Examination procedure
Written and oral exam
Prerequisites
First year students in Biology and Chemistry
Syllabus
Basics of Mathematics
Logic and sets; natural, integer, rational and real numbers.
Analysis
- 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.
- Differential Canculus Rolle, Cauchy, Lagrange and De L'Hopital theorems;
- Taylor formula and series.
- Integration Riemann integral;
- Fundamental theorem of calculus.
Multivariate Calculus
- Continuity. Partial and directional derivatives.
- Differentiability. Total differential theorem.
- Rules of calculus
- Dini theorem.
- Parametric curves, curve length.
- Conservative vector fields.
- First elements on differential equations.
Linear Algebra
- vector spaces, linear dependence (bases, dimension…), linear transformations
- matrices, vectors and correspondence with the intrinsic concepts of point 1); change of basis
- determinants (basics)
- eigenvalues and eigenvectors, algebraic and geometric multiplicity
- inner product, orthogonalization, unitary matrices, orthogonal projectors
- diagonalization and Jordan form
- Schur form and the spectral theorem
Bibliographical references
Notes given by the teacher.
Mariano Giaquinta, Giuseppe Modica, Analisi matematica, Volume 1: Funzioni di una variabile. Pitagora, 1998.
Carlo Domenico Pagani, Sandro Salsa, Analisi matematica 1, Second edition. Zanichelli, 2015