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## MATHEMATICAL ANALYSIS 2

CODE 57048 2021/2022 12 cfu during the 2nd year of 8758 FISICA (L-30) - GENOVA 8 cfu during the 2nd year of 8766 STATISTICA MATEM. E TRATTAM. INFORMATICO DEI DATI (L-35) - GENOVA MAT/05 Italian GENOVA Annual Prerequisites You can take the exam for this unit if you passed the following exam(s): PHYSICS 8758 (coorte 2020/2021) MATHEMATICAL ANALYSIS 1 52474 Prerequisites (for future units) This unit is a prerequisite for: PHYSICS 8758 (coorte 2020/2021) MATHEMATICAL METHODS IN PHYSICS 61734 AULAWEB

## 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

At the end of the first semester, 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.

Physics students will be able to apply vector calculus: use double, triple and line integrals in applications, including Green's Theorem, Stokes' Theorem and Divergence Theorem.

### PREREQUISITES

First year calculus (derivatives and integrals for functions of a single variable, sequences and numerical series). 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

Differential Calculus

1. Vectors, scalar product, norm, distance
2. Elements of Topology: open and closed sets, bounded, compact and connected sets, isolated, cluster and boundary points. Heine-Borel's Theorem
3. Functions: examples and graphs, level sets. Continuous functions and their local properties. Algebra of continuous functions.
4. Limits.
5. Global properties of continuous functions: Weierstrass Theorem and Intermediate value Theorem
6. Differential calculus: partial derivatives, gradient, jacobian matrix. Tangent vector to a curve. Tangent plane to the graph of a 2 variable function. Chain rule. Functions with null gradient. Second derivatives,  Schwarz' Theorem. Hessian and Taylor formula of order 2. Relative maxima and minima. Diffeomorphisms. Inverse Function Theorem. Implicit functions. Lagrange Multipliers.

Integral Calculus

1. 2 dimensional Riemann integral: definition. Fubini's Theorem. Normal domains. Change of variables.
2. Convergence of improper integrals.
3. Triple integrals. Spherical and cylindrical coordinates.
4. Parametric integrals.
5. Elements of Lebesgue measure. Monotone convergence Theorem and Lebesgue Theorem.

Sequences and series

1. Function sequences: point and uniform convergence. Relation with continuity, integration and differentiation of the limit function.
2. Function series: Weierstrass' test.
3. Power series.
4. Taylor series.
5. Fourier series

Differential geometry of curves and surfaces.

1. Curves: speed and velocity. Length, arc length, curvature. Line integrals.
2. Parametric surfaces: tangent plane, oriented surfaces. Area. Surface integrals and fluxes.
3. Vector fields in R^2. Bounded regular domains and their boundaries. Green's Theorem. Vector fields in R^3. Curl and divergence. Gauss and Stokes' Theorems.
4. Potential Theory

Differential equations

1. Ordinary differential equations and IVPs (Initial Value Problems).
2. Systems of differential equations.
3. Existence and uniqueness for IVPs (local and global)
4. Linear systems: structure of solutions. Constant coefficient systems.

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

## TEACHERS AND EXAM BOARD

### Exam Board

FRANCESCA ASTENGO (President)

CESARE MOLINARI

FILIPPO DE MARI CASARETO DAL VERME (President Substitute)

## LESSONS

### LESSONS START

Classes will start according to the academic calendar.

### Class schedule

All class schedules are posted on the EasyAcademy portal.

## EXAMS

### EXAM DESCRIPTION

Written and oral exam

### ASSESSMENT METHODS

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

### Exam schedule

Date Time Location Type Notes
14/01/2022 09:00 GENOVA Scritto
07/02/2022 09:00 GENOVA Scritto
16/06/2022 09:00 GENOVA Scritto
05/07/2022 09:00 GENOVA Scritto
18/07/2022 09:00 GENOVA Scritto
15/09/2022 09:00 GENOVA Scritto

### FURTHER INFORMATION

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