58048 - Mathematics with Exercises

Course Unit Page

Academic Year 2022/2023

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 in graphical form, to apply one-variable and multivariable calculus and to perform operations on vectors and matrices and will know how to use some basic concepts of scientific computing, such as error analysis, approximation of experimental data, interpolation, numerical integration, nonlinear equations and systems of linear equations.

Course contents

Limits and continuity, basic theorems.
Derivatives, basic theorems and applications: tangents to curves, increasing and decreasing functions, convexity, graphs of functions, Taylor's formula.
Integrals in one variable, primitives, integration of rational functions, integration by substitution and by parts.
Ordinary differential equations (ODEs), methods to solve first order ODEs, of linear type or separate variables type, and linear ODEs of higher order with constant coefficients.
First elements of differential calculus of several variables, partial derivatives, gradient and Hessian matrix, maxima and minima.
Double integrals: geometric meaning, computing double integrals as iterated integrals, change of variables, use of polar coordinates.

Complex numbers. Cartesian form and polar form. Nth roots of a complex number. Newton's binomial. Polynomials. Sum and product of polynomials. Euclidean division. Roots of polynomials and factorization. Ruffini's theorem. Linear systems and matrices. Systems of linear equations and set of solutions. Gauss algorithm (Gauss elimination method). Matrix of a linear system and reduction to the reduced step form. Application of the Gauss algorithm for the inversion of matrices. Vector spaces: sum of vectors and multiplication by a scalar. Linear combinations. Linear subspaces. Examples (polynomials, matrices, sequences). Intersection, sum and direct sum of subspaces. Vector subspace generated by a vector family. Free families, parent families. Bases and dimensions. Coordinates in a fixed base. Linear applications. Definition of linear application. Image and Kernel of a linear application. Rank theorem. Injectivity and surjectivity. Matrix of a linear application from a starting base to a destination base. Eigenvalues and eigenvectors of a linear map. Definition of a diagonalizable matrix. Reduction of endomorphisms (search for eigenvalues and eigenvectors). Calculation of determinants.


M. Bramanti, C. D. Pagani, S. Salsa: 2a ed., Zanichelli, Bologna, 2004. [http://www.zanichelli.it/ricerca/prodotti/matematica-calcolo-infinitesimale-e-algebra-lineare]

M. Bramanti, C. D. Pagani, S. Salsa: 1 Zanichelli, Bologna, 2014. [http://www.zanichelli.it/ricerca/prodotti/analisi-matematica-1-bramanti-pagani-salsa]

M. Bramanti, C. D. Pagani, S. Salsa: 2, 2. Zanichelli, Bologna, 2009. [http://www.zanichelli.it/ricerca/prodotti/analisi-matematica-2]

S. Salsa, A. Squellati: Esercizi di Analisi matematica 1, 2 (two volumes), Zanichelli, Bologna, 2011.

E. Steiner: The Chemistry Maths Book, Second Edition. Oxford University Press, Oxford, 2008.

M.R. Spiegel: Theory and Problems of Advanced Calculus. Schaum's Outline Series, McGraw-Hill, 1974.

P. Negrini: Equazioni differenziali. Pitagora editrice, Bologna, 1999.

Rita Fioresi, Marta Morigi "Introduzione all'algebra lineare". Zanichelli

Teaching methods

Lessons accompanied by exercise classes with tutor.

Assessment methods

The course assessment consists of a 3 hour open book examination (5 exercises on the topics covered in the course) followed by an oral examination.
Each exercise of the written examination is graded on a 6-point scale, and the pass mark is 50%, that is, 15 points in total. The validity of the written exam is limited to one examination session. The oral exam aims to test knowledge acquisition and to discuss exercises. The final mark, on a 30-point scale, is based on both parts of the examination.

The examination is unique for both modules of the course.

Teaching tools

Alma Mathematica [https://almaorienta.unibo.it/AlmaMathematica] : an online math-bridge course which with its diagnostic tests offers students the possibility to complete the missing pieces and refresh the material necessary for a successful study of mathematics. This self-study course is complemented by a virtual tutorial where students can get instant help by skype, e-mail and telephone.

Office hours

See the website of Paolo Negrini

See the website of Luca Marchese