


Politecnico di Torino  
Academic Year 2015/16  
05QPVLZ, 05QPVJM, 05QPVLI, 05QPVLM, 05QPVLN, 05QPVLP, 05QPVLS, 05QPVLX, 05QPVMA, 05QPVMB, 05QPVMC, 05QPVMH, 05QPVMK, 05QPVMN, 05QPVMO, 05QPVMQ, 05QPVNX, 05QPVOA, 05QPVOD, 05QPVPC, 05QPVPI, 05QPVPL Mathematical analysis II  Talenti 

1st degree and Bachelorlevel of the Bologna process in Aerospace Engineering  Torino 1st degree and Bachelorlevel of the Bologna process in Mechanical Engineering  Torino 1st degree and Bachelorlevel of the Bologna process in Automotive Engineering  Torino Espandi... 





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. 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. Finally equations and systems of differential equations are introduced and studied by the Laplace transform and by qualitative methods.

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. In particular, limits, sequences, differential and integral calculus for functions of one variable, differential equations.

Contents
Definition and convergence criteria for numerical series. Power series. Fourier series.
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 and constrained extrema. Double and triple integrals, center of mass. Parametric curves and surfaces. Length of a curve and area of a surface. Line and surface integrals, circulation and flux of a vector field. Conservative vector fields. Green, Gauss and Stokes theorems. Ordinary differential equations of order n. First order systems, Laplace transform and qualitative studies. Applications to mechanics (onedimensional systems, coupled oscillators). Stability of stationary solutions. 
Delivery modes
Theoretical lessons: 60 hours. Exercise hours: 40 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:
 C. Canuto, A. Tabacco, "Analisi Matematica II  Teoria e esercizi", seconda edizione, Springer, 2014  S. Lang, Calculus of several variables, third edition, Springer, 1987  C. Pagani, S. Salsa, "Analisi Matematica II", Zanichelli, 2016. Other material will be avalaible on the Portale della Didattica. 
Assessment and grading criteria
The goal of the exam is to test the knowledge of the candidate on the topics included in the official program of the course and to verify the computational and theoretical skills in solving problems. Marks range from 0 to 30 cum laude (corresponding to 33) and the exam is succesful if the mark is at least 18.
The exam is written and consists of 8 exercises with closed answer and one exercise with open answer on the topics presented in the course. Questions cover also theoretical aspects. Marks are given according to the following rules. Closed answer exercises: 3 points if correct, 0 points if blank, 1 point if wrong. Open answer exercise: 9 points. The exam lasts two hours. During the exam it is forbidden to use notes, books, exercise sheets, pocket calculators and any other electronice device. 
