.

0 (hodnocen0 x )

BK

2nd ed.

London : Springer, c2010

ix, 473 s. : il. ; 24 cm

ISBN 978-1-84882-890-2 (brož.)

Springer undergraduate mathematics series, ISSN 1615-2085

Obsahuje rejstřík

Popsáno dle dotisku vydaného v roce 2012

000248896

Contents // Preface // Contents // 1. Curves in the plane and in space // 1.1 What is a curve?... 1 // 1.2 Arc-length... 9 // 1.3 Reparametrization... 13 // 1.4 Closed curves... 19 // 1.5 Level curves versus parametrized curves... 23 // 2. How much does a curve curve? // 2.1 Curvature... 29 // 2.2 Plane curves... 34 // 2.3 Space curves... 46 // 3. Global properties of curves // 3.1 Simple closed curves... 55 // 3.2 The isoperimetric inequality... 58 // 3.3 The four vertex theorem... 62 // 4. Surfaces in three dimensions // 4.1 What is a surface?... 67 // 4.2 Smooth surfaces... 76 // 4.3 Smooth maps... 82 // 4.4 Tangents and derivatives... 85 // 4.5 Normals and orientability... 89 // x // Contents // 5. Examples of surfaces // 5.1 Level surfaces... 95 // 5.2 Quadric surfaces ... 97 // 5.3 Ruled surfaces and surfaces of revolution...IO4 // 5.4 Compact surfaces...IO9 // 5.5 Triply orthogonal systems...? // 5.6 Applications of the inverse function theorem...116 // 6. The first fundamental form // 6.1 Lengths of curves on surfaces...121 // 6.2 Isometries of surfaces...126 // 6.3 Conformal mappings of surfaces...I33 // 6.4 Equiareal maps and a theorem of Archimedes...I39 // 6.5 Spherical geometry...148 // 7. Curvature of surfaces // 7.1 The second fundamental form...I59 // 7.2 The Gauss and Weingarten maps...162 // 7.3 Normal and geodesic curvatures...165 // 7.4 Parallel transport and covariant derivative...170 // 8. Gaussian, mean and principal curvatures // 8.1 Gaussian and mean curvatures...179

// 8.2 Principal curvatures of a surface...187 // 8.3 Surfaces of constant Gaussian curvature...196 // 8.4 Flat surfaces...201 // 8.5 Surfaces of constant mean curvature...206 // 8.6 Gaussian curvature of compact surfaces...212 // 9. Geodesics // 9.1 Definition and basic properties...215 // 9.2 Geodesic equations...220 // 9.3 Geodesics on surfaces of revolution ...227 // 9.4 Geodesics as shortest paths...235 // 9.5 Geodesic coordinates...242 // 10. Gauss’ Theorema Egregium // 10.1 The Gauss and Codazzi-Mainardi equations...247 // 10.2 Gauss’ remarkable theorem...252 // 10.3 Surfaces of constant Gaussian curvature...257 // 10.4 Geodesic mappings...263 // XI // 11. Hyperbolic geometry // 111 Upper half-plane model ... // 11.2 Isometries of  ... // 11.3 Poincaré disc model... // 11.4 Hyperbolic parallels... // 11.5 Beltrami-Klein model... // 12. Minimal surfaces // 12.1 Plateau’s problem... // 12.2 Examples of minimal surfaces... // 12.3 Gauss map of a minimal surface... // 12.4 Conformal parametrization of minimal surfaces... // 12.5 Minimal surfaces and holomorphic functions... // 13. The Gauss-Bonnet theorem // 13.1 Gauss-Bonnet for simple closed curves... // 13.2 Gauss-Bonnet for curvilinear polygons... // 13.3 Integration on compact surfaces... // 13.4 Gauss-Bonnet for compact surfaces... // 13.5 Map colouring... // 13.6 Holonomy and Gaussian curvature... // 13.7 Singularities of vector fields... // 13.8 Critical points ... // AO. Inner product spaces and self-adjoint

linear maps // Al. Isometries of Euclidean spaces // A2. Möbius transformations // Hints to selected exercises // Solutions // 270 // 277 // 283 // 290 // 295 // 305 // 312 // 320 // 322 // 325 // 335 // 342 // 346 // 349 // 357 // 362 // 365 // 372 // Index