Complementi di matematica per chimici e biologi

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