Mathematical Analysis 2

Credits

Prerequisites

Mathematical Analysis 1

Scientific-disciplinary sector (SSD)

MAT/05 Mathematical Analysis.

Examination method

Written exam (exercises and numerical problems, possibly multiple-choice) and oral exam.

Learning
objectives

The course aims to provide an introduction to the study of functions of several variables and the related multiple integrals, as well as to the elementary theory of curves and surfaces and to the theory of ordinary differential equations.

Syllabus

Sequences and series of functions, power series, Taylor series, analytic functions. Topology of the spaces Rⁿ. Continuity and differentiability of functions of several variables: level curves, gradient field. Maxima and minima of functions of several variables. Taylor’s formula. Vector-valued functions. Elementary theory of curves, with particular regard to plane and space curves. Length of a curve. Line integrals. Area of a solid of revolution. Differential forms. Circulation of a field along a closed curve. Conservative fields and potential of a field. Irrotational fields. Double integrals: reduction formulae, Gauss-Green formulae and change of variables. Computation of volumes. Triple integrals: reduction formulae and change of variables. Parametrised surfaces in space. Computation of the area of a surface, surface integrals. Flux of a field through a surface. Stokes’ formula and the divergence theorem. Dini’s theorem, implicit functions and systems of implicit equations. Local and global invertibility. Constrained maxima and minima. Ordinary differential equations. Local and global existence and uniqueness theorems. Solution methods for linear equations and systems and for some particular non-linear equations.

Expected learning
outcomes

At the end of the course, students must demonstrate that they:

know and understand the issues related to differential and integral calculus for functions of several variables;
are able to apply the knowledge acquired to the study of functions of several variables and the related multiple integrals
are able to communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
are able to identify the most appropriate methods to analyse and solve a problem related to the course topics and to interpret the results correctly.

Learning outcomes
to be assessed

Command of the knowledge acquired, clarity of presentation, rigour in the use of language, confidence in using the notions acquired.

Mathematical Analysis 2