PORTALE DELLA DIDATTICA

### Mathematical analysis II

02KXULI, 02KXUJM

A.A. 2023/24

Course Language

Inglese

Course degree

1st degree and Bachelor-level of the Bologna process in Ingegneria Dell'Autoveicolo (Automotive Engineering) - Torino
1st degree and Bachelor-level of the Bologna process in Ingegneria Meccanica (Mechanical Engineering) - Torino

Course structure
Teaching Hours
Teachers
Teacher Status SSD h.Les h.Ex h.Lab h.Tut Years teaching
Teaching assistant
Context
SSD CFU Activities Area context
MAT/05 6 A - Di base Matematica, informatica e statistica
2022/23
This course first completes the theory of functions of one variable which was developed in the Mathematical Analysis course, by introducing the student to basic concepts of numerical series, power series, Fourier series, and Laplace transform. The course then introduces basic topics in the mathematical analysis of functions of several variables including differential calculus in several variables, the theory of multiple integration, line, and surface integration.
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 acquired knowledge: - Definition, properties, and convergence criteria for numerical series. - Definition, properties, and convergence criteria for sequences and series of functions, power series, and Fourier series. - Definition and main properties of Laplace transform. - Properties and analysis of functions of several variables. - Definition and properties of double, triple, line, and surface integrals. - Conservative vector field; Green, Gauss, and Stokes theorems. Expected acquired ablities: - to classify the asymptotic behaviour (e.g., converge or the lack thereof) of numerical series, sequences of functions, and series of functions using the criteria introduced in class; - to infer the set and type of convergence of sequences and series of functions; - to compute the power series approximation of a function and the Fourier series approximation of a periodic function; - to compute the Laplace transform of simple functions and compare their properties; - to analyze a function of several variables; find and classify its critical points; - to compute double, triple, line, and surface integrals; - to reproduce, discuss, and explain theoretical results presented in class and apply them to solve simple problems.
- Understanding of the subjects of the course and computational skills in applying the mathematical tools presented in the course. - Familiarity with the mathematical contents 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.
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.
- Laplace transform. - Numerical series: definition and convergence criteria. Power series and Taylor series. Fourier series. - Review on vectors and elementary notions 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, unconstrained maxima and minima. - Double and triple integrals, center of mass. Length of a curve and area of the region under the graph of a function. Line and surface integrals (for graphs of functions only). Circulation and flux of a vector field. - Conservative vector fields. Green, Gauss, and Stokes theorems.
- 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 (15 hrs). - 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 (25 hrs). - Definition and convergence criteria for numerical series. Power series. Fourier series (20 hrs).