Ingegneria Edile-Architettura | ANALISI MATEMATICA II

cod. 0660100013

ANALISI MATEMATICA II

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

PERCORSO COMUNE

	OBBLIGATORIO
	YEAR OF COURSE 2
	YEAR OF DIDACTIC SYSTEM 2012
	PRIMO SEMESTRE

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/05	6	60	LESSONS	BASIC COMPULSORY SUBJECTS

	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

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.

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