69260 - Mathematics

Learning outcomes

At the end of the course, the student will have acquired basic knowledge of differential and integral calculus for functions of one real variable, vector calculus and linear algebra, introductory elements of calculus for functions of several variables, complex numbers, and the most elementary methods for solving differential equations. In particular, the student will be able to represent data or functions graphically, apply differential and integral calculus to functions of one or more real variables, perform operations with vectors and matrices, and solve systems of linear equations.

Course contents

Prerequisites: Elementary set theory, algebra of real numbers, algebraic equations and inequalities, elementary geometry of the Euclidean plane.

Contents:

Real numbers, inequalities, absolute value. Elementary functions: powers, roots, exponentials and logarithms, trigonometric and hyperbolic functions and their inverses. Complex numbers, complex numbers in trigonometric form, De Moivre’s formula and Euler’s formula.

Linear systems, augmented matrix and coefficient matrix of a system, row echelon reduction, rank of a matrix, Rouché–Capelli theorem, solution of a system by row reduction, also known as Gaussian elimination, determinant of a square matrix.

Vector space structure of (\mathbb{R}^n), linear dependence and independence of vectors, connection with the rank of suitable matrices, bases of subspaces, dimension of subspaces, linear transformations from (\mathbb{R}^n) to (\mathbb{R}^m), kernel and image, matrix representation of a linear transformation, linear transformations from (\mathbb{R}^n) into itself. Composition and inversion. The determinant. Eigenvalues and eigenvectors, diagonalizability.

Limits and continuity, main theorems.

Derivatives, main theorems and applications: tangents to curves, increasing and decreasing functions, convexity, graphing functions, Taylor’s formula.

Integrals for functions of one variable, antiderivatives, integration of rational functions, integration by substitution and by parts.

Ordinary differential equations, solution methods for first-order linear differential equations, separable differential equations, and higher-order linear differential equations with constant coefficients.

Introductory elements of differential calculus for functions of several variables, partial derivatives, gradient and Hessian matrix, maximum and minimum points, determination of the absolute minimum and maximum of a function of two variables on a closed and bounded domain.

Double integrals: geometric meaning, reduction formula; change of variables, with particular attention to polar coordinates.

Readings/Bibliography

Strongly recommended texts:

• R. Fioresi, M. Morigi, Introduzione all’algebra lineare, Zanichelli, Bologna, 2015.
• M. Bramanti, C. D. Pagani, S. Salsa, Matematica. Calcolo infinitesimale e algebra lineare, 2ª ed., Zanichelli, Bologna, 2004.

Recommended text:

• S. Salsa, A. Squellati, Esercizi di Analisi matematica 1, 2, due volumi, Zanichelli, Bologna, 2011.

Any additional teaching and supplementary material will be uploaded to the course page on the Virtuale platform. Access is granted through institutional account.

Teaching methods

Lectures at the blackboard and exercise sessions

Assessment methods

Assessment is carried out through a final written examination followed by an oral examination. The written examination consists of the analytical solution of two or more exercises.

The oral examination aims to assess the acquisition of the knowledge covered by the course syllabus. The final grade, expressed out of thirty, takes into account the marks obtained in both examinations.

Grading Scale and Assessment Criteria

• < 18: insufficient preparation;
• 18–23: satisfactory preparation, but limited to a restricted portion of the topics covered in the course syllabus;
• 24–27: adequate preparation, although with some gaps in the topics covered in the course syllabus;
• 28–30: thorough mastery of all topics covered in the course syllabus;
• 30 cum laude (30L): outstanding knowledge of all topics covered in the course syllabus.

Students with specific learning disabilities, or with temporary or permanent disabilities, are advised to contact the relevant University office in good time (https://site.unibo.it/studenti-con-disabilita-e-dsa/it ). The office will propose any necessary accommodations to the students concerned. These accommodations must in any case be submitted to the lecturer for approval at least 15 days in advance; the lecturer will assess their suitability also in relation to the learning objectives of the course.

As regards assessment, the use of AI is prohibited. Any use of AI constitutes a violation of academic integrity.

Teaching tools

Exercises, teaching support notes provided by the lecturers, and the class log will be available online.

Office hours

See the website of Lorenzo Cerboni-Baiardi

See the website of Daniele Morbidelli