Differential geometry:
Preface -- Differentiable Curves -- Regular Surfaces -- The Geometry of the Gauss Map -- The Intrinsic Geometry of Surfaces -- The Global Geometry of Surfaces -- References -- Index
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English Portuguese |
| Veröffentlicht: |
Cham, Switzerland
Springer Nature
[2024]
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | Preface -- Differentiable Curves -- Regular Surfaces -- The Geometry of the Gauss Map -- The Intrinsic Geometry of Surfaces -- The Global Geometry of Surfaces -- References -- Index This textbook provides a concise introduction to the differential geometry of curves and surfaces in three-dimensional space, tailored for undergraduate students with a solid foundation in mathematical analysis and linear algebra. The book emphasizes the geometric content of the subject, aiming to quickly cover fundamental topics such as the isoperimetric inequality and the Gauss-Bonnet theorem. This approach allows the author to extend beyond the typical content of introductory books and include additional important geometric results, such as curves and surfaces of constant width, the classification of complete surfaces of non-negative constant curvature, and Hadamard's theorem on surfaces of non-positive curvature. This range of topics offers greater variety for an introductory course |
| Beschreibung: | viii, 185 Seiten Illustrationen |
| ISBN: | 3031623835 9783031623837 3031623851 |
Internformat
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| 520 | 3 | |a Preface -- Differentiable Curves -- Regular Surfaces -- The Geometry of the Gauss Map -- The Intrinsic Geometry of Surfaces -- The Global Geometry of Surfaces -- References -- Index | |
| 520 | 3 | |a This textbook provides a concise introduction to the differential geometry of curves and surfaces in three-dimensional space, tailored for undergraduate students with a solid foundation in mathematical analysis and linear algebra. The book emphasizes the geometric content of the subject, aiming to quickly cover fundamental topics such as the isoperimetric inequality and the Gauss-Bonnet theorem. This approach allows the author to extend beyond the typical content of introductory books and include additional important geometric results, such as curves and surfaces of constant width, the classification of complete surfaces of non-negative constant curvature, and Hadamard's theorem on surfaces of non-positive curvature. This range of topics offers greater variety for an introductory course | |
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Datensatz im Suchindex
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|---|---|
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Contents 1 Differentiable Curves. 1.1 Velocity and Arc Length. . 1.2 Acceleration, Curvature and the Frenet Trihedron. 1.3 Planar Curves . 1.4 Contact of Curves. 1.5 Convex Curves . 1.6 Curves of Constant Width. 1.7 Theorem of the Four Vertices. 1.8 The Isoperimetric Inequality . 1 1 5 9 12 15 19 25 26 2 Regular Surfaces. 2.1 Definition and Examples. 2.2 Change of Parameters, Level Surfaces. 2.3 Differentiable Functions on Surfaces, Tangent Space. 2.4 Orientability. 2.5 Areas, Lengths, and Angles: The First Fundamental Form . 31 31 37 42 46 51 3 The Geometry of the Gauss Map. 3.1 The Gauss Map
and its Derivative. 3.2 The Second Fundamental Form. 3.3 Vector Fields. 59 59 67 76 4 The Intrinsic Geometry of Surfaces. 83 4.1 Conformal Mappings and Isometries. 83 4.2 Gauss’s Theorema Egregium. 90 4.3 Covariant Derivative, Parallel Transport,Geodesic Curvature. 92 4.4 The Divergence Theorem. First Variationof Area. 101 4.5 The Gauss-Bonnet Theorem. 110 4.6 Minimizing Properties of Geodesics. 121 Appendix: Rotation Index. 133 vii
viii 5 Contents The Global Geometry of Surfaces. 139 5.1 Complete Surfaces. 139 5.2 Coverings. 144 5.3 Complete Surfaces of Non-Positive Curvature. 149 5.4 Ovals (First Part): The Rigidity of the Sphere. 152 5.5 Ovals: Areas and Volumes; Surfaces of Constant Width. 157 5.6 Abstract Surfaces. The Hyperbolic Plane. 163 5.7 Complete Surfaces of Constant Curvature. 172 References. 181 Index. 183 |
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| spelling | Araújo, Paulo Ventura ca. 20./21. Jh. Verfasser (DE-588)1352770180 aut Geometria Diferencia Differential geometry Paulo Ventura Araújo Cham, Switzerland Springer Nature [2024] viii, 185 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Preface -- Differentiable Curves -- Regular Surfaces -- The Geometry of the Gauss Map -- The Intrinsic Geometry of Surfaces -- The Global Geometry of Surfaces -- References -- Index This textbook provides a concise introduction to the differential geometry of curves and surfaces in three-dimensional space, tailored for undergraduate students with a solid foundation in mathematical analysis and linear algebra. The book emphasizes the geometric content of the subject, aiming to quickly cover fundamental topics such as the isoperimetric inequality and the Gauss-Bonnet theorem. This approach allows the author to extend beyond the typical content of introductory books and include additional important geometric results, such as curves and surfaces of constant width, the classification of complete surfaces of non-negative constant curvature, and Hadamard's theorem on surfaces of non-positive curvature. This range of topics offers greater variety for an introductory course Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Geometry, Differential Géométrie différentielle Differentialgeometrie (DE-588)4012248-7 s DE-604 978-3-031-62384-4 Erscheint auch als Online-Ausgabe Araújo, Paulo Ventura Differential Geometry 1st ed. 2024 Cham : Springer International Publishing, 2024 1 Online-Ressource(VIII, 185 p.) 9783031623844 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=035407384&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Araújo, Paulo Ventura ca. 20./21. Jh Differential geometry Differentialgeometrie (DE-588)4012248-7 gnd |
| subject_GND | (DE-588)4012248-7 |
| title | Differential geometry |
| title_alt | Geometria Diferencia |
| title_auth | Differential geometry |
| title_exact_search | Differential geometry |
| title_full | Differential geometry Paulo Ventura Araújo |
| title_fullStr | Differential geometry Paulo Ventura Araújo |
| title_full_unstemmed | Differential geometry Paulo Ventura Araújo |
| title_short | Differential geometry |
| title_sort | differential geometry |
| topic | Differentialgeometrie (DE-588)4012248-7 gnd |
| topic_facet | Differentialgeometrie |
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