# Ingegneria Edile-Architettura | ANALISI MATEMATICA II

## Ingegneria Edile-Architettura ANALISI MATEMATICA II

 0660100013 DIPARTIMENTO DI INGEGNERIA CIVILE CORSO DI LAUREA MAGISTRALE A CICLO UNICO DI 5 ANNI BUILDING ENGINEERING - ARCHITECTURE 2014/2015

 OBBLIGATORIO YEAR OF COURSE 2 YEAR OF DIDACTIC SYSTEM 2012 PRIMO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/05 6 60 LESSONS BASIC COMPULSORY SUBJECTS
 MARIA EMILIA AMENDOLA T
Objectives
THE AIM OF THIS COURSE IS TO ACHIEVE SOME BASIC TOOLS SUCH AS: SEQUENCES AND SERIES OF FUNCTIONS, FUNCTIONS OF SEVERAL VARIABLES, DIFFERENTIAL EQUATIONS, CURVES AND LINE INTEGRALS, DIFFERENTIAL FORMS, INTEGRAL CALCULUS OF FUNCTIONS OF SEVERAL VARIABLES, SURFACES AND SURFACE INTEGRALS.
Prerequisites
NUMBER SEQUENCES AND SERIES: DEFINITION, CONVERGENCE CRITERIA, CAUCHY SEQUENCES.
REAL VALUED FUNCTIONS OF REAL VARIABLES: DEFINITION, DOMAIN, COMPOSITE FUNCTIONS, INVERTIBLE FUNCTIONS, ELEMENTARY FUNCTIONS.
LIMIT OF A FUNCTION:DEFINITION, OPERATIONS WITH LIMITS, NOTABLE SPECIAL LIMITS, INDETERMINATE FORMS.
CONTINUOUS FUNCTIONS: DEFINITION, CONTINUITY AND DISCONTINUITY, WEIERSTRASS THEOREM, ZEROS THEOREM, BOLZANO THEOREM. UNIFORM CONTINUITY.
DERIVATIVE OF A FUNCTION: DEFINITION, LEFT AND RIGHT DERIVATIVES, GEOMETRIC MEANING. TANGENT LINE TO THE GRAPH OF A FUNCTION, DIFFERENTIABILITY AND CONTINUITY. DERIVATION RULES. DERIVATIVES OF ELEMENTARY FUNCTION, DERIVATIVES OF COMPOSITE AND INVERSE FUNCTIONS. HIGHER ORDER DERIVATIVES. DIFFERENTIAL OF A FUNCTION AND ITS GEOMETRIC MEANING.
GRAPH OF A FUNCTION.
INTEGRAL OF A FUNCTION: RULES AND METHODS OF INTEGRATION, INTEGRATION OF RATIONAL FUNCTIONS
Contents
SEQUENCES AND SERIES OF FUNCTIONS:
SEQUENCES: DEFINITION, POINT WISE AND UNIFORM CONVERGENCE, EXAMPLES AND COUNTEREXAMPLES, UNIFORM CONVERGENCE THEOREM, CAUCHY'S CRITERION FOR UNIFORM CONVERGENCE, TERM BY TERM INTEGRATION AND DIFFERENTIATION OF SEQUENCES; SERIES OF FUNCTIONS: DEFINITION, POINTWISE, UNIFORM AND TOTAL CONVERGENCES, CAUCHY'S CRITERION, TERM BY TERM INTEGRATION AND DIFFERENTIATION THEOREM FOR SERIES; POWER SERIES: DEFINITION, CONVERGENCE SET AND RADIUS OF CONVERGENCE, CAUCHY -HADAMARD'S THEOREM, D'ALEMBERT'S THEOREM, RADIUS OF CONVERGENCE WHEN INTEGRATING AND DIFFERENTIATING POWER SERIES, TOOL AND UNIFORM CONVERGENCES, TERM BY TERM INTEGRATION AND DERIVATION, EXAMPLES AND COUNTEREXAMPLES.
SEVERAL VARIABLES FUNCTIONS: DEFINITION, LIMITS AND CONTINUITY, WEIERSTRASS THEOREM, INTERMEDIATE VALUE THEOREM, PARTIAL DERIVATIVES, SCHWARTZ' THEOREM, DIFFERENTIABILITY, THEOREM (SUFFICIENT CONDITION ON DIFFERENTIABILITY), COMPOSITE FUNCTIONS, THEOREM (DERIVATIVES OF COMPOSITE FUNCTIONS), DIFFERENTIABILITY OF COMPOSITE FUNCTIONS, INTEGRAL FUNCTIONS, TAYLOR'S FORMULA, HIGHER ORDER DIFFERENTIALS, MAXIMA AND MINIMA POINTS IN A NEIGHBORHOOD, VECTOR VALUED FUNCTIONS, GRADIENT, DIVERGENCE AND CURL, DIRECTIONAL DERIVATIVES.
DIFFERENTIAL EQUATIONS: DEFINITION, PARTICULAR INTEGRAL AND GENERAL SOLUTION, CAUCHY PROBLEM, EXISTENCE AND UNIQUENESS THEOREMS, FIRST ORDER DIFFERENTIAL EQUATIONS, DIFFERENTIAL EQUATIONS OF ORDER N, ALGEBRAIC STRUCTURE OF THE SET OF SOLUTIONS, CONSTANT COEFFICIENT LINEAR DIFFERENTIAL EQUATIONS, RESOLUTION METHODS.
CURVES AND LINE INTEGRALS: DEFINITION, SMOOTH CURVES, ARC LENGTH OF A CURVE, THEOREM ON RECTIFIABLE CURVES, LINE INTEGRAL OF A FUNCTION.
DIFFERENTIAL FORMS: DEFINITION, VECTOR FIELD, LINE INTEGRAL OF A LINEAR DIFFERENTIAL FORM, CLOSED AND EXACT DIFFERENTIAL FORMS, EXACTNESS CRITERION, RELATION BETWEEN CLOSEDNESS AND EXACTNESS, CLOSED FORM DEFINED IN A RECTANGLE OR IN STAR SHAPED OPEN SETS, CLOSED FORMS IN SIMPLY CONNECTED OPEN SETS.
INTEGRATION OF FUNCTIONS OF SEVERAL VARIABLES: DEFINITION, SURFACE AREA AND VOLUME, FIRST PAPPUS-GULDIN THEOREM, NORMAL DOMAINS, ITERATED INTEGRAL FORMULAE, GAUSS-GREEN THEOREM, CHANGING OF VARIABLES.
SURFACES AND SURFACE INTEGRALS: DEFINITION, EXAMPLES, PROPERTIES, PARAMETRIC REPRESENTATION, SURFACE AREA AND SURFACE INTEGRALS, SURFACES WITH BOUNDARY, SECOND PAPPUS-GULDIN THEOREM, DIVERGENCE THEOREM, STOKES' THEOREM.
Teaching Methods
THERE ARE FRONTAL LECTURES, DEVOTED TO SHOW THE CONTENTS OF THE COURSE, AND EXERCISES SOLVED INVOLVING THE STUDENTS AS A REALLY ACTIVE PART, USEFUL TO GIVE THE FUNDAMENTAL TOOLS FOR SOLVING PROBLEMS RELATED TO THE SUBJECT
Verification of learning
THE FIXED OBJECTIVES WILL BE TEST BY A WRITTEN PROOF AND AN ORAL INTERVIEW. THE FINAL MARK WILL BE EXPRESS IN THIRTIETHS.
Texts
SEE THE ITALIAN VERSION FOR SUGGESTED READINGS IN ITALIAN.
BETA VERSION Data source ESSE3 [Ultima Sincronizzazione: 2016-09-30]
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