


Politecnico di Torino  
Academic Year 2017/18  
23ACIOD, 23ACIMQ Mathematical analysis II 

1st degree and Bachelorlevel of the Bologna process in Physical Engineering  Torino 1st degree and Bachelorlevel of the Bologna process in Mathematics For Engineering  Torino 





Subject fundamentals
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 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.

Expected learning outcomes
Understanding of the subjects of the course and computational skill. Familiarity with the mathematical content of engineering disciplines.

Prerequisites / Assumed knowledge
The topics contained in the courses of Mathematical Analysis I and Linear Algebra and Geometry. In particular, limits, sequences, differential and integral calculus for functions of one variable, differential equations, linear algebra, geometry of curves.

Contents
Laplace transform.
Definition and convergence criteria for numerical series. Power series. Fourier series. 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. 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. 
Delivery modes
Theoretical lessons: 50 hours. Exercise hours: 30 hours. Theoretical lessons are devoted to the presentation of the topics, with definitions, properties and the proofs which are believed to facilitate the learning process. Every theoretical aspect is associated with introductory examples. The exercise hours are devoted to the analysis and the methods required for solving exercises with the aim of preparing the student to the exam.

Texts, readings, handouts and other learning resources
The teacher of the course will suggest some textbooks among the following during the lectures:
 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 Other material will be avalaible on the Portale della Didattica. 
Assessment and grading criteria
The written exam consists of 7 exercises with closed answer and one exercise with open answer on the topics presented in the course. Questions cover also theoretical aspects.
The exam lasts two hours. During the exam it is forbidden to use notes, books, exercise sheets, pocket calculators and any other electronice device. 
