# MATHEMATICAL ANALYSIS 2

8 credits during the 2nd year of 8766 Mathematical Statistics and Data Management (L-35) GENOVA

OVERVIEW

Basic topics in calculus of severable variables are treated. The course is split into two semesters. The first part is devoted to differential calculus, integration theory in two (or several) variables and function series. The second part deals with integration along curves and surfaces, Gauss and Stokes' theorems and their consequences, and an introduction to systems of differential equations. SMID students need to take only part I (first semester - 8 credits). Both semesters are mandatory for Physics students.

## AIMS AND CONTENT

LEARNING OUTCOMES

Students will become acquainted with the most important topics in several (real) variables and how they are used in practice. We only present proofs that illustrate fundamental principles and are free of technicalities.

Applications to Physics and Probability are emphasised.

AIMS AND LEARNING OUTCOMES

Students will be able to manipulate functions of several variables and solve basic optimization problems. Moreover they will be at their ease with mean and conditional expectation.

PREREQUISITES

First year calculus (derivatives and integrals for functions of a single variable). Vector spaces, eigevalues and eigenvectors are frequently used.

Teaching methods

Both theory and exercises are presented by the teacher. The first semester consists of 12 weeks with four hours of theory and two hours of exercises per week. The second semester consists of 12 weeks with three hours of theory and two hours of exercises per week.

SYLLABUS/CONTENT

This course covers differential, integral and vector calculus for functions of several variables. Topics include:

- Sequences and series (numerical, function series, power series,...)
- Functions of several variables: continuity, partial derivatives, differentiability, local maxima and minima, implicit functions, optimization.
- Parametrized curves, arc length, curvature, torsion.
- Vector fields, gradient, curl, divergence.
- Multiple integrals, change of variables, line integrals, surface integrals.
- Stokes' theorem in one, two, and three dimensions.
- Systems of constant coefficient linear ordinary differential equations

RECOMMENDED READING/BIBLIOGRAPHY

Serge Lang - Calculus of Several Variables, Third Edition, Undergraduate Texts in Mathematics, Springer, 1987.

## TEACHERS AND EXAM BOARD

**Ricevimento:** Questions during or at the end of lectures are welcome. Meetings will be organized upon email request.

**Ricevimento:** Weekly office hours will be communicated. Meetings upon email requests will also be considered.

## LESSONS

Teaching methods

Both theory and exercises are presented by the teacher. The first semester consists of 12 weeks with four hours of theory and two hours of exercises per week. The second semester consists of 12 weeks with three hours of theory and two hours of exercises per week.

LESSONS START

Classes will start according to the accademic calendar.

## EXAMS

Exam description

Written and oral exam

Assessment methods

The written examination consists in solving four/five exercises covering the topics of the course. It is approved with grade at least 15/30.

If your written exam is approved with grade at least 18/30, then the oral exam is optional. In case you do not want to take the oral part

- the final mark is the grade of the written exam if less than or equal to 24
- the final mark is 24 if the grade of the written exam is greater than 24

If you take also the oral part, the final mark is a weighted average of the written and oral tests.

## FURTHER INFORMATION

For further information, please send a message to astengo@dima.unige.it