SM/0093 - MATHEMATICS 3
Academic Year 2021/2022
Free text for the University
IRENE IGNAZIA ONNIS (Tit.)
- Teaching style
- Lingua Insegnamento
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1. Acquiring knowledge and understanding.
The aim of the course is to provide students with the basic concepts of mathematical analysis for functions in several real variables, which will then be used in most of their subsequent studies. The theoretical structure of the course consists in the development of the topics of the program through a rigorous theoretical justification, motivated by significant examples and accompanied by exercises and some applications.
2. Applying knowledge and understanding.
At the end of the course, the student must be able to use the notions and techniques learned in standard exercises and new problems, which require the autonomous development of a strategy. In particular, the student must be able to apply the methods studied to solve particular problems arising in other disciplines like in Chemistry and Physics.
3. Making informed judgements and choices.
This course allows assiduous students to achieve knowledge and understanding sufficient for applying the techniques studied to the solution of mathematical problems which typically appeared in chemical sciences.
4. Communicating knowledge and understanding.
The evaluation of the written test takes into account the ability of the student to give a methodical and consistent exposition of the topics of the program necessary to find the solutions of the assigned exercises.
5. Abilities to continue learning.
This course allows assiduous students to acquire a basic expertise which provides the basis to attend the consecutive exams of Mathematics as well as the disciplines, like chemistry, whose language is expressed in a mathematical way.
It is necessary that the student has already passed "Matematica 1" and "Matematica 2". In particular, the knowledge of the basic notions of linear algebra, analytic geometry and the theory of real functions in a single real variable (limits, derivatives, integrals and important results related to these topics) is required.
1. Functions of several variables: scalar functions, vector functions and vector fields. Review of topology of the n-dimensional Euclidean space. Vector-valued functions, limits and continuity. Regular (piecewise) curves and vector differential calculus. Examples. Parametrization and length of a curve. Regular surfaces in the three-dimensional space. Parametrization of a surface. Surfaces that are the graph of a function of two variables and rotational surfaces. Examples.
2. Differential calculus for real functions of several variables. Graphs and level sets. Limits and continuity. Properties of limits and continuity. Partial derivatives and gradient. Partial derivatives of composite functions. Partial derivatives of higher order and the Schwarz Theorem. Differentiability, tangent plane and directional derivatives.
3. Applications of differential calculus. Classification of the critical points. Hessian matrix. Relative maxima and minima, saddle points.
4. Integral calculus for functions of several variables. Double integrals on regular domains: definition and properties. Area of a bounded sets of the plane. Reduction formulas for double integrals. Theorem of change of variables in double integrals; the case of the polar coordinates. Applications. Area of a surface. Triple integrals: definition and properties. Volume of limited subsets of the three-dimensional space. The reduction theorems for triple integrals. Change of variables in triple integrals; formula for triple integrals in cylindrical and spherical coordinates.
5. Vector fields. Gradient, rotor operator and divergence operator. Path integrals of a vector field: definition and physical meaning. Conservative fields and potentials. Gauss-Green formula in the plane. Examples and applications.
6. Differential Equations. Introduction and terminology. First order differential equation. Some particular type of first order differential equations: linear differential equations and separable equations. Cauchy problem for first order differential equation. Existence and uniqueness. Second order differential equations with constant coefficients.
Compatibly with the teaching methods provided for in the in the Manifesto Accademico 2021-22, as a consequence of the COVID-19 emergency, the tools used for the frontal lessons will mainly be the blackboard with, possibly, a synchronous streaming system via internet using the Microsoft Teams platform.
For the student's preparation at home, the teacher updates a website dedicated to students (https://www.unica.it/unica/page/it/ireneionnis)
where they can find teacher notes and weekly exercises to do at home. In order to make the teaching as much efficient as possible, the theoretical topics are immediately supported by exercises.
The teacher offers constant assistance to the students during the whole year both by personal interviews and by means of e-mail messages.
Verification of learning
The written test lasts three hours and consists of some questions relating to the topics learned during the course and in carrying out exercises similar to those discussed during the lectures or assigned as homework, inherent all the topics of the course, which require both operational skills and theoretical knowledge to be solved. The test is passed if the student obtains an evaluation of at least 18/30.
An alternative grading mode is available for those students who regularly attend the lectures and consists of two intermediate written tests. The first one concerns the topics studied in the first 24 lecturing hours and it takes place immediately after that the first half of the lectures has been delivered. The second written test concerns the remaining topics (but it also requires a global understanding of the program of the course) and it takes place, approximately, one week before the first official grading date. The final vote is the arithmetic average between the evaluations of the two written tests.
1) G.C. Barozzi, G. Dore, E. Obrecht, Elementi di Analisi Matematica, volume 2, Zanichelli
2) M. Bramanti, C.D. Pagani, S. Salsa, Analisi Matematica 2, Zanichelli
3) C. Canuto, A. Tabacco, Analisi Matematica II, Teoria ed esercizi con complementi in rete, Springer
4) N. Fusco, P. Marcellini, C. Sbordone, Elementi di Analisi Matematica due, Liguori Editore
5) P. Marcellini, C. Sbordone, Esercitazioni di Analisi Matematica Due, prima e seconda parte, Zanichelli
6) J. Stewart, Calcolo - Funzioni di più variabili, Apogeo Education, Maggioli Editore
In the site https://www.unica.it/unica/page/it/ireneionnis
the student find other informations about the course, the exercises proposed by the teacher and the lesson diary.
Our University provides support for students with specific learning disability (SLD). Those interested can find more informations at this link: