


Politecnico di Torino  
Academic Year 2012/13  
22ACIMK, 22ACIMA Mathematical analysis II 

1st degree and Bachelorlevel of the Bologna process in Energy Engineering  Torino 1st degree and Bachelorlevel of the Bologna process in Biomedical Engineering  Torino 





Subject fundamentals
The main goal of this course is to present 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. The course also presents the theory of numerical, power and Fourier series.

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
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. Definition and convergence criteria for numerical series. Power series. Fourier series. 
Delivery modes
Theoretical lessons: 40 hours. Exercises: 20 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 execise hours are devoted to the analysis and the methods reqired for solving exercises with the aim of preparing the student to the exam.

Texts, readings, handouts and other learning resources
The following lists collects some textbooks covering the topics of the course.
 C. Bianca, L. Mazzi "Pillole di Analisi Matematica II", CLUT, 2014  S. Lancelotti, "Lezioni di Analisi Matematica II", Celid, 2017.  S. Lancelotti, "Esercizi e quiz di Analisi Matematica II", Celid, 2017.  V. Barutello, M. Conti, D. Ferrario, S. Terracini, G. Verzini, "Analisi Matematica", volume 2, Apogeo, 2013.  C. Canuto, A. Tabacco, "Analisi Matematica II", Springer, 2014. 
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 and the exam is succesful if the mark is at least 18.
The exam is written and 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. 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. An additional point is reserved to notational clarity and rigour in the exposition and allows the student to obtain a cum laude mark. The exam lasts two hours. 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 together with the date in which the students can see their tests and ask for explanations. 
