The Springer Undergraduate Mathematics Series (SUMS) is designed for undergraduates in the mathematical sciences. From core foundational material to final year topics, SUMS books take a fresh and modern approach and are ideal for self-study or for a one- or two-semester course. Each book includes numerous examples, problems and fully-worked solutions.

Preface 
Contents 
Curves in the plane and in space 
What is a curve? p. 1
Arc-length p. 9
Reparametrization p. 13
Closed curves p. 19
Level curves versus parametrized curves p. 23
How much does a curve curve? 
Curvature p. 29
Plane curves p. 34
Space curves p. 46
Global properties of curves 
Simple closed curves p. 55
The isoperimetric inequality p. 58
The four vertex theorem p. 62
Surfaces in three dimensions 
What is a surface? p. 67
Smooth surfaces p. 76
Smooth maps p. 82
Tangents and derivatives p. 85
Normals and orientability p. 89
Examples of surfaces 
Level surfaces p. 95
Quadric surfaces p. 97
Ruled surfaces and surfaces of revolution p. 104
Compact surfaces p. 109
Triply orthogonal systems p. 111
Applications of the inverse function theorem p. 116
The first fundamental form 
Lengths of curves on surfaces p. 121
Isometries of surfaces p. 126
Conformal mappings of surfaces p. 133
Equiareal maps mid a theorem of Archimedes p. 139
Spherical geometry p. 148
Curvature of surfaces 
The second fundamental form p. 159
The Gauss and Weingarten maps p. 162
Normal and geodesic curvatures p. 165
Parallel transport and covariant derivative p. 170
Gaussian, mean and principal curvatures 
Gaussian and mean curvatures p. 179
Principal curvatures of a surface p. 187
Surfaces of constant Gaussian curvature p. 196
Flat surfaces p. 201
Surfaces of constant mean curvature p. 206
Gaussian curvature of compact surfaces p. 212
Geodesics 
Definition and basic properties p. 215
Geodesic equations p. 220
Geodesics on surfaces of revolution p. 227
Geodesics as shortest paths p. 235
Geodesic coordinates p. 242
Gauss' Theorema Egregium 
The Gauss and Codazzi-Mainardi equations p. 247
Gauss' remarkable theorem p. 252
Surfaces of constant Gaussian curvature p. 257
Geodesic mappings p. 263
Hyperbolic geometry 
Upper half-plane model p. 270
Isometries of H p. 277
Poincar?disc model p. 283
Hyperbolic parallels p. 290
Beltrami-Klein model p. 295
Minimal surfaces 
Plateau's problem p. 305
Examples of minimal surfaces p. 312
Gauss map of a minimal surface p. 320
Conformal parametrization of minimal surfaces p. 322
Minimal surfaces and holomorphic functions p. 325
The Gauss-Bonnet theorem 
Gauss-Bonnet for simple closed curves p. 335
Gauss-Bonnet for curvilinear polygons p. 342
Integration on compact surfaces p. 346
Gauss-Bonnet for compact surfaces p. 349
Map colouring p. 357
Holonomy and Gaussian curvature p. 362
Singularities of vector fields p. 365
Critical points p. 372
Inner product spaces and self-adjoint linear maps 
Isometries of Euclidean spaces 
M?ius transformations 
Hints to selected exercises 
Solutions 
Index