Antonio VITOLO | FUCNTION THEORY

cod. 0512300023

FUCNTION THEORY

	0512300023
	DIPARTIMENTO DI MATEMATICA

	MATHEMATICS
	2015/2016

CURRICULUM MATEMATICA AD INDIRIZZO APPLICATIVO

	YEAR OF COURSE 3
	YEAR OF DIDACTIC SYSTEM 2010
	PRIMO SEMESTRE

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/05	6	48	LESSONS	SUPPLEMENTARY COMPULSORY SUBJECTS

PROFESSORS

	ANTONIO VITOLO T Curriculum
	PAOLA CAVALIERE Curriculum

	Objectives
	1. KNOWLEDGE AND UNDERSTANDING: THE COURSE IS FOCUSED ON THE FUNDAMENTAL CONCEPTS OF MEASURE AND INTEGRATION THEORY IN RN TOGETHER WITH CALCULUS AND APPLICATIONS. 2. APPLYING KNOWLEDGE AND UNDERSTANDING: THE STUDENT WOULD BE FAMILIAR WITH THE METHODS TYPICAL OF MEASURE AND INEGRATION THEORY, AND WITH THE REDUCTION AND APPROXIMATION TECHNIQUE OF INTEGRAL CALCULUS. 3. MAKING JUDGEMENTS: THE STUDENT WILL BE ABLE TO COMPARE ALTERNATIVE PROOFS, TO UNDERSTAND THE ROLE OF THE ASSUMPTIONS AND TO CHOOSE THE MOST APPROPRIATE WAY TO OBTAIN A RESULT, LEARNING CRITICALLY. 4. COMMUNICATION SKILLS: THE STUDENT WILL BE ABLE TO COMMUNICATE AND EXPRESS WITH RIGOROUS AND EFFECTIVE EVIDENCE THE RESULTS OF THE LEARNED TOPICS AND TO PRESENT THEM WITH THE SUITABLE MOTIVATIONS, EXAMPLES AND COUNTEREXAMPLES. 5. LEARNING SKILLS: THE STUDENT WILL GET A MATHEMATICAL BACKGROUND WHICH ALLOWS HIM TO READ MORE ADVANCED TEXTS OF MATHEMATICAL ANALYSIS AND IN GENERAL OF SCIENTIFIC SUBJECTS WHICH USE MATHEMATICAL TOOLS.

1. KNOWLEDGE AND UNDERSTANDING: THE COURSE IS FOCUSED ON THE FUNDAMENTAL CONCEPTS OF MEASURE AND INTEGRATION THEORY IN RN TOGETHER WITH CALCULUS AND APPLICATIONS.
2. APPLYING KNOWLEDGE AND UNDERSTANDING: THE STUDENT WOULD BE FAMILIAR WITH THE METHODS TYPICAL OF MEASURE AND INEGRATION THEORY, AND WITH THE REDUCTION AND APPROXIMATION TECHNIQUE OF INTEGRAL CALCULUS.
3. MAKING JUDGEMENTS: THE STUDENT WILL BE ABLE TO COMPARE ALTERNATIVE PROOFS, TO UNDERSTAND THE ROLE OF THE ASSUMPTIONS AND TO CHOOSE THE MOST APPROPRIATE WAY TO OBTAIN A RESULT, LEARNING CRITICALLY.
4. COMMUNICATION SKILLS: THE STUDENT WILL BE ABLE TO COMMUNICATE AND EXPRESS WITH RIGOROUS AND EFFECTIVE EVIDENCE THE RESULTS OF THE LEARNED TOPICS AND TO PRESENT THEM WITH THE SUITABLE MOTIVATIONS, EXAMPLES AND COUNTEREXAMPLES.
5. LEARNING SKILLS: THE STUDENT WILL GET A MATHEMATICAL BACKGROUND WHICH ALLOWS HIM TO READ MORE ADVANCED TEXTS OF MATHEMATICAL ANALYSIS AND IN GENERAL OF SCIENTIFIC SUBJECTS WHICH USE MATHEMATICAL TOOLS.

	Prerequisites
	BASIC CONCEPTS ON FUNCTIONS ONE AND SEVERAL COMPLEX VARIABLE.

	Contents
	METRIC SPACES: DISTANCE AND TOPOLOGY. BOUNDEDNESS AND COMPLETENESS. COMPACTNESS. BANACH SPACES. PEANO-JORDAN MEASURE AND RIEMANN INTEGRAL. LEBESGUE MEASURABLE FUNCTIONS. LEBESGUE MEASURE AND INTEGRATION. COMPARISON WITH PEANO-JORDAN MEASURE AND RIEMANN INTEGRAL. THEOREMS ON TAKING LIMITS UNDER THE INTEGRAL SIGN. INTEGRATION TERM BY TERM. DERIVATION UNDER INTEGRAL SIGN. MEASURE REDUCTION FORMULAS AND INTERCHANGING THE ORDER OF INTEGRATION: THEOREMS OF TONELLI AND FUBINI. APPLICATIONS.

Contents

METRIC SPACES: DISTANCE AND TOPOLOGY. BOUNDEDNESS AND COMPLETENESS. COMPACTNESS. BANACH SPACES.
PEANO-JORDAN MEASURE AND RIEMANN INTEGRAL. LEBESGUE MEASURABLE FUNCTIONS. LEBESGUE MEASURE AND INTEGRATION. COMPARISON WITH PEANO-JORDAN MEASURE AND RIEMANN INTEGRAL.
THEOREMS ON TAKING LIMITS UNDER THE INTEGRAL SIGN. INTEGRATION TERM BY TERM. DERIVATION UNDER INTEGRAL SIGN.
MEASURE REDUCTION FORMULAS AND INTERCHANGING THE ORDER OF INTEGRATION: THEOREMS OF TONELLI AND FUBINI.
APPLICATIONS.

	Teaching Methods
	FRONTAL LECTURES.

	Verification of learning
	ORAL EXAM.

	Texts
	1. E. GIUSTI, ANALISI MATEMATICA 2, BOLLATI BORINGHIERI. 2. LECTURE NOTES.

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Antonio VITOLO | FUCNTION THEORY