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 multivariable functions, multiple integrals, number and power series, Fourier series, Laplace transforms, equation and systems of linear differential equations, curves, surfaces and related integrals, conservative vector fields and Gauss-Green formulas.

AIMS AND CONTENT

LEARNING OUTCOMES

The course aims to provide the essential tools for the study of power series, Fourier series, the Laplace transform, as well as multiple integration, systems of linear differential equations, and an introduction to the integral-differential calculus of vector fields. The course also provides students with the basic knowledge of mathematical analysis related to the theory of real functions of multiple real variables.

AIMS AND LEARNING OUTCOMES

Topics of this course include the study of multivariable functions, multiple integrals, series of real numbers, power series, 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

80 hours of lecture classes and exercise classes, where the definitions and the theorems will be presented with heuristic examples together with the solution of related exercises. Attendance is not compulsory but strongly recommended..

Students having disabilities or Specific Learning Disorders can request suitable aids for the examinations. Such aids will be defined according to specific needs, together with the Referent for the Polytechnic School of the Committe for the inclusion of Students with Disabilites and with SLD. Students in such conditions are invited to get in touch (via e-mail) with the Teacher a sufficient time before the examination, inserting in copy the Referent prof. Federico Scarpa - federico.scarpa@unige.it (https://unige.it/en/commissioni/comitatoperlinclusionedeglistudenticondisabilita.html), without sending any document about their disabilites.

SYLLABUS/CONTENT

Functions of several variables. Continuity, directional and partial derivatives, gradient.  Differentiability and tangent plane. Level sets. Local minima and maxima: second order derivatives and the Hessian. Schwarz theorem.

Multiple integrals, double and triple; center of gravity, moment of inertia.

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

Specific guidelines on the reference bibliography will be provided by the professor at the beginning of the course.

In general, lecture notes and the materials available for download from the course webpage are sufficient for exam preparation. More specifically, the following materials may be useful:

• Professor Maurizio Romeo's theory handouts, available for free download from the course's AulaWeb page;
• Worksheets containing links to webpages with various solved exercises, available for free download from the course's AulaWeb page;
• "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.
• C. Canuto, A. Tabacco, Analisi Matematica 2, 2a edizione, Springer-Verlag Italia, 2014;
• M. Baronti, M., F. De Mari,  R. van der Putten, I. Venturi,  Calculus Problems, Springer International Publishing Switzerland, 2016.

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

https://corsi.unige.it/en/corsi/11976/studenti-orario

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam consists of a written and an oral examination.

The written test consists in solving exercises on the various topics of the course. The written exam must be done before the oral exam and can be taken either in previous sessions or in the same session in which the student intends to take the oral exam. Only students who have previously passed the written exam with a score greater than or equal to 16/30 can access the oral exam. At least 2 exam sessions will be available for the winter session (mid-January and February) and 3 exam sessions for the summer session (June, July and September).

To enroll the exam you must register by the deadline on the website https://servizionline.unige.it/studenti/esami/prenotazione

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 simple exercises.

FURTHER INFORMATION

Ask the professor for other information not included in the teaching schedule.