Chiara ESPOSITO | GEOMETRY IV

cod. 0512300013

GEOMETRY IV

	0512300013
	DEPARTMENT OF MATHEMATICS
	EQF6
	MATHEMATICS
	2024/2025

PERCORSO SHARED STUDY PATHWAY

	OBBLIGATORIO
	YEAR OF COURSE 3
	YEAR OF DIDACTIC SYSTEM 2018
	AUTUMN SEMESTER

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/03	6	48	LESSONS	COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS

EXAMS

Exam	Date	Session
GEOMETRIA IV	20/01/2025 - 11:00	SESSIONE ORDINARIA
GEOMETRIA IV	20/01/2025 - 11:00	SESSIONE DI RECUPERO
GEOMETRIA IV	07/02/2025 - 11:00	SESSIONE ORDINARIA
GEOMETRIA IV	07/02/2025 - 11:00	SESSIONE DI RECUPERO
GEOMETRIA IV	28/02/2025 - 11:00	SESSIONE ORDINARIA
GEOMETRIA IV	28/02/2025 - 11:00	SESSIONE DI RECUPERO

	Objectives
	COURSE AIMS TO PROVIDE STUDENTS WITH BOTH A SOLID THEORETICAL FOUNDATION IN DIFFERENTIAL GEOMETRY AND THE ABILITY TO APPLY THESE KNOWLEDGES IN PRACTICAL CONTEXTS THROUGH EXERCISES AND PROBLEMS. KNOWLEDGE AND UNDERSTANDING: STUDENTS WILL ACQUIRE A THOROUGH UNDERSTANDING OF THE FUNDAMENTAL CONCEPTS OF DIFFERENTIAL GEOMETRY, INCLUDING CONCEPTS SUCH AS SUBVARIETIES IN EUCLIDEAN SPACES, CURVES, AND SURFACES IN R3. STUDENTS WILL BE ABLE TO UNDERSTAND HOW TO APPLY THE NOTIONS OF ANALYSIS AND LINEAR ALGEBRA PREVIOUSLY LEARNED TO STUDY THE INFINITESIMAL PROPERTIES OF GEOMETRIC OBJECTS. THIS WILL ENABLE THEM TO RECOGNIZE HOW THE INFINITESIMAL PROPERTIES REFLECT IN THE GLOBAL AND TOPOLOGICAL PROPERTIES OF THE GEOMETRIC OBJECTS. ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING: STUDENTS WILL BE ABLE TO CONCRETELY APPLY THE THEORETICAL NOTIONS LEARNED. THIS MAY INCLUDE THE ABILITY TO SOLVE PRACTICAL PROBLEMS INVOLVING CONCEPTS OF DIFFERENTIAL GEOMETRY. PARTICULAR ATTENTION WILL BE GIVEN TO EXERCISES, WHICH CONSTITUTE A SIGNIFICANT PART OF THE COURSE, ALLOWING STUDENTS TO PUT THEIR KNOWLEDGES INTO PRACTICE THROUGH THE PRACTICAL APPLICATION OF THEORETICAL CONCEPTS. TRANSVERSAL OBJECTIVES, SUCH AS JUDGMENT AUTONOMY AND COMMUNICATIVE SKILLS, CAN BE INTEGRAL PART OF THE COURSE AIMS. HERE'S HOW THEY CAN BE FORMULATED AS SPECIFIC COURSE OBJECTIVES: DEVELOP JUDGMENT AUTONOMY: AIM: TO PROMOTE STUDENTS' JUDGMENT AUTONOMY SO THAT THEY ARE ABLE TO CRITICALLY ANALYZE CONCEPTS AND THEORIES OF DIFFERENTIAL GEOMETRY. METHODS: ENCOURAGE STUDENTS TO EVALUATE THE EVIDENCE AND ARGUMENTS PRESENTED DURING THE COURSE AND TO DEVELOP THE ABILITY TO MAKE DECISIONS IN MATHEMATICAL CONTEXTS. ENHANCE COMMUNICATIVE SKILLS: AIM: TO IMPROVE STUDENTS' COMMUNICATIVE SKILLS SO THAT THEY CAN CLEARLY EXPRESS THEIR OWN IDEAS AND ARGUMENTS, BOTH ORALLY AND IN WRITING, USING APPROPRIATE TECHNICAL LANGUAGE. METHODS: PROVIDE OPPORTUNITIES FOR STUDENTS TO PARTICIPATE ACTIVELY DURING CLASSES, ENCOURAGING THEM TO EXPRESS THEIR IDEAS, ASK QUESTIONS, AND ENGAGE IN DISCUSSIONS. ALSO, ENCOURAGE WRITING DEMONSTRATIONS THAT REQUIRE A CLEAR AND WELL-STRUCTURED COMMUNICATION OF MATHEMATICAL IDEAS. PROMOTE AUTONOMOUS LEARNING: AIM: TO FAVOR STUDENTS' AUTONOMOUS LEARNING SO THAT THEY CAN DEVELOP EFFECTIVE STUDY STRATEGIES, DEEPEN THEIR UNDERSTANDING OF CONCEPTS, AND APPLY THE ACQUIRED KNOWLEDGES IN A CREATIVE WAY. METHODS: PROVIDE RESOURCES AND TEACHING MATERIALS THAT ALLOW STUDENTS TO DEEPEN THEIR UNDERSTANDING OF CONCEPTS BEYOND FRONT CLASSES, SUCH AS RECOMMENDED READINGS, EXTRA EXERCISES, AND ADVANCED OPTIONAL TOPICS. ALSO, ENCOURAGE ACTIVE PARTICIPATION OF STUDENTS THROUGH CRITICAL ANALYSIS OF MATHEMATICAL PROBLEMS AND INDEPENDENT RESEARCH OF SOLUTIONS. FORMULATING THE COURSE AIMS IN A WAY THAT INCLUDES THESE TRANSVERSAL ASPECTS CONTRIBUTES TO ENSURING A COMPREHENSIVE AND MEANINGFUL LEARNING OF STUDENTS.

COURSE AIMS TO PROVIDE STUDENTS WITH BOTH A SOLID THEORETICAL FOUNDATION IN DIFFERENTIAL GEOMETRY AND THE ABILITY TO APPLY THESE KNOWLEDGES IN PRACTICAL CONTEXTS THROUGH EXERCISES AND PROBLEMS.

KNOWLEDGE AND UNDERSTANDING:

STUDENTS WILL ACQUIRE A THOROUGH UNDERSTANDING OF THE FUNDAMENTAL CONCEPTS OF DIFFERENTIAL GEOMETRY, INCLUDING CONCEPTS SUCH AS SUBVARIETIES IN EUCLIDEAN SPACES, CURVES, AND SURFACES IN R3. STUDENTS WILL BE ABLE TO UNDERSTAND HOW TO APPLY THE NOTIONS OF ANALYSIS AND LINEAR ALGEBRA PREVIOUSLY LEARNED TO STUDY THE INFINITESIMAL PROPERTIES OF GEOMETRIC OBJECTS. THIS WILL ENABLE THEM TO RECOGNIZE HOW THE INFINITESIMAL PROPERTIES REFLECT IN THE GLOBAL AND TOPOLOGICAL PROPERTIES OF THE GEOMETRIC OBJECTS.

ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING:

STUDENTS WILL BE ABLE TO CONCRETELY APPLY THE THEORETICAL NOTIONS LEARNED. THIS MAY INCLUDE THE ABILITY TO SOLVE PRACTICAL PROBLEMS INVOLVING CONCEPTS OF DIFFERENTIAL GEOMETRY. PARTICULAR ATTENTION WILL BE GIVEN TO EXERCISES, WHICH CONSTITUTE A SIGNIFICANT PART OF THE COURSE, ALLOWING STUDENTS TO PUT THEIR KNOWLEDGES INTO PRACTICE THROUGH THE PRACTICAL APPLICATION OF THEORETICAL CONCEPTS.

TRANSVERSAL OBJECTIVES, SUCH AS JUDGMENT AUTONOMY AND COMMUNICATIVE SKILLS, CAN BE INTEGRAL PART OF THE COURSE AIMS. HERE'S HOW THEY CAN BE FORMULATED AS SPECIFIC COURSE OBJECTIVES:

DEVELOP JUDGMENT AUTONOMY:

AIM: TO PROMOTE STUDENTS' JUDGMENT AUTONOMY SO THAT THEY ARE ABLE TO CRITICALLY ANALYZE CONCEPTS AND THEORIES OF DIFFERENTIAL GEOMETRY.

METHODS: ENCOURAGE STUDENTS TO EVALUATE THE EVIDENCE AND ARGUMENTS PRESENTED DURING THE COURSE AND TO DEVELOP THE ABILITY TO MAKE DECISIONS IN MATHEMATICAL CONTEXTS.

ENHANCE COMMUNICATIVE SKILLS:
AIM: TO IMPROVE STUDENTS' COMMUNICATIVE SKILLS SO THAT THEY CAN CLEARLY EXPRESS THEIR OWN IDEAS AND ARGUMENTS, BOTH ORALLY AND IN WRITING, USING APPROPRIATE TECHNICAL LANGUAGE.

METHODS: PROVIDE OPPORTUNITIES FOR STUDENTS TO PARTICIPATE ACTIVELY DURING CLASSES, ENCOURAGING THEM TO EXPRESS THEIR IDEAS, ASK QUESTIONS, AND ENGAGE IN DISCUSSIONS. ALSO, ENCOURAGE WRITING DEMONSTRATIONS THAT REQUIRE A CLEAR AND WELL-STRUCTURED COMMUNICATION OF MATHEMATICAL IDEAS.

PROMOTE AUTONOMOUS LEARNING:
AIM: TO FAVOR STUDENTS' AUTONOMOUS LEARNING SO THAT THEY CAN DEVELOP EFFECTIVE STUDY STRATEGIES, DEEPEN THEIR UNDERSTANDING OF CONCEPTS, AND APPLY THE ACQUIRED KNOWLEDGES IN A CREATIVE WAY.

METHODS: PROVIDE RESOURCES AND TEACHING MATERIALS THAT ALLOW STUDENTS TO DEEPEN THEIR UNDERSTANDING OF CONCEPTS BEYOND FRONT CLASSES, SUCH AS RECOMMENDED READINGS, EXTRA EXERCISES, AND ADVANCED OPTIONAL TOPICS. ALSO, ENCOURAGE ACTIVE PARTICIPATION OF STUDENTS THROUGH CRITICAL ANALYSIS OF MATHEMATICAL PROBLEMS AND INDEPENDENT RESEARCH OF SOLUTIONS.

FORMULATING THE COURSE AIMS IN A WAY THAT INCLUDES THESE TRANSVERSAL ASPECTS CONTRIBUTES TO ENSURING A COMPREHENSIVE AND MEANINGFUL LEARNING OF STUDENTS.

	Prerequisites
	LINEAR ALGEBRA: THIS INCLUDES CONCEPTS SUCH AS VECTOR SPACES, LINEAR TRANSFORMATIONS, DUAL SPACES, EIGENVALUES AND EIGENVECTORS, AND MATRIX DIAGONALIZATION, EUCLIDEAN VECTOR SPACES, DOT PRODUCT, CROSS PRODUCT. THESE CONCEPTS ARE FUNDAMENTAL FOR UNDERSTANDING MANY OF THE NOTIONS OF DIFFERENTIAL GEOMETRY, ESPECIALLY WHEN DEALING WITH TANGENT AND NORMAL VECTOR SPACES, AND LINEAR TRANSFORMATIONS USED TO DESCRIBE GEOMETRIC PROPERTIES. ANALYTICAL GEOMETRY: THIS MAY INCLUDE KNOWLEDGE OF CARTESIAN COORDINATES, DISTANCES AND LENGTHS, LINE AND PLANE EQUATIONS, CONICS, AND OTHER CURVES. ANALYTICAL GEOMETRY PROVIDES THE BASIS FOR UNDERSTANDING HOW THE NOTIONS OF DIFFERENTIAL GEOMETRY INTEGRATE WITH CLASSICAL GEOMETRY, ESPECIALLY WHEN DESCRIBING CURVES AND SURFACES IN THREE-DIMENSIONAL SPACE. DIFFERENTIAL AND INTEGRAL CALCULUS: THIS INCLUDES THE RULES OF DIFFERENTIATION AND INTEGRATION. THESE CONCEPTS ARE FUNDAMENTAL FOR UNDERSTANDING THE DEFINITIONS AND PROPERTIES OF CURVES AND SURFACES, AS WELL AS THE ANALYSIS OF FUNCTIONS OF SEVERAL VARIABLES.

Prerequisites

LINEAR ALGEBRA: THIS INCLUDES CONCEPTS SUCH AS VECTOR SPACES, LINEAR TRANSFORMATIONS, DUAL SPACES, EIGENVALUES AND EIGENVECTORS, AND MATRIX DIAGONALIZATION, EUCLIDEAN VECTOR SPACES, DOT PRODUCT, CROSS PRODUCT. THESE CONCEPTS ARE FUNDAMENTAL FOR UNDERSTANDING MANY OF THE NOTIONS OF DIFFERENTIAL GEOMETRY, ESPECIALLY WHEN DEALING WITH TANGENT AND NORMAL VECTOR SPACES, AND LINEAR TRANSFORMATIONS USED TO DESCRIBE GEOMETRIC PROPERTIES.

ANALYTICAL GEOMETRY: THIS MAY INCLUDE KNOWLEDGE OF CARTESIAN COORDINATES, DISTANCES AND LENGTHS, LINE AND PLANE EQUATIONS, CONICS, AND OTHER CURVES. ANALYTICAL GEOMETRY PROVIDES THE BASIS FOR UNDERSTANDING HOW THE NOTIONS OF DIFFERENTIAL GEOMETRY INTEGRATE WITH CLASSICAL GEOMETRY, ESPECIALLY WHEN DESCRIBING CURVES AND SURFACES IN THREE-DIMENSIONAL SPACE.

DIFFERENTIAL AND INTEGRAL CALCULUS: THIS INCLUDES THE RULES OF DIFFERENTIATION AND INTEGRATION. THESE CONCEPTS ARE FUNDAMENTAL FOR UNDERSTANDING THE DEFINITIONS AND PROPERTIES OF CURVES AND SURFACES, AS WELL AS THE ANALYSIS OF FUNCTIONS OF SEVERAL VARIABLES.

	Contents
	THE COURSE CONSISTS OF A SINGLE MODULE, FOR A TOTAL OF 48 HOURS OF CLASSROOM TEACHING, DIVIDED IN 36 HOURS OF THEORY AND 12 TUTORIALS. BELOW IS THE DETAILED PROGRAM: LOCAL THEORY OF CURVES. DIFFERENTIABLE CURVES IN R^3; ARC LENGTH OF A CURVE AND CURVILINEAR ABSCISSA. OSCULATING SPACES, FRENET TRIHEDRAL, FRENET EQUATIONS; RECONSTRUCTABILITY OF A CURVE FROM ITS CURVATURES. CURVES IN R^2 AND R^3: FRENET FORMULAS WITH A GENERIC PARAMETER; GEOMETRIC INTERPRETATION OF CURVATURE AND TORSION. DIFFERENTIAL GEOMETRY OF SURFACES. FIRST FUNDAMENTAL FORM AND INTRINSIC GEOMETRY OF A SURFACE. COVARIANT DERIVATIVE. SHAPE OPERATOR AND SECOND FUNDAMENTAL FORM; NORMAL CURVATURES, PRINCIPAL CURVATURES AND DIRECTIONS, TOTAL CURVATURE AND MEAN CURVATURE; HYPERBOLIC, ELLIPTIC, AND PARABOLIC POINTS; ROTATIONAL SURFACES. GAUSS'S THEOREMA EGREGIUM. PARALLEL TRANSPORT; GEODESIC CURVATURE; GEODESIC CURVES AND THEIR MINIMAL PROPERTIES. GAUSS-BONNET THEOREM. INTRODUCTION TO DIFFERENTIABLE MANIFOLDS, DIFFERENTIABLE MAPS, AND FIRST EXAMPLES.

Contents

THE COURSE CONSISTS OF A SINGLE MODULE, FOR A TOTAL OF 48 HOURS OF CLASSROOM TEACHING, DIVIDED IN 36 HOURS OF THEORY AND 12 TUTORIALS. BELOW IS THE DETAILED PROGRAM:

LOCAL THEORY OF CURVES.

DIFFERENTIABLE CURVES IN R^3; ARC LENGTH OF A CURVE AND CURVILINEAR ABSCISSA. OSCULATING SPACES, FRENET TRIHEDRAL, FRENET EQUATIONS; RECONSTRUCTABILITY OF A CURVE FROM ITS CURVATURES. CURVES IN R^2 AND R^3: FRENET FORMULAS WITH A GENERIC PARAMETER; GEOMETRIC INTERPRETATION OF CURVATURE AND TORSION.

DIFFERENTIAL GEOMETRY OF SURFACES.

FIRST FUNDAMENTAL FORM AND INTRINSIC GEOMETRY OF A SURFACE. COVARIANT DERIVATIVE. SHAPE OPERATOR AND SECOND FUNDAMENTAL FORM; NORMAL CURVATURES, PRINCIPAL CURVATURES AND DIRECTIONS, TOTAL CURVATURE AND MEAN CURVATURE; HYPERBOLIC, ELLIPTIC, AND PARABOLIC POINTS; ROTATIONAL SURFACES. GAUSS'S THEOREMA EGREGIUM. PARALLEL TRANSPORT; GEODESIC CURVATURE; GEODESIC CURVES AND THEIR MINIMAL PROPERTIES. GAUSS-BONNET THEOREM.

INTRODUCTION TO DIFFERENTIABLE MANIFOLDS, DIFFERENTIABLE MAPS, AND FIRST EXAMPLES.

	Teaching Methods
	DURING THE LECTURES THE RESULTS OF DIFFERENTIAL GEOMETRY LISTED IN SECTION "CONTENTS OF THE COURSE" WILL BE EXPOSED; IN THE PROBLEM SESSIONS IT WILL BE TAUGHT HOW TO APPLY SUCH RESULTS TO CONCRETE PROBLEMS; FURTHERMORE, THE PROOF OF SOME THEOREMS WILL BE SPLIT INTO AS A SEQUENCE OF EXERCISES, WHOSE SOLUTION WILL TRAIN THE STUDENT TO REASON AUTONOMOUSLY AND MAKE HIS/HER OWN PROOFS. PART OF THE PROBLEMS WILL BE DISCUSSED AND SOLVED IN THE CLASSROOM, WHILE OTHERS WILL BE ASSIGNED AS HOMEWORK, SO THAT STUDENTS CAN DEVELOP THEIR ABILITY TO SOLVE PROBLEMS AUTONOMOUSLY.

Teaching Methods

DURING THE LECTURES THE RESULTS OF DIFFERENTIAL GEOMETRY LISTED IN SECTION "CONTENTS OF THE COURSE" WILL BE EXPOSED; IN THE PROBLEM SESSIONS IT WILL BE TAUGHT HOW TO APPLY SUCH RESULTS TO CONCRETE PROBLEMS; FURTHERMORE, THE PROOF OF SOME THEOREMS WILL BE SPLIT INTO AS A SEQUENCE OF EXERCISES, WHOSE SOLUTION WILL TRAIN THE STUDENT TO REASON AUTONOMOUSLY AND MAKE HIS/HER OWN PROOFS. PART OF THE PROBLEMS WILL BE DISCUSSED AND SOLVED IN THE CLASSROOM, WHILE OTHERS WILL BE ASSIGNED AS HOMEWORK, SO THAT STUDENTS CAN DEVELOP THEIR ABILITY TO SOLVE PROBLEMS AUTONOMOUSLY.

	Verification of learning
	THE EXAM STRUCTURE AIMS TO ASSESS BOTH STUDENTS' ABILITY TO SOLVE PRACTICAL PROBLEMS AND THEIR THEORETICAL UNDERSTANDING OF THE CONCEPTS DISCUSSED DURING THE COURSE. THE EXAM CONSISTS OF A WRITTEN TEST AND AN ORAL TEST. THE WRITTEN TEST LASTING 2 HOURS AND 30 MINUTES. KNOWLEDGE AND UNDERSTANDING: THE WRITTEN PART OF THE EXAM, BASED ON EXERCISES SIMILAR TO THOSE ASSIGNED AS HOMEWORK DURING THE YEAR, ALLOWS STUDENTS TO DEMONSTRATE THEIR PRACTICAL UNDERSTANDING OF DIFFERENTIAL GEOMETRY CONCEPTS. THE ORAL PART, WHICH COVERS THEORY QUESTIONS ON THE DEFINITIONS, THEOREMS, AND PROOFS DISCUSSED DURING THE COURSE, EVALUATES THE DEPTH OF STUDENTS' THEORETICAL UNDERSTANDING. ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING: THE WRITTEN EXAM ALLOWS STUDENTS TO DEMONSTRATE THEIR ABILITY TO APPLY THEORETICAL KNOWLEDGE ACQUIRED DURING THE COURSE TO SOLVE PRACTICAL PROBLEMS, SIMILAR TO THOSE ENCOUNTERED DURING LECTURES AND HOMEWORK ASSIGNMENTS. DURING THE ORAL EXAM, STUDENTS ARE ASKED TO RESPOND TO THEORY QUESTIONS THAT REQUIRE THEM TO APPLY THE ACQUIRED NOTIONS TO EXPLAIN CONCEPTS, DEFINITIONS, AND PROOFS. LEARNING ABILITY: THE COMBINATION OF WRITTEN AND ORAL EXAMS ALLOWS STUDENTS TO DEMONSTRATE THEIR ABILITY TO CRITICALLY AND INDEPENDENTLY LEARN, BOTH IN SOLVING PRACTICAL PROBLEMS AND IN THEORETICAL UNDERSTANDING OF CONCEPTS. THE POSSIBILITY OF OBTAINING LAUDE REWARDS STUDENTS WHO DEMONSTRATE FULL MASTERY OF THE SUBJECT AND INDEPENDENT THOUGHT, ENCOURAGING THEM TO PURSUE ACADEMIC EXCELLENCE. THE MINIMUM GRADE IS OBTAINED BY PASSING THE WRITTEN EXAM AND SHOWING FULL UNDERSTANDING OF THE BASIC NOTIONS AS REGULAR SURFACE, TANGENT PLANE, GEODESIC, MANIFOLD. COMMUNICATION SKILLS: DURING THE ORAL EXAM, STUDENTS MUST BE ABLE TO CLEARLY COMMUNICATE THEIR THEORETICAL KNOWLEDGE AND UNDERSTANDING ACQUIRED DURING THE COURSE. THEY MUST BE ABLE TO EXPLAIN CONCEPTS, DEFINITIONS, THEOREMS, AND PROOFS IN A COHERENT AND UNDERSTANDABLE MANNER. IN CONCLUSION, THIS EXAMINATIONAL APPROACH PROVIDES A BALANCED EVALUATION OF STUDENTS' PRACTICAL AND THEORETICAL SKILLS IN DIFFERENTIAL GEOMETRY.

Verification of learning

THE EXAM STRUCTURE AIMS TO ASSESS BOTH STUDENTS' ABILITY TO SOLVE PRACTICAL PROBLEMS AND THEIR THEORETICAL UNDERSTANDING OF THE CONCEPTS DISCUSSED DURING THE COURSE. THE EXAM CONSISTS OF A WRITTEN TEST AND AN ORAL TEST. THE WRITTEN TEST LASTING 2 HOURS AND 30 MINUTES.

KNOWLEDGE AND UNDERSTANDING:
THE WRITTEN PART OF THE EXAM, BASED ON EXERCISES SIMILAR TO THOSE ASSIGNED AS HOMEWORK DURING THE YEAR, ALLOWS STUDENTS TO DEMONSTRATE THEIR PRACTICAL UNDERSTANDING OF DIFFERENTIAL GEOMETRY CONCEPTS.
THE ORAL PART, WHICH COVERS THEORY QUESTIONS ON THE DEFINITIONS, THEOREMS, AND PROOFS DISCUSSED DURING THE COURSE, EVALUATES THE DEPTH OF STUDENTS' THEORETICAL UNDERSTANDING.

ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING:
THE WRITTEN EXAM ALLOWS STUDENTS TO DEMONSTRATE THEIR ABILITY TO APPLY THEORETICAL KNOWLEDGE ACQUIRED DURING THE COURSE TO SOLVE PRACTICAL PROBLEMS, SIMILAR TO THOSE ENCOUNTERED DURING LECTURES AND HOMEWORK ASSIGNMENTS.
DURING THE ORAL EXAM, STUDENTS ARE ASKED TO RESPOND TO THEORY QUESTIONS THAT REQUIRE THEM TO APPLY THE ACQUIRED NOTIONS TO EXPLAIN CONCEPTS, DEFINITIONS, AND PROOFS.

LEARNING ABILITY:
THE COMBINATION OF WRITTEN AND ORAL EXAMS ALLOWS STUDENTS TO DEMONSTRATE THEIR ABILITY TO CRITICALLY AND INDEPENDENTLY LEARN, BOTH IN SOLVING PRACTICAL PROBLEMS AND IN THEORETICAL UNDERSTANDING OF CONCEPTS.
THE POSSIBILITY OF OBTAINING LAUDE REWARDS STUDENTS WHO DEMONSTRATE FULL MASTERY OF THE SUBJECT AND INDEPENDENT THOUGHT, ENCOURAGING THEM TO PURSUE ACADEMIC EXCELLENCE. THE MINIMUM GRADE IS OBTAINED BY PASSING THE WRITTEN EXAM AND SHOWING FULL UNDERSTANDING OF THE BASIC NOTIONS AS REGULAR SURFACE, TANGENT PLANE, GEODESIC, MANIFOLD.

COMMUNICATION SKILLS:
DURING THE ORAL EXAM, STUDENTS MUST BE ABLE TO CLEARLY COMMUNICATE THEIR THEORETICAL KNOWLEDGE AND UNDERSTANDING ACQUIRED DURING THE COURSE.
THEY MUST BE ABLE TO EXPLAIN CONCEPTS, DEFINITIONS, THEOREMS, AND PROOFS IN A COHERENT AND UNDERSTANDABLE MANNER.

IN CONCLUSION, THIS EXAMINATIONAL APPROACH PROVIDES A BALANCED EVALUATION OF STUDENTS' PRACTICAL AND THEORETICAL SKILLS IN DIFFERENTIAL GEOMETRY.

	Texts
	- LECTURE NOTES - M. ABATE, F. TOVENA, CURVES AND SURFACES. SPRINGER 2006 - M. DO CARMO: DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES, SECOND EDITION (DOVER 2016) - W. KLINGENBERG, A COURSE IN DIFFERENTIAL GEOMETRY -L. VITAGLIANO LECTURE NOTES

	More Information
	CHESPOSITO@UNISA.IT

Lessons Timetable

BETA VERSION Data source ESSE3 [Ultima Sincronizzazione: 2024-11-18]

Chiara ESPOSITO | GEOMETRY IV

GEOMETRY IV

PERCORSO SHARED STUDY PATHWAY

TEACHING UNITS

PROFESSORS

EXAMS

SHEET

-

Chiara ESPOSITO | GEOMETRY IV