Complementi di matematica per chimici e biologi
Period of duration of course
First year students in Biology and Chemistry
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.
- Differential Canculus Rolle, Cauchy, Lagrange and De L'Hopital theorems;
- Taylor formula and series.
- Integration Riemann integral;
- Fundamental theorem of 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.
- 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
Goal of the lectures is to introduce students to basic topics but under an advanced and stimulating viewpoint, offering relevant additional elements with respect to traditional courses.
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