OVERVIEW

The course "Mathematical Analysis I" aims to provide students with some basic mathematical tools, both theoretical and computational, useful for engineering and application-oriented topics.

The course will be focused on functions of one and several real variables, on the related differential and integral calculus, on the resolution of ordinary differential equations and series of functions.

AIMS AND CONTENT

LEARNING OUTCOMES

The course introduces general mathematical notions and tools at the basis of engineering modeling, related to the study of the functions of one or more real variables. In particular, the concept of limit and continuity, the differential and integral calculus, also of functions of several real variables, the resolution of ordinary differential equations, the analysis of curves and surfaces, and the study of the convergence of numerical series and series of functions.

AIMS AND LEARNING OUTCOMES

Topics of this course include the study of functions of one and several variables, including limits, continuity, differentials and integrals, gradients, Hessian matrices, max/min unconstrained optimization. Moreover, sequences, first order differential equations, higher order linear differential equations,  series of real numbers, series of functions.

PREREQUISITES

Elementary algebra: literal calculus, polynomials, equations and inequalities, trigonometry.

TEACHING METHODS

72 hours of theoretical lessons, 48 hours of classroom practices. During the theoretical lessons the definitions and the theorems will be presented with many examples and applications. During the other part of the course many exercises will be solved. 

In addition, a tutor will solve some exercises in extra (optional) lesson hours.

SYLLABUS/CONTENT

Sets, logic, real numbers, infimum and supremum.

Functions of one real variable, elementary functions, limits, infinitesimals and infinities, continuous functions, derivable functions, differentiable functions. Taylor’s formula, expansion of elementary functions.

Primitives and indefinite integrals, methods of indefinite integration, definite integrals, fundamental theorem of integral calculus. Improper integrals.

First order differential equations, Cauchy’s problem and theorem, resolution of linear first order differential equations and separable variables equations, linear differential equations with constant coefficients of order n.

Functions of several variables (scalar and vectorial fields), limits and continuity. Directional derivatives. Differentiable functions. Necessary and sufficient conditions for differentiability. Derivatives of composite functions. Derivatives of higher order, Schwarz Theorem and Taylor polynomial in several variables. Unconstrained maxima and minima of scalar fields, necessary and sufficient conditions, Hessian matrix.

Differential equations of the first order, with separable variables, linear, homogeneous, Bernoulli and Riccati types. Existence and uniqueness theorem (hints). Linear differential equations. Linear differential equations of higher order with constant coefficients, homogeneous and non-homogeneous,

Numerical series. Convergence criteria for constant sign numerical series. Numerical alternating series and absolutely convergent series. Series of functions. Pointwise, absolute, uniform and total convergence criteria. Derivation and integration of several functions (hints). Power series, radius of convergence.

RECOMMENDED READING/BIBLIOGRAPHY

Handouts: "MATHEMATICS I"  and "MATHEMATICS II" by prof. Maurizio Romeo, downloadable for free from the web page of the course.

Sheets containing links to web pages with different solved exercises, downloadable for free from the web page of the course.

Workbook: Laura Recine - Maurizio Romeo, Esercizi di analisi matematica - Volume II, Maggioli Editore.

P. Marcellini – C. Sbordone: Calcolo, Liguori Editore, Napoli, or any other good text of mathematical analysis.

M.Baronti – F.De Mari – R.Van Der Putten – I.Venturi: Calculus Problems, Springer

TEACHERS AND EXAM BOARD

Exam Board

CLAUDIO ESTATICO (President)

MARCO BARONTI (President Substitute)

ULDERICO FUGACCI (President Substitute)

LESSONS

LESSONS START

Lessons start according to the academic calendar.

Class schedule

MATHEMATICAL ANALYSIS I

EXAMS

EXAM DESCRIPTION

The final exam consists of a written test and an oral exam. The student must obtain an evaluation of at least 16/30 in the written test to access the oral exam.

ASSESSMENT METHODS

During the written test the student will have to solve some exercises concerning the arguments of the course. 

During the oral examination the student must highlight critical analytical skills and must be able to apply the main theorems for the solution of easy exercises.

Exam schedule

FURTHER INFORMATION

Attendance is not compulsory but strongly recommended to all students.