Fisica | CALCULUS II

cod. 0512600002

CALCULUS II

	0512600002
	DEPARTMENT OF PHYSICS "E. R. CAIANIELLO"
	EQF6
	PHYSICS
	2022/2023

PERCORSO SHARED STUDY PATHWAY

	OBBLIGATORIO
	YEAR OF COURSE 2
	YEAR OF DIDACTIC SYSTEM 2017
	AUTUMN SEMESTER

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/05	9	72	LESSONS	BASIC COMPULSORY SUBJECTS

EXAMS

	Exam	Date	Session
	APPELLO DI APRILE 2023	05/04/2023 - 10:00	SESSIONE ORDINARIA

	Objectives
	AIM OF THE COURSE IS TO PROVIDE THE STUDENTS WITH SOME OF THE BASIC NOTIONS OF DIFFERENTIAL AND INTEGRAL CALCULUS IN MORE DIMENSIONS, OF CERTAIN KINDS OF ORDINARY DIFFERENTIAL EQUATIONS AND OF SEQUENCES AND SERIES OF FUNCTIONS. KNOWLEDGE AND UNDERSTANDING THE COURSE WILL FOCUS ON THE BASIC NOTIONS OF MATHEMATICAL ANALYSIS NECESSARY FOR A SECOND YEAR STUDENT OF THE DEGREE IN PHISICS. CERTAIN KINDS OF ORDINARY DIFFERENTIAL EQUATIONS, SEQUENCES AND SERIES OF FUNCTIONS, INFINITESIMAL, DIFFERENTIAL AND INTEGRAL CALCULUS, ELEMENTARY THEORY OF CURVES AND SURFACES AND DIFFERENTIAL FORMS WILL BE STUDIED. A SECOND POURPOSE OF THE COURSE IS TO GET THE STUDENT ACCUSTOMED WITH RIGOROUS ARGUMENTS AND WITH A CRITICAL USE OF THE TECHNIQUES THOUGHT. APPLYING KNOWLEDGE AND UNDERSTANDING THE STUDENT IS SUPPOSED TO LEARN THEORETICAL ASPECTS OF THE COURSE AND TO EXPLOIT THEM IN ORDER TO SOLVE EXERCISES AND PROBLEMS INCLUDING THOSE THAT ARE RELATED TO PRACTICAL APPLICATIONS OF THE SCIENCE. AT THE SAME TIME THE COURSE IS DESIGNED TO ENABLE STUDENTS TO INTERPRET THE MAIN CONCEPTS THOUGHT ANALYTICALLY, GRAPHICALLY AND VERBALLY, DEVELOP THEIR ABILITY TO THINK IN A CRITICAL MANNER, IMPROVE THEIR SKILLS IN ACQUIRING NEW UNDERSTANDING AND EXPERIENCE.

AIM OF THE COURSE IS TO PROVIDE THE STUDENTS WITH SOME OF THE BASIC NOTIONS OF DIFFERENTIAL AND INTEGRAL CALCULUS IN MORE DIMENSIONS, OF CERTAIN KINDS OF ORDINARY DIFFERENTIAL EQUATIONS AND OF SEQUENCES AND SERIES OF FUNCTIONS.

KNOWLEDGE AND UNDERSTANDING

THE COURSE WILL FOCUS ON THE BASIC NOTIONS OF MATHEMATICAL ANALYSIS NECESSARY FOR A SECOND YEAR STUDENT OF THE DEGREE IN PHISICS. CERTAIN KINDS OF ORDINARY DIFFERENTIAL EQUATIONS, SEQUENCES AND SERIES OF FUNCTIONS, INFINITESIMAL, DIFFERENTIAL AND INTEGRAL CALCULUS, ELEMENTARY THEORY OF CURVES AND SURFACES AND DIFFERENTIAL FORMS WILL BE STUDIED. A SECOND POURPOSE OF THE COURSE IS TO GET THE STUDENT ACCUSTOMED WITH RIGOROUS ARGUMENTS AND WITH A CRITICAL USE OF THE TECHNIQUES THOUGHT.

APPLYING KNOWLEDGE AND UNDERSTANDING

THE STUDENT IS SUPPOSED TO LEARN THEORETICAL ASPECTS OF THE COURSE AND TO EXPLOIT THEM IN ORDER TO SOLVE EXERCISES AND PROBLEMS INCLUDING THOSE THAT ARE RELATED TO PRACTICAL APPLICATIONS OF THE SCIENCE.

AT THE SAME TIME THE COURSE IS DESIGNED TO ENABLE STUDENTS TO INTERPRET THE MAIN CONCEPTS THOUGHT ANALYTICALLY, GRAPHICALLY AND VERBALLY, DEVELOP THEIR ABILITY TO THINK IN A CRITICAL MANNER, IMPROVE THEIR SKILLS IN ACQUIRING NEW UNDERSTANDING AND EXPERIENCE.

	Prerequisites
	KNOWLEDGE OF THE TOPICS TREATED IN “CALCULUS I”.

	Contents
	DIFFERENTIAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES (T:8H; P:2H) LIMITS AND CONTINUITY. DIRECTIONAL DERIVATIVES, DIFFERENTIAL, TANGENT PLANE. HIGHER ORDER PARTIAL DERIVATIVES. COMPOSITE FUNCTIONS. FREE EXTREMA. ODES (T:6H; P:6H) DIFFERENTIAL MODELS. FIRST ORDER, SEPARABLE, LINEAR, BERNOULLI, HOMOGENEOUS ODES. CAUCHY PROBLEM. N-TH ORDER ODES. INTEGRAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES (T:4H; P:5H) DOUBLE INTEGRALS. TRIPLE INTEGRALS. DERIVATIVE OF THE INTEGRAL. INFINITESIMAL CALCULUS FOR CURVES (T:8H; P:2H) VECTOR VALUED FUNCTIONS, LIMITS AND CONTINUITY. REGULAR CURVES, VECTORIAL DIFFERENTIAL CALCULUS. LENGHT OF A CURVE. LINE INTEGRALS. ELEMENTS OF DIFFRENTIAL GEOMETRY FOR CURVES. PHYSICAL APPLICATIONS. VECTOR FIELDS (T:10H; P:3H) FIELD LINES. GRADIENT, CURL AND DIVERGENCE. LINE INTEGRAL OF A VECTOR FIELD. WORK AND CIRCULATION. CONSERVATIVE FIELDS AND POTENTIALS. DIFFERENTIAL FORMS. GAUSS-GREEN FORMULA. AREA AND SURFACE INTEGRALS. DIVERGENCE AND CURL THEOREMS. IMPLICIT FUNCTIONS (T:5H; P:3H) DINI THEOREM. CONSTRAINED EXTREMA. LAGRANGE'S THEOREM. SERIES OF FUNCTIONS (T:7H; P:3H) SEQUENCES AND SERIES OF FUNCTIONS. POWER SERIES. TAYLOR SERIES.

Contents

DIFFERENTIAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES (T:8H; P:2H)
LIMITS AND CONTINUITY. DIRECTIONAL DERIVATIVES, DIFFERENTIAL, TANGENT PLANE. HIGHER ORDER PARTIAL DERIVATIVES. COMPOSITE FUNCTIONS. FREE EXTREMA.

ODES (T:6H; P:6H)
DIFFERENTIAL MODELS. FIRST ORDER, SEPARABLE, LINEAR, BERNOULLI, HOMOGENEOUS ODES. CAUCHY PROBLEM. N-TH ORDER ODES.

INTEGRAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES (T:4H; P:5H)
DOUBLE INTEGRALS. TRIPLE INTEGRALS. DERIVATIVE OF THE INTEGRAL.

INFINITESIMAL CALCULUS FOR CURVES (T:8H; P:2H)
VECTOR VALUED FUNCTIONS, LIMITS AND CONTINUITY. REGULAR CURVES, VECTORIAL DIFFERENTIAL CALCULUS. LENGHT OF A CURVE. LINE INTEGRALS. ELEMENTS OF DIFFRENTIAL GEOMETRY FOR CURVES. PHYSICAL APPLICATIONS.

VECTOR FIELDS (T:10H; P:3H)
FIELD LINES. GRADIENT, CURL AND DIVERGENCE. LINE INTEGRAL OF A VECTOR FIELD. WORK AND CIRCULATION. CONSERVATIVE FIELDS AND POTENTIALS. DIFFERENTIAL FORMS. GAUSS-GREEN FORMULA. AREA AND SURFACE INTEGRALS. DIVERGENCE AND CURL THEOREMS.

IMPLICIT FUNCTIONS (T:5H; P:3H)
DINI THEOREM. CONSTRAINED EXTREMA. LAGRANGE'S THEOREM.

SERIES OF FUNCTIONS (T:7H; P:3H)
SEQUENCES AND SERIES OF FUNCTIONS. POWER SERIES. TAYLOR SERIES.

	Teaching Methods
	THE COURSE IS STRUCTURED AS A COMBINATION OF THEORETICAL LECTURES (48 HOURS) AND EXERCISES SESSIONS (24 HOURS). THE THEORETICAL TOOLS WILL BE EXPLOITED ALSO IN VARIOUS PHYSICAL APPLICATIONS.

	Verification of learning
	THE FINAL EXAM CONSISTS OF A WRITTEN TEST AND AN ORAL EXAMINATION. THE EXAM IS DESIGNED TO EVALUATE AS A WHOLE: THE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED DURING THE COURSE, THE MASTERY OF THE MATHEMATICAL LANGUAGE, THE SKILL OF PROVING THEOREMS, THE SKILL OF SOLVING EXERCISES, THE ABILITY TO IDENTIFY AND APPLY THE BEST AND MORE EFFICIENT METHOD IN EXERCISES SOLVING, THE ABILITY TO USE THE ACQUIRED KNOWLEDGE. THE FINAL MARK, EXPRESSED IN THIRTIETHS (EVENTUALLY WITH LAUDE), DEPENDS ON THE GLOBAL VALUTATION OF THE STUDENT.

Verification of learning

THE FINAL EXAM CONSISTS OF A WRITTEN TEST AND AN ORAL EXAMINATION. THE EXAM IS DESIGNED TO EVALUATE AS A WHOLE: THE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED DURING THE COURSE, THE MASTERY OF THE MATHEMATICAL LANGUAGE, THE SKILL OF PROVING THEOREMS, THE SKILL OF SOLVING EXERCISES, THE ABILITY TO IDENTIFY AND APPLY THE BEST AND MORE EFFICIENT METHOD IN EXERCISES SOLVING, THE ABILITY TO USE THE ACQUIRED KNOWLEDGE.

THE FINAL MARK, EXPRESSED IN THIRTIETHS (EVENTUALLY WITH LAUDE), DEPENDS ON THE GLOBAL VALUTATION OF THE STUDENT.

	Texts
	"LEZIONI DI ANALISI MATEMATICA 2", N. FUSCO, P. MARCELLINI, C. SBORDONE, ZANICHELLI, 2020 "ESERCITAZIONI DI ANALISI MATEMATICA 2", P. MARCELLINI, C. SBORDONE, ZANICHELLI, 2017