Complements of Linear Algebra

Matrixes as representations of linear transformations.

Eigenvalues and eigenvectors of a square matrix. Eigenspaces and algebraic multiplicity of an eigenvalue.

Characteristic polynomial and geometric multiplicity of an eigenvalue.

Eigenvalues of symmetric matrixes.

Application to the classification of quadratic forms.

Sequences and Elementary topology of

Euclidian distance; neighborhoods and open balls; interior, exterior, boundary; Open subsets of R ⁿ .

Closure of a set and closed sets. Compact sets. Sequences in of : bounded sequences and convergent sequences.

Topology from the point of view of sequences.

Functions of several real variables

Some generalities: domain, range, graph and level curves.

Limit of a real function of several variables. The algebra of limits.

Som classical theorems.

Limit with respect to a subset of the domain; Directional limits and main properties.

Notion of continuity. Operations with continuous functions. The Weierstrass Theorem.

Differential calculus

Directional derivatives and partial derivatives: definition and geometric interpretation.

Linear approximation of a function: the notion of differentiability.

Differentiability of vector valued functions. The Jacobian matrix.

Partial derivation and function composition: the chain rule.

Link between differentiability and the existence of directional and partial derivatives.

Space of continuously differentiable functions. The Schwarz Theorem.

Hessian Matrix and Taylor's formula.

Application of the differential calculus to optimization

Critical points. Extremal values and saddle points.

Second order criteria for the classification of critical points in an open set.

Higher order criteria.

Extrema with constraints and Lagrange multipliers.

Study if extremal functions of a continuous function in a compact set.

Integral calculus in  R ²

Definition and first properties.

The Fubini theorem. Integration over simple regions. Application to the calculus of areas and volumes.

Brief notions of Integration in R ⁿ

Differential equations

First definitions and examples.

First order differential equations.

Linear second order differential equations.

Non-homogeneous second order equations. The variation of constants method.