# Introductory Real Analysis

- Unit Coordinator: Rosella Sampalmieri, Antonio Esposito
- Programme: Double Degrees
- ECTS Credits: 9
- Semester: 1
- Year: 1
- Campus: University of L'Aquila
- Language: English
- Aims:
The purpose of this introductory course is to give a uniform background

of basic knowledge in mathematical analysis along with a common

mathematical language. Indeed, due to the international nature of the

course, Students come from different countries, with different study

curricula and mathematical training. Furthermore, over the years it has

been noted that most of the Students have a training in analysis that is

more applied than theoretical (calculus-type courses), which prevents the

full understanding of the more theoretical topics addressed in subsequent

courses. Therefore numerous fundamental topics, traditionally present in

mathematical analysis courses will be resumed and deepened. The

Student who has successfully attended the course should be able to

easily understand the presentation of more advanced and sophisticated

mathematical arguments and proofs and be able to apply the theory to

the proposed problems.

On successful completion of this course, the Student should

1-have a clear understanding of the basic notions and concepts of the

real analysis

2-know and know how to apply the theoretical results learned to the

proposed problems, using appropriate mathematical language;

3-understand and consciously reproduce a demonstration;

4- be able to read and understand scientific texts on related topics. - Content:
THE COURSE INCLUDES THE FOLLOWING TOPICS:

The set of real numbers and related axioms. Inferior and supreme.

The topology of real numbers: intervals, half lines, open sets, closed sets.

The topology of the Euclidean space R^n: spherical neighborhoods, open,

closed, connected, convex sets. Compact sets in Euclidean space.

Numerical sequences: limits of sequences and related theorems.

Real functions of real variable: limits of functions, continuous functions,

continuity and compactness, continuity and connection, discontinuity, monotonic functions.

Derivation in IR: the derivative of a real function, rules of derivation,

Taylor's theorem, study of the graph of a function.

The Riemann integral in IR: definition and existence of the integral,

properties of the integral, integration techniques.

Functions of several variables: limits of functions of several variables,

continuity, differentiability and differentiability, higher order derivatives,

necessary and sufficient conditions for the existence of local and global

extrema, constrained extrema, implicit function theorem (various

versions), functions vectors, transformations of coordinates.

Multiple integrals: double and triple integrals, reduction formulas, change

of variables in multiple integrals.

Curves and surfaces: curves, regular curves, length of a curve, rectifiable

curves, curvilinear integrals.

Surfaces: regular surfaces, regular surface area, surface integrals.

Vector fields: conservative vector fields, Green's formulas, Stokes's

theorem, Gauss's theorem. - Pre-requisites:
The prerequisites to this course are the following:

Propositional logic. Propositional calculus.

Sets, set operations, relations, functions. Cardinality of sets, countable sets, uncountable sets. Elementary number sets. Integers and rationals. Induction principle.

More on functions: injective and surjective functions, invertible functions, image and pre-image.

The set of real numbers. Separation axiom, Dedekind cuts. Infimum and supremum. Archimedean property. Complex numbers: cartesian and trigonetric form, basic properties, powers, complex roots, fundamental theorem of algebra.

Introduction to functions of real numbers. Elementary functions: exponential and logarithmic function, trigonometric functions, irrational functions. Monotone functions.

The topology of real numbers: intervals, half lines, open sets, closed sets. The topology of the Euclidean space Rn: balls, open and closed sets. Compact sets in the Euclidean space.

The prerequisites to this course can also be found in the section ''Real Analisys: Foundations ''of the Pre-Master's Foundation Programme (PMFP) in Applied Mathematics whose purpose is homogenising the competencies portfolios of prospective students of the two Master's Programmes in Mathematical Modelling and Mathematical Engineering, in particular in Mathematical Analisys. - Reading list:
-An Introduction to Real Analysis

John K. Hunter,

Department of Mathematics, University of California at Davis- Calculus of several variables,

Serge Lang,

Yale University,

Addison-Wesley Publishing Company-Principles of mathematical analysis -W.Rudin - McGraw-Hill 1976

- Additional info:
**Teaching methods**In addition to the classic lectures, the Students will be involved in informal discussions on the subjects presented, in order to stimulate a deeper understanding. Students will be asked to propose strategies to solve problems and exercises during the lessons. Moreover home-works will be assigned regularly and commented during the classes.

A support tutoring activity is also organized for the students.**Assessment methods**As far as summative assessment is concerned, there will be a written test followed by an oral examination.

There will also be 2/3 optional intermediate written tests, which aim to cover all the different parts of the program, whose passing will be equivalent to passing the written test.

The written tests will focus on both calculus and more theoretical exercises. A written test with a grade of 18/30 is considered sufficient. The oral tests will focus on definitions, statements of theorems and examples. The proof of 10 theorems presented in class will also be required, at the student's choice but within the whole program. The oral exam can add a maximum of 4 points to the mark of the written exam or decrease or cancel this mark if negative.