Basics of Mathematics
Logic and sets; natural, integer, rational and real numbers.
Sequences and Series
Limit and convergence; Cauchy sequences.
Metric and Topological Spaces
Open and closed sets; neighborhoods.
Limits and Continuity
Limits of real functions; lim sup and lim inf; continuity and Weiestrass theorem.
Rolle, Cauchy, Lagrange and De L'Hopital theorems; Taylor formula and series.
Riemann integral; fundamental theorem of calculus.
Functions of Many Variables
Continuity; partial derivatives; gradient; differential and differentiability; critical points and Hessian.
Vector spaces over a field, subspaces.
Linearly dependent or independent vectors.
Dimension of a vector space.
First applications to linear systems, homogeneous and not.
Linear maps between vector spaces.
Matrices and linear maps.
Positive definite scalar products. Orthonormal bases and Gram-Schmidt orthogonalization process.
Linear systems: rank and dimension of the space of solutions.
Determinant of a matrix.
Eigenvectors and eigenvalues of an operator. Characteristic polynomial.
Jordan's canonical form.
W. Rudin, Principles of Mathematical Analysis, third edition.
S. Lang, Linear Algebra, third edition.