Fisica | CALCULUS II

cod. 0512600002

CALCULUS II

	0512600002
	DIPARTIMENTO DI FISICA "E.R. CAIANIELLO"
	EQF6
	PHYSICS
	2020/2021

PERCORSO SHARED STUDY PATHWAY

	OBBLIGATORIO
	YEAR OF COURSE 2
	YEAR OF DIDACTIC SYSTEM 2017
	PRIMO SEMESTRE

TEACHING UNITS

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

	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
	PREREQUISITE IS THE COURSE OF “ANALISI MATEMATICA I”.

	Contents
	ORDINARY DIFFERENTIAL EQUATIONS - ODE -(LECTURE HOURS 8; EXERCISES HOURS 4) DIFFERENTIAL MODELS. FIRST ORDER ODE’S: SEPARABLE ODE’S, LINEAR ODE’S. SECOND ORDER ODE’S: HOMOGENEOUS WITH CONSTANT COEFFICIENTS, NON HOMOGENEOUS. CAUCHY PROBLEM. N-TH ORDER ODE’S. INFINITESIMAL CALCULUS FOR CURVES (LECTURE HOURS 7; EXERCISES HOURS 3) 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. DIFFERENTIAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES (LECTURE HOURS 7; EXERCISES HOURS 3) GRAPHS AND LEVEL SETS. LIMITS AND CONTINUITY. PARTIAL DERIVATIVES TANGENT PLANE, DIFFRENTIAL. SECOND ORDER PARTIAL DERIVATIVES. OPTIMIZATION. FREE EXTREMA. INTEGRAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES (LECTURE HOURS 6; EXERCISES HOURS 3) DOUBLE INTEGRALS. TRIPLE INTEGRALS. DERIVATIVE OF THE INTEGRAL. VECTOR FIELDS (LECTURE HOURS 9; EXERCISES HOURS 4) 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 (LECTURE HOURS 5; EXERCISES HOURS 3) DINI THEOREM FOR EQUATIONS. DINI THEOREM FOR SYSTEMS. IMPLICIT FUNCTIONS. CONSTRAINED EXTREMA. LAGRANGE'S THEOREM. POWER SERIES (LECTURE HOURS 6; EXERCISES HOURS 4) SEQUENCES AND SERIES OF FUNCTIONS. POWER SERIES. TAYLOR SERIES

Contents

ORDINARY DIFFERENTIAL EQUATIONS - ODE -(LECTURE HOURS 8; EXERCISES HOURS 4)
DIFFERENTIAL MODELS. FIRST ORDER ODE’S: SEPARABLE ODE’S, LINEAR ODE’S. SECOND ORDER ODE’S: HOMOGENEOUS WITH CONSTANT COEFFICIENTS, NON HOMOGENEOUS. CAUCHY PROBLEM. N-TH ORDER ODE’S.

INFINITESIMAL CALCULUS FOR CURVES (LECTURE HOURS 7; EXERCISES HOURS 3)
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.

DIFFERENTIAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES (LECTURE HOURS 7; EXERCISES HOURS 3)
GRAPHS AND LEVEL SETS. LIMITS AND CONTINUITY. PARTIAL DERIVATIVES TANGENT PLANE, DIFFRENTIAL. SECOND ORDER PARTIAL DERIVATIVES. OPTIMIZATION. FREE EXTREMA.

INTEGRAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES (LECTURE HOURS 6; EXERCISES HOURS 3)
DOUBLE INTEGRALS. TRIPLE INTEGRALS. DERIVATIVE OF THE INTEGRAL.

VECTOR FIELDS (LECTURE HOURS 9; EXERCISES HOURS 4)
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 (LECTURE HOURS 5; EXERCISES HOURS 3)
DINI THEOREM FOR EQUATIONS.
DINI THEOREM FOR SYSTEMS.
IMPLICIT FUNCTIONS.
CONSTRAINED EXTREMA. LAGRANGE'S THEOREM.

POWER SERIES (LECTURE HOURS 6; EXERCISES HOURS 4)
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 EXAM CONSISTS OF TWO PARTS. THE FIRST ONE IS A WRITTEN EXAMINATION, TO BE COMPLETED WITHIN TWO HOURS, ON THE TOPICS STUDIED DURING THE EXERCISES SESSIONS. SCORES ARE ASSIGNED TO THE WRITTEN TEST: 'TOP MARK', 'GOOD', 'FAIR', 'PASSING', 'POOR', 'DISSUADED'. THE MARK 'DISSUADED' MEANS THAT THE STUDENT IS ADVISED AGAINST TO TAKE THE ORAL EXAMINATION, WITH THE OTHER MARKS THE ORAL EXAMINATION CAN BE TAKEN. THE ORAL EXAMINATION IS INTENDED TO TEST THE ABILITY TO PRESENT AND CONNECT THEOREMS AND PROBLEMS RELATED TO THE THEORETICAL PART OF COURSE AND EVENTUALLY A DISCUSSION OF THE WRITTEN TEST. THE FINAL EVALUATION IS OBTAINED AS AN OVERALL EVALUATION OF THE TWO PARTS OF THE EXAMINATION. THE MINIMUM SCORE (18) IS ATTRIBUTED TO STUDENTS SHOWING LIMITED KNOWLEDGE BOTH OF THE APPLIED AND THEORETICAL ASPECTS OF THE COURSE. THE MAXIMUM GRADE (30) IS ATTRIBUTED TO STUDENTS SHOWING DEEP AND CRITICAL KNOWLEDGE BOTH OF THE APPLIED AND THEORETICAL ASPECTS OF THE COURSE. THE COMENDATION (LODE) IS ATTRIBUTED TO STUDENTS SHOWING RIGOROUS KNOWLEDGE OF THE THEORETICAL ASPECTS OF THE PROGRAM AND AN AUTONOMOUS AND CRITICAL APPROACH TO EXERCISES THAT PRESENT FURTHER DIFFICOULTIES COMPARED TO THOSE STUDIED DURING THE COURSE.

Verification of learning

THE EXAM CONSISTS OF TWO PARTS. THE FIRST ONE IS A WRITTEN EXAMINATION, TO BE COMPLETED WITHIN TWO HOURS, ON THE TOPICS STUDIED DURING THE EXERCISES SESSIONS.
SCORES ARE ASSIGNED TO THE WRITTEN TEST: 'TOP MARK', 'GOOD', 'FAIR', 'PASSING', 'POOR', 'DISSUADED'. THE MARK 'DISSUADED' MEANS THAT THE STUDENT IS ADVISED AGAINST TO TAKE THE ORAL EXAMINATION, WITH THE OTHER MARKS THE ORAL EXAMINATION CAN BE TAKEN.

THE ORAL EXAMINATION IS INTENDED TO TEST THE ABILITY TO PRESENT AND CONNECT THEOREMS AND PROBLEMS RELATED TO THE THEORETICAL PART OF COURSE AND EVENTUALLY A DISCUSSION OF THE WRITTEN TEST.
THE FINAL EVALUATION IS OBTAINED AS AN OVERALL EVALUATION OF THE TWO PARTS OF THE EXAMINATION.

THE MINIMUM SCORE (18) IS ATTRIBUTED TO STUDENTS SHOWING LIMITED KNOWLEDGE BOTH OF THE APPLIED AND THEORETICAL ASPECTS OF THE COURSE.

THE MAXIMUM GRADE (30) IS ATTRIBUTED TO STUDENTS SHOWING DEEP AND CRITICAL KNOWLEDGE BOTH OF THE APPLIED AND THEORETICAL ASPECTS OF THE COURSE.

THE COMENDATION (LODE) IS ATTRIBUTED TO STUDENTS SHOWING RIGOROUS KNOWLEDGE OF THE THEORETICAL ASPECTS OF THE PROGRAM AND AN AUTONOMOUS AND CRITICAL APPROACH TO EXERCISES THAT PRESENT FURTHER DIFFICOULTIES COMPARED TO THOSE STUDIED DURING THE COURSE.

	Texts
	"ANALISI MATEMATICA 2", M. BRAMANTI - C. PAGANI - S. SALSA, ZANICHELLI, 2019 "ESERCIZI DI ANALISI MATEMATICA 2", S. SALSA - A. SQUELLATI, ZANICHELLI, 2019 OTHER TEXTBOOKS: "CORSO DI MATEMATICA SUPERIORE II", V.I. SMIRNOV - EDITORI RIUNITI "ESERCITAZIONI DI ANALISI MATEMATICA DUE (PRIMA PARTE)", P. MARCELLINI - C. SBORDONE, ZANICHELLI, 2018 "ESERCITAZIONI DI ANALISI MATEMATICA DUE (SECONDA PARTE)", P. MARCELLINI - C. SBORDONE, ZANICHELLI, 2020

Texts

"ANALISI MATEMATICA 2", M. BRAMANTI - C. PAGANI - S. SALSA, ZANICHELLI, 2019

"ESERCIZI DI ANALISI MATEMATICA 2", S. SALSA - A. SQUELLATI, ZANICHELLI, 2019

OTHER TEXTBOOKS:

"CORSO DI MATEMATICA SUPERIORE II", V.I. SMIRNOV - EDITORI RIUNITI

"ESERCITAZIONI DI ANALISI MATEMATICA DUE (PRIMA PARTE)", P. MARCELLINI - C. SBORDONE, ZANICHELLI, 2018

"ESERCITAZIONI DI ANALISI MATEMATICA DUE (SECONDA PARTE)", P. MARCELLINI - C. SBORDONE, ZANICHELLI, 2020

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