Geometry from a Differentiable ViewpointCambridge University Press, 1994 - 308 σελίδες This book offers a new treatment of the topic, one which is designed to make differential geometry an approachable subject for advanced undergraduates. Professor McCleary considers the historical development of non-Euclidean geometry, placing differential geometry in the context of geometry students will be familiar with from high school. The text serves as both an introduction to the classical differential geometry of curves and surfaces and as a history of a particular surface, the non-Euclidean or hyperbolic plane. The main theorems of non-Euclidean geometry are presented along with their historical development. The author then introduces the methods of differential geometry and develops them toward the goal of constructing models of the hyperbolic plane. While interesting diversions are offered, such as Huygen's pendulum clock and mathematical cartography, the book thoroughly treats the models of non-Euclidean geometry and the modern ideas of abstract surfaces and manifolds. |
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Αποτελέσματα 1 - 5 από τα 46.
Σελίδα vii
... Euclid Euclid's theory of parallels Appendix . The Elements : Book I 3. The theory of parallels Uniqueness of parallels Equidistance and boundedness of parallels On the angle sum of a triangle Similarity of triangles 4. Non - Euclidean ...
... Euclid Euclid's theory of parallels Appendix . The Elements : Book I 3. The theory of parallels Uniqueness of parallels Equidistance and boundedness of parallels On the angle sum of a triangle Similarity of triangles 4. Non - Euclidean ...
Σελίδα viii
... Euclid revisited I : The Hopf - Rinow theorem 12. The Gauss - Bonnet theorem Euclid revisited II : Uniqueness of lines Compact surfaces A digression on curves 13. Constant - curvature surfaces Euclid revisited III : Congruences The work ...
... Euclid revisited I : The Hopf - Rinow theorem 12. The Gauss - Bonnet theorem Euclid revisited II : Uniqueness of lines Compact surfaces A digression on curves 13. Constant - curvature surfaces Euclid revisited III : Congruences The work ...
Σελίδα ix
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Σελίδα 3
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Περιεχόμενα
The theory of parallels | 24 |
NonEuclidean geometry I | 34 |
Curves | 63 |
Curves in space | 80 |
Surfaces | 95 |
8bis Map projections | 116 |
Curvature for surfaces | 131 |
Metric equivalence of surfaces | 145 |
Constantcurvature surfaces | 186 |
Abstract surfaces | 201 |
Modeling the nonEuclidean plane | 217 |
Where from here? | 242 |
On the hypotheses which lie at the foundations | 269 |
Notes on selected exercises | 279 |
Bibliography | 297 |
303 | |
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Συχνά εμφανιζόμενοι όροι και φράσεις
abstract surface angle defect angle sum arc length asymptotic axiom Beltrami Ca(s Chapter Christoffel symbols circle component functions compute congruent consider constant construct coordinate chart coordinate curves coordinate patch cos² cosh curve a(s defined Definition denote determined diffeomorphism differential equations differential geometry ds² dt dt Euclid Euclidean expression follows formula Gauss Gaussian curvature geodesic curvature given line horocycle implies inner product interior angles intersection isometry k₁ Lemma line element line segment linear fractional transformation manifold map projection matrix metric relations non-Euclidean geometry parallel parametrization perpendicular polar coordinates Postulate PROOF properties Proposition prove radius regular surface Riemann Riemann curvature tensor Riemannian metric right angles S₁ satisfies space sphere Suppose surface in R3 tangent plane tangent vector tensor theorem Tp(S unit-speed curve vector field ди
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