Lecturer: Simona Fornaro  

Course name: Calculus II (Mathematical analysis II)
Course code: 500121
Degree course: Bioingegneria, Ingegneria Elettronica e delle Telecomunicazioni, Ingegneria Informatica
Disciplinary field of science: MAT/05
University credits: CFU 9
Course website: n.d.

Specific course objectives

The aim of the present course is to provide the Students with the notions of numerical and power series and, above all, with the basic knowledge of differential and integral calculus for both real- and vector-valued functions depending on several real variables. The course will focus on the comprehension of the definitions and the main results. Besides, some theorems will be proved in full details. A large part of the course will be devoted to examples and exercises. At the end of the course, the Students should be able to solve, without any problem, numerical series, power series, to compute partial derivatives, multiple integrals, surface and curve integrals and to know the main theoretical notions.

Course programme

Numerical and power series
Numerical series: definition and examples. Series with positive terms. Tests for convergence. Absolute convergence. Alternating series. Power series on the real line. Convergence and arithmetic for power series. Calculus of power series. Taylor polynomials. Taylor formulas. Taylor series. MacLaurin developments of some elementary functions.

Differential calculus
Real-valued functions of several variables. Graphs and level sets. Limits and continuity. Partial and directional derivatives. Differentiability and gradient. Higher order derivatives. The chain rule. Second order Taylor formula. Vector-valued functions of several variables. Jacobian matrix. Relative maxima and minima. Classification of critical points. Constrained extrema and Lagrange multipliers.

Multiple integrals
Integrals in two and three dimensions: definition and main properties. Applications to Geometry and Physics. Iterated integrals. Change of variables formula. Polar coordinates. Spherical and cylindrical coordinates.

Curves and surfaces
Definition of a curve. Tangent line. Lenght of a curve. Rectifiable curves. Parametric representation of a surface. Surface area. Tangent plane. Revolution surfaces. Line integrals and their physical meaning. Divergence and curl of a vector field. Conservative fields and potential functions. Green's theorem. Stokes' theorem. Divergence theorems.

Course entry requirements

The course presupposes a general background in linear algebra, analytical geometry in the plane and in the space, complex numbers, differential and integral calculus for functions depending on one real variable. The contents of the courses of Calculus 1 and Linear Algebra are strongly recommended.

Course structure and teaching

Lectures (hours/year in lecture theatre): 45
Practical class (hours/year in lecture theatre): 45
Practicals / Workshops (hours/year in lecture theatre): 0

Suggested reading materials

C. Canuto, A. Tabacco. Analisi Matematica II. Springer, Milano, 2008.

Testing and exams

The exam consists of a written part, which is made of exercises concerning the program of the course, and an oral one, which focus on the theoretical aspects of the subjects. Only the Student who succeeds in the written test can be admitted to the oral part.