Annamaria MIRANDA | GEOMETRY III
Annamaria MIRANDA GEOMETRY III
cod. 0512300009
GEOMETRY III
| 0512300009 | |
| DIPARTIMENTO DI MATEMATICA | |
| EQF6 | |
| MATHEMATICS | |
| 2016/2017 |
| OBBLIGATORIO | |
| YEAR OF COURSE 2 | |
| YEAR OF DIDACTIC SYSTEM 2010 | |
| SECONDO SEMESTRE |
| SSD | CFU | HOURS | ACTIVITY | |
|---|---|---|---|---|
| MAT/03 | 6 | 48 | LESSONS |
| Objectives | |
|---|---|
| TRAINING PURPOSES WHAT IS A GEOMETRY? THE COURSE OF GEOMETRY III HAS BEEN THOUGHT OF AS GIVING AN ANSWER TO THIS NATURAL QUERY BY CONSTRUCTING SEVERAL EXAMPLES OF GEOMETRIES FOLLOWING THE ERLANGEN PROGRAM BY F. KLEIN. MOREOVER, TO GENERATE INTEREST AND CURIOSITY AROUND THEOREMS GENERALLY CONSIDERED HARD, AN INTRODUCTION TO THE EUCLIDEAN AND HYPERBOLIC GEOMETRIES BOTH AS A “PLAY OF MIRRORS” IS GIVEN. -KNOWLEDGE AND UNDERSTANDING A PURPOSE OF THIS COURSE IS TO DEEPEN THE KNOWLEDGE OF THE EUCLIDEAN GEOMETRY BY REFINING THE “EUCLIDEAN EYE” IN PARTICULAR IN DIMENSION TWO AND THREE. ANOTHER TRAINING PURPOSE, WORKING EFFICIENTLY FOR DROPPING EUCLIDEAN PREJUDICES, IS TO INTERPRETER THE SAME REALITY BY USING THE “ AFFINE EYE” OR ALSO THE “AFFINE CONFORMAL EYE”. A NEXT OBJECTIVE IS THE CONSTRUCTION OF MODELS OF PLANE GEOMETRIES OF HYPERBOLIC TYPE, AS THE POINCARÉ PLANE AND POINCARÉ HALF-PLANE, OR OF ELLIPTIC TYPE AS THE SPHERICAL SURFACE AND THE REAL PROJECTIVE PLANE. TO THESE TOPICS THE STUDY OF CONICS WITH RELATIVE PROJECTIVE, AFFINE AND METRIC CLASSIFICATION IS ADDED. ITS MAIN SCOPE IS THE ANALYTIC RECASTING OF SYNTHETIC PROPERTIES BY USING ALREADY ACQUIRED MATHEMATICAL KNOWLEDGE. THE ARGUMENTS, TOOLS AND METHODS ARE TARGETED AT INTEGRATING EXPERIENCE, KNOWLEDGE, LEARNING, CURIOSITY AND SKILLS. ASSIGNMENTS, HINTS, SUGGESTIONS ARE GIVEN TO IMPROVE APPLYING ABILITY AND INVENTION IN DEMONSTRATION. AT THE END OF THE COURSE THE STUDENTS SHOULD HAVE • ACQUIRED THE BASIC INTRODUCED CONCEPTS, AS , FOR EXAMPLE, THE EUCLIDEAN GEOMETRY AND NON-EUCLIDEAN GEOMETRIES, PROJECTIVE AND HYPERBOLIC GEOMETRIES • UNDERSTOOD THE RELATIONS AMONG THEM JOINTLY WITH THE USED TECHNIQUES TO STATE THEM, AS, FOR EXAMPLE, IN THE CONSTRUCTION OF NON-EUCLIDEAN MODELS • ACQUIRED A DEEP KNOWLEDGE AND UNDERSTANDING OF THE EUCLIDEAN WORLD BUT ALSO THE ABILITY TO LOOK AT THE THINGS WITH DIFFERENT “ GEOMETRIC EYES” • TO CONSIDER THE EUCLIDEAN AND HYPERBOLIC GEOMETRY AS “ A MIRROR PLAY “ -APPLYING KNOWLEDGE AND UNDERSTANDING PROBLEMS ARE SUGGESTED AND TYPICAL MATHEMATICAL APPROACHES ARE CONSIDERED TO IMPROVE THE APPLYING ABILITY AND CAPACITY OF INVENTION IN DEMONSTRATION. AT THE END OF THE COURSE THE STUDENTS MUST BE ABLE : • TO USE EFFICIENTLY THE PROPOSED TECHNIQUES BY THEIR APPLICATION IN CONSTRUCTING SIGNIFICANT EXAMPLES AND IN SOLVING PROBLEMS AND EXERCISES • TO ANALYZE PROBLEMS, EXPLAIN CONCEPTS AND MAKE PROOFS. |
| Prerequisites | |
|---|---|
| FINITE-DIMENSIONAL VECTOR SPACES. LINEAR EQUATIONS. LINEAR MAPS AND MATRICES. BILINEAR AND QUADRATIC FORMS. DIAGONALIZATION. |
| Contents | |
|---|---|
| EUCLIDEAN SPACES AND THEIR VECTOR, AFFINE AND METRIC STRUCTURE. CLASSIFICATION AND DECOMPOSITION OF ORTHOGONAL TRANSFORMATIONS IN LOW DIMENSIONS: EULER THEOREM AND CARTAN-DIEUDONNÉ THEOREM. THE EUCLIDEAN ISOMETRIES AND EUCLIDEAN GEOMETRY: THE FUNDAMENTAL DECOMPOSITION THEOREM. CHASLES THEOREM. THE CARTAN-DIEUDONNÉ THEOREM FOR ISOMETRIES: EUCLIDEAN GEOMETRY AS A PLAY OF MIRRORS. ELEMENTARY GEOMETRICAL FIGURES AND THEIR SYMMETRIES. SIMILARITIES AND THEIR DECOMPOSITION. PROJECTIVE REAL PLANE:CONSTRUCTION OF MODELS OF THE REAL PROJECTIVE PLANE. CONICS AND THEIR PROJECTIVE, AFFINE, METRIC CLASSIFICATION. AN INTRODUCTION TO HYPERBOLIC GEOMETRY: THE POINCARÉ PLANE AND HALF-PLANE. CROSS-RATIO. HYPERBOLIC DISTANCE. CIRCULAR INVERSIONS. |
| Teaching Methods | |
|---|---|
| TEACHING METHODS ARE ESSENTIALLY BASED ON LESSONS MADE TO REDUCE COMPLEXITY TO SIMPLICITY AND, CONTEMPORANEOUSLY, DISPLAY AND MAKE GRADUALLY ACCESSIBILE STANDARD DEMONSTRATIVE TECHNIQUES. FURTHER, TO MAKE EASY TO UNDERSTAND AND APPRECIATE HOW FUNDAMENTALS THEOREMS PERFORM SIMPLICITY. THE TOOLS AND METHODS ARE TARGETED AT INTEGRATING EXPERIENCE, KNOWLEDGE, LEARNING, CURIOSITY AND SKILLS. ASSIGNMENTS, HINTS, SUGGESTIONS ARE GIVEN TO IMPROVE APPLYING ABILITY AND INVENTION IN DEMONSTRATION. |
| Verification of learning | |
|---|---|
| A FINAL WRITTEN AND ORAL EXAMINATION TO VALUE THE KNOWLEDGE OF THE ARGUMENTS TREATED IN THE COURSE, THE LEVEL OF UNDERSTANDING OF PERFORMED MATHEMATICAL APPROACHES, THE COMMUNICATION SKILLS, THE OPENING IN DISCUSSION, THE ORIGINALITY IN ARGUMENTATION AND THE INVENTION IN DEMONSTRATION. |
| Texts | |
|---|---|
| 1] E. AGAZZI- D. PALLADINO, LE GEOMETRIE NON EUCLIDEE E I FONDAMENTI DELLA GEOMETRIA DAL PUNTO DI VISTA ELEMENTARE, LA SCUOLA, 1998. [2] D. HILBERT- S. COHN-VOSSEN, GEOMETRIA INTUITIVA, BORINGHIERI, 2000. [3] E. SERNESI GEOMETRIA 1 BOLLATI-BORINGHIERI 2000. [4] E. SERNESI GEOMETRIA 2 BOLLATI- BORINGHIERI, 2001. [5] G. TALLINI STRUTTURE GEOMETRICHE LIGUORI EDITORE. [6] S. WILLARD, GENERAL TOPOLOGY , ED. DOVER 2004. |
| More Information | |
|---|---|
| E-MAIL ADDRESS: AMIRANDA@UNISA.IT |
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