74790 - Mathematics - a Course with Exercises

Academic Year 2021/2022

  • Moduli: Giovanni Mongardi (Modulo 1) Daniele Morbidelli (Modulo 2)
  • Teaching Mode: Traditional lectures (Modulo 1) Traditional lectures (Modulo 2)
  • Campus: Rimini
  • Corso: First cycle degree programme (L) in Chemistry and Technologies for the Environment and Materials (cod. 8514)

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

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

Course content.

Real numbers, inequalities, absolute value. Elementary real functions: power functions, roots, exponential and logarithm, circular and hyperbolic functions and their inverses.
Systems of linear equations, coefficient matrix and augmented matrix of a system of linear equations, (Gauss-Jordan) row reduction, rank of a matrix, Rouche'-Capelli theorem, solving systems of linear equations by reducing the system to row echelon form (Gaussian elimination).
Vector space structure of R^n, linear dependence and independence of vectors, connection with the rank of suitable matrices, bases of subspaces, dimension of subspaces, linear transformations from R^n to R^m, kernel and image, matrix of a linear transformation, linear transformations from R^n to itself. Composition and inversion. Determinant, eigenvalues and eigenvectors, eigenbases.
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.

Readings/Bibliography

M. Bramanti, C. D. Pagani, S. Salsa : Matematica. Calcolo infinitesimale e algebra lineare. 2a ed., Zanichelli, Bologna, 2004.

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

Teaching methods

Front lectures and exercise sessions.

Assessment methods

The course assessment consists of a written examination followed by an oral examination. The written part consists of a series of exercises. 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 is based on both parts of the examination.

Teaching tools

Teaching tools consisting in updated diary of the lectures, exercises with solutions, and notes of the lecturers
will be available online

Office hours

See the website of Daniele Morbidelli

See the website of Giovanni Mongardi