A First Course in Geometric Topology and Differential Geometry (Modern Birkhäuser Classics) - Hardcover

Bloch, Ethan D.

Hardcover

ISBN 10: 0817638407 ISBN 13: 9780817638405

Verlag: Birkhäuser, 1997

Inhaltsangabe

I. Topology of Subsets of Euclidean Space.- 1.1 Introduction.- 1.2 Open and Closed Subsets of Sets in ?n.- 1.3 Continuous Maps.- 1.4 Homeomorphisms and Quotient Maps.- 1.5 Connectedness.- 1.6 Compactness.- II. Topological Surfaces.- 2.1 Introduction.- 2.2 Arcs, Disks and 1-spheres.- 2.3 Surfaces in ?n.- 2.4 Surfaces Via Gluing.- 2.5 Properties of Surfaces.- 2.6 Connected Sum and the Classification of Compact Connected Surfaces.- Appendix A2.1 Proof of Theorem 2.4.3 (i).- Appendix A2.2 Proof of Theorem 2.6.1.- III. Simplicial Surfaces.- 3.1 Introduction.- 3.2 Simplices.- 3.3 Simplicial Complexes.- 3.4 Simplicial Surfaces.- 3.5 The Euler Characteristic.- 3.6 Proof of the Classification of Compact Connected Surfaces.- 3.7 Simplicial Curvature and the Simplicial Gauss-Bonnet Theorem.- 3.8 Simplicial Disks and the Brouwer Fixed Point Theorem.- IV. Curves in ?3.- 4.1 Introduction.- 4.2 Smooth Functions.- 4.3 Curves in ?3.- 4.4 Tangent, Normal and Binormal Vectors.- 4.5 Curvature and Torsion.- 4.6 Fundamental Theorem of Curves.- 4.7 Plane Curves.- V. Smooth Surfaces.- 5.1 Introduction.- 5.2 Smooth Surfaces.- 5.3 Examples of Smooth Surfaces.- 5.4 Tangent and Normal Vectors.- 5.5 First Fundamental Form.- 5.6 Directional Derivatives - Coordinate Free.- 5.7 Directional Derivatives - Coordinates.- 5.8 Length and Area.- 5.9 Isometries.- Appendix A5.1 Proof of Proposition 5.3.1.- VI. Curvature of Smooth Surfaces.- 6.1 Introduction and First Attempt.- 6.2 The Weingarten Map and the Second Fundamental Form.- 6.3 Curvature - Second Attempt.- 6.4 Computations of Curvature Using Coordinates.- 6.5 Theorema Egregium and the Fundamental Theorem of Surfaces.- VII. Geodesics.- 7.1 Introduction - "Straight Lines" on Surfaces.- 7.2 Geodesics.- 7.3 Shortest Paths.- VIII. TheGauss-Bonnet Theorem.- 8.1 Introduction.- 8.2 The Exponential Map.- 8.3 Geodesic Polar Coordinates.- 8.4 Proof of the Gauss-Bonnet Theorem.- 8.5 Non-Euclidean Geometry.- Appendix A8.1 Geodesic Convexity.- Appendix A8.2 Geodesic Triangulations.- Further Study.- References.- Hints for Selected Exercises.- Index of Notation.

Über diesen Titel

Über die Autorin bzw. den Autor

Dr. Ethan D. Bloch of Bard College is the author of two Springer publications "A First Course in Geometric Topology and Differential Geometry," and the first and second editions of, "Proofs and Fundamentals: A First Course in Abstract Mathematics." More information about Dr. Ethan D. Bloch can be found on his person web page: http://math.bard.edu/bloch

Details zur Ausgabe

• Verlag: Birkhäuser
• Erscheinungsdatum: 1997
• Sprache: Englisch
• ISBN 10: 0817638407
• ISBN 13: 9780817638405
• Einband: Gebundene Ausgabe
• Anzahl der Seiten: 421
• Kontakt zum Hersteller: Springer Nature Customer Service Center GmbH; ProductSafety@springernature.com; ; Europaplatz 3; Heidelberg; 69115; Deutschland