# Fisica | MATHEMATICAL ANALYSIS I

## Fisica MATHEMATICAL ANALYSIS I

 0512600001 DIPARTIMENTO DI FISICA "E.R. CAIANIELLO" PHYSICS 2015/2016

 OBBLIGATORIO YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2010 ANNUALE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/05 12 96 LESSONS BASIC COMPULSORY SUBJECTS
 PAOLA CAVALIERE T
Objectives
THE COURSE IS ONE-YEAR COURSE IN MATHEMATICAL ANALYSIS. IT PROVIDES THE STUDENT COMPLETE THE TRANSITION FROM PURELY MANIPULATIVE TO RIGOROUS MATHEMATICS, INTRODUCING THE MAIN TOPICS FROM BASIC SET THEORY TO THE THEORY OF LIMIT, CONTINUITY, DIFFERENTIABILITY AND INTEGRABILITY OF REAL ONE-VARIABLE FUNCTIONS.
THE STUDENT ARE EXPECTED TO GAIN A GOOD UNDERSTANDING OF THEORETICAL ARGUMENTS AND TO BE ABLE TO APPLY THEM IN CONCRETE PROBLEMS. UPON COMPLETION OF THIS COURSE, THE STUDENT SHOULD:
-DEMONSTRATE AN UNDERSTANDING OF BASIC MATHEMATICAL SKILLS USED IN CALCULUS;
-APPLY THE SHOWN PRINCIPLES OF CALCULUS TO SOLVE APPLIED PROBLEMS RELATED IN MATHEMATICS AS WELL AS IN OTHER DISCIPLINES;
-UNDERSTAND AND COMMUNICATE CLEARLY AND EFFECTIVELY THE PRINCIPLES OF CALCULUS, USING PROPER VOCABULARY AND NOMENCLATURE;
-DEVELOP EFFECTIVE STUDY SKILLS IN ORDER TO MASTER COURSE CONTENT AND OBJECTIVES; THINK CRITICALLY IN READING AND WRITING MATHEMATICS, DETERMINING IN PARTICULAR IF CONCLUSIONS OR SOLUTIONS ARE REASONABLE.
Prerequisites
THE PREREQUISITES ARE HIGH SCHOOL ALGEBRA AND TRIGONOMETRY.
Contents
FOUNDATIONS - LOGIC, SET NOTATIONS. RELATIONS. FUNCTIONS. EQUIVALENCE AND ORDER RELATIONS.

THE REAL NUMBER SYSTEM - THE AXIOMS OF THE REAL NUMBER FIELD. THE SUBSETS N OF NATURAL NUMBERS (AND THE PRINCIPLE OF INDUCTION), OF THE INTEGERS Z AND OF THE RATIONAL NUMBERS Q. INTERVALS OF R. GEOMETRIC REPRESENTATION OF R AND OF R2.

REAL FUNCTIONS – BOUNDS AND GRAPHIC OF ONE-VARIABLE REAL FUNCTION. MONOTONE FUNCTIONS. EVEN, ODD AND PERIODIC FUNCTIONS. REAL SEQUENCES. MONOTONE SEQUENCES. THE NEPER NUMBER E.

ELEMENTARY FUNCTIONS – LINEAR FUNCTIONS. ABSOLUTE VALUE FUNCTION. N-POWER AND N-SQUARE FUNCTIONS. EXPONENTIAL AND LOGARITHMIC FUNCTIONS. REAL POWER FUNCTIONS. TRIGONOMETRIC FUNCTIONS AND THEIR LOCALLY INVERSES.

THE COMPLEX FIELD - DEFINITION, PROPERTIES AND GEOMETRICAL REPRESENTATION. DE MOIVRE FORMULAS. ROOTS OF A COMPLEX NUMBER.

LIMITS OF FUNCTIONS – EXTENDED REAL NUMBERS AND ITS TOPOLOGY. CONCEPT OF LIMIT AND ITS PROPERTIES: UNIQUENESS, ALGEBRA OF LIMITS, INDETERMINATE FORMS, COMPARISON THEOREMS. MONOTONE FUNCTIONS AND THEIR LIMITS. COMPOSITE FUNCTIONS AND THEIR LIMITS. LIMITS OF SEQUENCES.

CONTINUOUS FUNCTIONS. CONCEPT OF CONTINUITY AT A POINT. DISCONTINUITY POINTS. THE WEIERSTRASS THEOREM AND THE INTERMEDIATE-VALUE THEOREM. UNIFORMLY CONTINUOUS FUNCTIONS.

THE NOTION OF DERIVATIVE – DEFINITIONS, EXAMPLES AND GEOMETRICAL MEANING. DERIVATION AND ALGEBRAIC OPERATIONS. DERIVATIONS AND COMPOSITE FUNCTIONS. DERIVATION OF ELEMENTARY FUNCTIONS.

APPLICATION OF DIFFERENTIAL CALCULUS. QUALITATIVE STUDY OF FUNCTIONS – LOCALLY MAXIMUM AND MINIMUM POINTS. THE FERMAT THEOREM. THE THEOREMS OF ROLLE, LAGRANGE AND CAUCHY: RELATIONSHIPS AND CONSEQUENCES. CONVEX AND CONCAVE FUNCTIONS: DEFINITIONS AND PROPERTIES. DE L’HOPITAL THEOREMS. QUALITATIVE STUDY OF THE GRAPH OF A FUNCTION.

RIEMANN INTEGRAL – DEFINITIONS AND EXAMPLES. GEOMETRICAL MEANING OF DEFINITE INTEGRAL. MAIN PROPERTIES OF FINITE INTEGRATION. INTEGRABILITY OF CONTINUOUS FUNCTIONS AS WELL AS OF MONOTONE FUNCTIONS. MEAN VALUE THEOREMS.

INDEFINITE INTEGRAL – PRIMITIVE FUNCTIONS. THE FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS. INDEFINITE INTEGRAL AND ITS PROPERTIES. INTEGRATION METHODS: DECOMPOSITION IN SUMS, INTEGRATION BY PARTS AND SUBSTITUTION.

NUMERICAL SERIES– DEFINITIONS, EXAMPLES AND PRELIMINARY RESULTS. SERIES OF POSITIVE AND NEGATIVE TERMS. GEOMETRIC AND HARMONIC SERIES. TESTS FOR CONVERGENCE AND DIVERGENCE. ALTERNATING SERIES AND LEIBNIZ THEOREM. ABSOLUTELY CONVERGENT SERIES.
Teaching Methods
THE COURSE IS STRUCTURED AS A COMBINATION OF LECTURES AND PRATICAL SESSIONS. THE FEEDBACK FROM STUDENTS WILL ALWAYS BE HIGHLY APPRECIATED.
Verification of learning
ABILITY IN PROBLEM SOLVING AND IT IS A TWO HOUR-LONG WRITTEN EXAM. DURING IT, STUDENT WILL BE NOT ALLOWED TO USE ANY TYPE OF BOOK, NOTE OR CALCULATOR. THE FINAL ORAL EXAMINATION IS AIMED TO EVALUATE THE COMPREHENSIVE KNOWLEDGE OF THE MATTER. ONLY STUDENTS REACHING A POSITIVE GRADE AT WRITTEN TEST ARE ALLOWED TO TAKE THE ORAL EXAMINATION. EXAM COVERS ALL COURSE MATERIAL.
Texts
W. RUDIN: PRINCIPLES OF MATHEMATICAL ANALYSIS, HARDCOVER 1986.