74790 - Mathematics - a Course with Exercises

Learning outcomes

On successful completion of the course, students will have acquired the basic knowledge of one-variable calculus, vector calculus and linear algebra, the first elements of multivariable calculus, complex numbers and the most elementary methods for solving ordinary differential equations. In particular, students will be able to represent data or functions graphically, to apply one-variable and multivariable calculus, to perform operations on vectors and matrices and to solve systems of linear equations.

Course contents

Module #1: Complex numbers, Euler's formula, vectors and matrices, linear transformations, eigenvalues and eigenvectors, least squares, linear regression, approximation of functions, Taylor series, scalar and vector functions of several variables and partial derivatives, gradient divergenze rotor and Laplacian operators, Hessian matrix, maxima and minima of functions of several variables, bases of orthogonal functions, Fourier series and transform.

Module #2: Refresher of elementary functions, polynomials, logarithms, exponential functions, trigonometric functions and their inverses, limits and continuity, derivatives, Rolle Lagrange and de l'Hopital theorems.

Module #3: Curve sketching, integrals, integration by parts and by substitution, definite integrals. ordinary differential equations (ODE) first-order linear, separable, and linear to nth order with constant coefficients.

Prerequisites: algebra, algebraic equations, trigonometry, logarithms and exponentials, inequalities.

A detailed description of the course, including references, and final assessment methods can be found in the syllabus given at the start of the class.

Readings/Bibliography

Claudio Canuto, Anita Tabacco, ``Analisi Matematica I'', 4a edizione (Springer--Verlag, Milano 2014) --- ISBN13:978-88-470-5722-7.

Tom M. Apostol, ``Calcolo'', volume II ``Geometria'', (Bollati Boringhieri, Torino 1979) --- ISBN13:978-88-339-5034-1.

Handouts and notes provided at lecture time.

Teaching methods

Lectures, classroom exercises, and audience/student response system (clickers).

Assessment methods

All exams and test are written; there are no oral exams. Each test consists of a collection of numerical exercises regarding the topics of the three modules. The evaluation of each module contributes proportionally the overall grade.

During the lectures the students can enroll to three written partial tests (at about 1/3, 2/3, and at the end of the course). The arithmetic mean of the partial tests is the final grade. Alternatively, students can take a comprehensive written examination on the subjects of the three modules.

All outcomes of examinations booked via Almaesami are recorded.

Teaching tools

Beamer, interactive display, and classroom response system (clickers).

Office hours

See the website of Roberto Berardi

See the website of Marco Garavelli

See the website of Ivan Rivalta