OVERVIEW

The course is aimed at sophomore students and needs basic skills in Calculus, Linear Algebra and Geometry.

AIMS AND CONTENT

LEARNING OUTCOMES

The course provides basic notions about multiple integrals, line integrals, surface integrals and vector fields. It provides also basic skills about  holomorphoic functions, Laplace transforms together with some appplications to ODE's.

AIMS AND LEARNING OUTCOMES

Students are expected to master the main techniques for calculating double and triple integrals, line integrals of scalar and vector fields, the basic properties of vector fields and the classical theorems of the differential calculus in Euclidean space (divergence, curl, Stokes, Gauss-Green).

At the end of the course students will be able to recall and present the theoretical notions that have been presnted during the lectures. Applying the various techniques introducted in the course and justifying every step, they will be able to:

• compute a double or triple integral using the main integration techniques over regions with significant geometry (portions of cones, cylinders, spheres, elipsoids);
• compute line and surface integrals and use tclassical theorems such as the divergence or the Gauss-Green theorems;
• sestablish if a vector field is conservative, and in the affirmative case, compute its potentials;
• use the main properties of functions of a complex variable to compute line integrals (the residue theorem)
• know and use the basic properties of the Laplace transform

-solve ODE's by using Laplace transform.

PREREQUISITES

Basic Calculus, Linear algebra and Geometry.

TEACHING METHODS

Traditional lecture (60 h) unless otherwise required by the pandemic emergency

SYLLABUS/CONTENT

• Riemann integral for  functions of  2/3 variables. Measure in  R^2 andR^3. Reduction formulae for double and triple integrals.Change of variables. Polar, cylindrical and spherical coordinates.
• Curves in R^n. Length of a curve. Line integrals w.r.t. arclength.
• Parametric surfaces in R^3. Area of a surface, surface integrals. 
• Vector fields. Irrotational vector fields, conservative vector fields. Divergence theorem, Gauss-Green formulae.
• Functions of one complex variable. Holomorphic functions,  Cauchy Riemann equations. Cauchy’s theorem and  intergal formula. Isolated singularities and residue theorem.  Definition and basic properties of the Laplace transform, simple applications.

RECOMMENDED READING/BIBLIOGRAPHY

•     Canuto-Tabacco, Analisi Matematica II
• Pagani-Salsa, Analisi Matematica 2

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

https://corsi.unige.it/8716/p/studenti-orario

Class schedule

MATHEMATICAL ANALYSIS II

EXAMS

Exam schedule