Code: DT1303   |   ECTS Credits: 3   |   Delivery: Online   |   Taught hours: 18   |   Language: English

Unit Coordinator: Mario Santilli

Content:

Linear spaces, linear dependence, bases of a linear space, dimension of a linear space, linear subspaces.

Matrices, basic operations with matrices, change of coordinates, determinants, rank. A brief account on linear systems and Gauss elimination.

Diagonalisation of squared matrices, eigenvalues, eigenvectors. Inner products, bilinear forms and quadratic forms.

Code: DT0708   |   ECTS Credits: 9   |   Delivery: Online   |   Taught hours: 54   |   Language: English

Unit Coordinator: Maicol Caponi, Gennaro Ciampa, Simone Fagioli

Aims:

Part I – Sequences, Series, Limits, and Continuity
• Numerical sequences: convergence, divergence, limits of sequences, and basic limit theorems.
• Numerical series: convergence, divergence, and convergence tests.
• Real-valued functions: limits of functions, continuity, extreme value property, and intermediate
value property.

Part II – Differentiation and Multivariable Calculus
• Derivatives and differentials, rules of differentiation, and fundamental theorems of differential
calculus.
• Higher-order derivatives and Taylor expansions.
• Qualitative study of functions: monotonicity, local and global extrema, concavity and
convexity, asymptotic behaviour, graph sketching.
• Functions of several variables: limits in Rn, continuity, partial derivatives, and differentiability.

Part III – Integration
• The Riemann integral: definition and existence, integrable functions, and properties of the
integral.
• Fundamental Theorem of Calculus.
• Techniques of integration: substitution, integration by parts, and integration of rational
functions using partial fractions.
• Improper integrals.
• Comprehensive review exercises.

Content:

To be announced soon

Code: DT0710   |   ECTS Credits: 3   |   Delivery: Online   |   Taught hours: 18   |   Language: English

Unit Coordinator: Bruno Rubino

Content:

General introduction to differential equations, Cauchy problems.

Existence and uniqueness of solutions. Peano’s and Cauchy’s theorems. Examples, Peano’s brush.

Introduction to linear differential equations. Examples.

A brief outline of qualitative analysis of Cauchy problems. Comparison of solutions, maximal solutions, global existence of solutions, blow-up of solutions. Examples.

   |   ECTS Credits: 3   |   Delivery: Online   |   Taught hours: 18   |   Language: English

Unit Coordinator: Raffaele D'Ambrosio

Content:

• Elements of floating point arithmetics. Representation of numbers with normalized mantissa and characteristic. Set of floating-point numbers. Overflow and underflow. Truncation, rounding and related errors. Machine precision. Propagation of round-off error in floating-point operations.
• Elements of iterative methods for linear systems. General form of an iterative method for linear systems. Jacobi and Gauss-Seidel iterations. Convergence.
• Elements of iterative methods for nonlinear equations. Bisection method. Local linearization. Regula falsi, secant method, Newton-Raphson method. Fixed point iterations. Convergence.
• Elements of polynomial interpolation. Construction of the interpolation polynomial: construction via Vandermonde matrices; Lagrange interpolation polynomial, fundamental Lagrange polynomials. Estimation of interpolation error.  
• Elements of numerical quadrature. Interpolatory quadrature, computation of weights and nodes. Newton-Cotes formulae. Trapezoidal quadrature rule and error estimate. Cavalieri-Simpson quadrature rule and error estimate. Degree of precision. Idea of composite quadrature.

   |   ECTS Credits: 3   |   Delivery: Online   |   Taught hours: 18   |   Language: English

Unit Coordinator: Ida Minelli