00013 - Mathematical Analysis

Learning outcomes

Students must develop three skills: 1) working with functions of one or more real variables (limits, differentiation, integration) and solving problems using mathematical analysis techniques; 2) understanding and applying the core definitions of mathematical analysis; 3) mastering elementary functions.
The ultimate goal is to provide a foundational grounding in mathematical analysis.

Course contents

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

Complex numbers. Definition and basic operations. Algebraic and exponential forms 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. Definition of the number e and some remarkable limits.

Functions of one real variable: limits and continuity. Elementary functions: powers, exponentials, logarithms, trigonometric functions, and their inverses. Definition of limit and main properties. Some remarkable limits. Continuity and main theorems on continuous functions.

Differential calculus. Definition of the derivative, rules of differentiation, and main theorems on differentiable functions (Rolle, Lagrange, de l'Hôpital, theorem on monotone functions). Higher-order derivatives and Taylor's formula. Local maxima and minima, Fermat's Theorem, and convex functions.

Integral calculus. The integral of continuous functions and its main properties. The Mean Value Theorem and the Fundamental Theorem of Calculus. Integration by parts and by substitution (change of variables). Integration of rational functions.

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

Differential calculus for functions of several real variables. Functions of two variables: examples, graphs, and level sets. Definition of limit and continuous functions. Weierstrass Theorem. Definition of partial derivative and differentiability. Tangent plane. Relationships between the notions of continuity, partial derivability, and differentiability. Directional derivatives: the gradient and its geometric meaning. Higher-order derivatives, Schwarz's Theorem, and the Hessian matrix. Taylor's formula. Critical points, local maxima and minima, Fermat's 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.

Teaching methods

Traditional classroom lectures supplemented by examples, counterexamples, and worked problems.

Assessment methods

The examination consists of a written test lasting 2.5 hours. This final exam is divided into a practical section with open-ended problems and a theoretical section with questions on the various topics covered in the course.

During the exam, students are not permitted to use calculators, textbooks, or notes.

 

Students with learning disorders and\or temporary or permanent disabilities: please, contact the office responsible (https://site.unibo.it/studenti-con-disabilita-e-dsa/en/for-students) as soon as possible so that they can propose acceptable adjustments. The request for adaptation must be submitted in advance (15 days before the exam date) to the lecturer, who will assess the appropriateness of the adjustments, taking into account the teaching objectives.

Teaching tools

Supplementary learning materials made available by the instructors on the Virtuale platform. Tutoring activities.

Office hours

See the website of Gregorio Chinni

See the website of Massimo Cicognani