OVERVIEW

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

The course will be focused on series of functions, Fourier series, Laplace transforms, equation and systems of linear differential equations, curves, surfaces and related integrals, conservative vector fields and Gauss-Green formulas.

LEARNING OUTCOMES

The main objective of the course is to provide students tools for differential and integral calculus and methods for numerical integration for Cauchy problems and  for definite integrals.

AIMS AND LEARNING OUTCOMES

Topics of this course include the study of series of real numbers, series of functions, Fourier series, Laplace transforms, equations and systems of linear differential equations, curves and surfaces, conservative vector fields and Gauss-Green formulas.

PREREQUISITES

All the topics of the I year courses “Analisi Matematica I” and “Geometria”.

Teaching methods

52 hours of lessons in classroom, where the definitions and the theorems will be presented with heuristic examples together with the solution of related exercises.

SYLLABUS/CONTENT

Improper integrals on unbounded domains, Improper integrals of functions with unbounded range.

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.

Fourier series. Derivative and integral of Fourier series. Gibbs phenomenon. Fourier series for the heat Equation Trigonometric systems.  

Linear differential equations of higher order with constant coefficients, homogeneous and non-homogeneous. Systems of Linear differential equations.

Laplace transform and its properties. Laplace antitransform. Examples and exercises. Application to  Linear differential equations.

Regular curves and length.

Regular surfaces. Curves on surfaces. Tangent plane. Surface area, Surface integrals.

Line integrals of scalar fields. Line integrals of linear differential forms.

Exact Differential Forms and conservative fields. Gauss-Green formulas

Conservative vector fields. Simply connected domains. Poincaré’s lemma. Computation of the potential field.

RECOMMENDED READING/BIBLIOGRAPHY

Handouts “Matematica II” e "Metodi matematici per l'ingegneria" by prof. Maurizio Romeo, downloadable for free from the web page of the course.

"Appunti sulle serie" by prof. Franco Parodi, downloadable for free from the web page of the course.

"Appunti sulla trasformata di Laplace" by prof. Paolo Tilli, 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.

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

Exam Board

CLAUDIO ESTATICO (President)

ROBERTUS VAN DER PUTTEN

MARCO BARONTI (President Substitute)

Teaching methods

52 hours of lessons in classroom, where the definitions and the theorems will be presented with heuristic examples together with the solution of related exercises.

LESSONS START

Lessons start according to the academic calendar.

ORARI

L'orario di tutti gli insegnamenti è consultabile su EasyAcademy.

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MATHEMATICAL ANALYSIS 2

Exam description

Written an oral examination.

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