This course first completes the theory of functions of one variable which was developed in Mathematical Analysis I, presenting the basic concepts of numerical series, power series and Fourier series. The basic notions of the Laplace transform and an introduction of the theory of analytic functions are also presented here. Then the course presents the basic topics in the mathematical analysis of functions of several variables. In particular, differential calculus in several variables, the theory of multiple integration, line and surface integration.

Understanding of the subjects of the course and computational skills in applying the mathematical tools presented in the course. Familiarity with the mathematical content of engineering disciplines. Ability in building a logical sequence of arguments using the tools introduced in the course.

The topics covered in the Mathematical Analysis I and Linear Algebra and Geometry courses. In particular, notions and properties of limits, numerical sequences, differential and integral calculus for functions of one variable, differential equations, linear algebra, geometry of curves.

Laplace transform (10 hours) Differentiating functions of several variables (20 hours) - Review on vectors and elements of topology of R^n. - Functions of several variables, vector fields. - Limits and continuity. - Partial and directional derivatives, Jacobian matrix. - Differentiability, gradient and tangent plane. - Second derivatives, Hessian matrix. - Taylor polynomial. - Critical points, free extrema. Integrating functions of several variables (30 hours) - Double and triple integrals, center of mass. - Length of a curve and area of a graph. - Line and surface integrals (graphs only), circulation and flux of a vector field. - Conservative vector fields. - Green, Gauss and Stokes theorems. Complex analysis and Series (40 hours) - Function theory of complex variable: differentiability, Cauchy-Riemann equations. - Complex line integrals, Cauchy integral formula. - Definition and convergence criteria for numerical series. - Power series (real and complex). Taylor series and Laurent series. Residue theorem - Fourier series.

The course consists in 60 hours of theoretical lectures and 40 hours of practice classes. Theoretical lectures are devoted to the presentation of the topics, with definitions, properties, introductory examples, as well as a number of selected proofs which are believed to facilitate the learning process. The practice classes are devoted to the analysis and the methods required for solving exercises with the aim of preparing the student to the exam.

The following lists collects some textbooks covering the topics of the course. Exercises, exams of preceding years and other material are available on the Home Page of the course in the Teaching Portal. - D. Bazzanella, P. Boieri, L. Caire, A. Tabacco, Serie di funzioni e trasformate, CLUT, 2001 - M. Bramanti, C.D. Pagani, S. Salsa, “Analisi matematica 2”, Zanichelli, 2009. - C. Canuto, A. Tabacco, "Analisi Matematica II", Springer, 2014 seconda edizione. - S. Salsa, A. Squellati, “Esercizi di Analisi matematica 2”, Zanichelli, 2011. - R.A. Adams, C. Essex, Calculus 2, Pearson Higher, 2019 - M. Codegone. Metodi matematici per l'ingegneria. Zanichelli, 1995. - G.C. Barozzi, Matematica per l'ingegneria dell'informazione, 2005, Zanichelli.

Modalita di esame:

Exam:

... The goal of the exam is to test the knowledge of the students on the topics included in the official program of the course and to verify their computational and theoretical skills in solving problems. Marks range from 0 to 30 and the exam is successful if the mark is at least 18. The exam consists in a written part and in an optional oral part. The written part consists of 10 exercises with closed answer on the topics presented in the course. Questions cover also theoretical aspects. The aim of the exam is to certify the Expected Learning Outcomes (see above). Questions cover both computational and theoretical aspects, to evaluate the ability in building a logical sequence of arguments using the tools introduced in the course. The exam lasts 100 minutes. Marks are given according to the following rules. Each exercise assigns: 3 points if correct, 0 points if blank, -1 point if wrong. The final mark is the sum of points obtained in each exercise plus 2 extra points. A cum Laude mark is assigned if the student answers correctly to all of the 10 questions. During the exam it is forbidden to use notes, books, exercise sheets and pocket calculators. The test results will be posted on the teaching portal. Students can request an optional oral part (upon request by the teacher or the student) that can alter both in the positive and in the negative the mark obtained in the written part. The optional oral part can only be requested in the same exam session of the written part. Students can request the optional oral part only if the mark they obtained in the written part is at least 18/30.

Gli studenti e le studentesse con disabilita o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unita Special Needs, al fine di permettere al/la docente la declinazione piu idonea in riferimento alla specifica tipologia di esame.