00013 - Mathematical Analysis

Learning outcomes

The student must develop three types of skills: 1) making computations relative to functions of one or more variables (limits, derivatives, integrals) and being able to solve some problems with the techniques of mathematical analysis (calculate the length of a curve, of a volume or solve some ordinary differential equations);  2) understanding and being able to use the basic definitions of mathematical analysis;3) knowing the elementary functions.

Course contents

Introduction. Basics on set theory, cartesian product, relations. Functions: definition, injective, surjective and bijective functions, composition between functions. Ordered sets: maximum, minimum, supremum, infimum. Completeness. The real numbers. Natural, integer and rational numbers. Density property. 

Complex numbers. Definition and basic operations. Algebraic and exponential form of a complex number. Roots and algebraic equations in C.

Real sequences. Limits of real sequences and their basic properties. Monotone sequences and their limits. Definitions of the number e and some remarkable limits.

Functions of one real variable, limits and continuity. Elementary functions: powers, exponential, logarithm, trigonometric functions and their inverse. Definition of limit and main properties. Some remakable limits. Continuity and main theorems on continuous functions.

Differential calculus. Definition of derivative, rules of derivations, main theorems on differentiable functions (Rolle, Lagrange, de l'Hôpital, theorem on monotone functions). Higher order derivatives, Taylor formula. Local maximum and minimum, Fermat Theorem; convex functions.  

Integral calculus. Integral of continuous functions and its main properties. The mean value Theorem an the Fundamental Theorem of calculus. Integration by parts and by change of variables. Integrations of rational functions.

Ordinary Differential equations. Linear first order differential equations, general integral for homogeneous and non-homogeneous equations. Cauchy problem. Second order linear differential equations with constant coefficients. Separable differential equations.

Basics of Linear Algebra. Vectors in R^n and operations. Lines and planes in R^3, cartesian and parametric equations. Matrices and operations between them. Quadratic forms.

Differential calculus for functions of more real variables. Functions of two variables, examples, their graphs and level sets. Definition of limit and of continuous function. Weierstrass Theorem. Definition of partial derivative and differentiability. Tangent plane. Relation between the notion of continuity, derivability and differentiability. Directional derivative; the gradient and its geometric meaning.Higher order derivatives, Schwarz Theorem, Hessian. Taylor formula. Critical points, local maximum and minimum, Fermat Theorem. Classification of critical points.

 

 

 

 

 

 

Readings/Bibliography

M.Bramanti, C.Pagani, S.Salsa, Matematica. Calcolo infinitesimale e algebra lineare,
Zanichelli Editore

M. Bertsch, R. Dal Passo, L. Giacomelli - Analisi Matematica, ed. McGraw Hill. (seconda edizione)

G.C. Barozzi, G. Dore, E. Obrecht: Elementi di Analisi Matematica, vol. 1, ed. Zanichelli.

Marcellini-Sbordone: Elementi di analisi matematica uno (versione semplificata per i nuovi corsi di laurea), Liguori Editore, Napoli 2002.

 

On the Teacher's website (link: http://www.dm.unibo.it/~eleonora.cinti5/Teaching.html) and on the page "Virtuale" some exercizes and past exams are available).

 

Teaching methods

The course will consist of both theoretical lectures and problem sessions with the aim of making the students familiar with the main concepts and techniques of the course. Moreover, some exercices will be given to the students as homework.

Assessment methods

The exam consists in two parts: a written one and an oral one. The written part, of the duration of 3 hours, consists in the resolution of exercises, the oral part on the theoretical aspects of the course, such as definitons and theorems. During the exam the use of calculator, books and notes is not allowed.

Teaching tools

The suggested Textbooks and other material that will be available online on the webpage of the teacher and on the page "Virtuale".

Office hours

See the website of Eleonora Cinti