors, diagonalization.
Scalar product on Rn, norm. Symmetrical matrices, orthogonal matrices.

5. Functions of several variables

Focus on functions of the type f(x, y), f(x, y, z), f(t, x, y, z).
Partial derivatives. Schwartz's theorem.
Taylor's formula of order 2.
Compound functions.
Surfaces given as a graph, implicitly or parametrically. Tangent plane. Calculation of tangent plane and normal. Position relative to the tangent.
Vector analysis: differential operators and their properties. Change of coordinate system.

6. multiple integrals

Double integrals: link with surface. Calculation, Fubini's theorem, change of variables.
Triple integrals: volume. Calculation by layers, by stack. Change of variables.
Surface integral: calculation of surface element, normal.
Formulas of Stokes, Green-Ostrogradski.
Differential forms of degree 1 and their integration. Green-Riemann formula.

Engineering Sciences and Techniques