Differential Geometry

John Wiley & Sons, Jan 18, 1989 - Mathematics - 432 pages
This classic work is now available in an unabridged paperback edition. Stoker makes this fertile branch of mathematics accessible to the nonspecialist by the use of three different notations: vector algebra and calculus, tensor calculus, and the notation devised by Cartan, which employs invariant differential forms as elements in an algebra due to Grassman, combined with an operation called exterior differentiation. Assumed are a passing acquaintance with linear algebra and the basic elements of analysis.

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Contents

 Operations with Vectors 1 Addition of vectors 2 Multiplication by scalars 4 Representation of a vector by means of linearly independent vectors 3 Scalar product 5 Vector product 6 Scalar triple product 8 Invariance under orthogonal transformations 7 Vector calculus 1 2 3 3 9 6 10
 34 98 Characterization of the sphere as a locus of umbilical points 99 Asymptotic lines 20 Torsion of asymptotic lines 100 Introduction of special parameter curves 101 Asymptotic lines and lines of curvature as parameter curves 23 Embedding a given arc in a system of parameter curves 103 Analogues of polar coordinates on a surface 105 41 108 Some Special Surfaces 109

 16 12 Regular curves 13 Change of parameters 15 Invariance under changes of parameter 16 Tangent lines and tangent vectors of a curve 17 Orientation of a curve 18 Length of a curve 19 Arc length as an invariant 20 Curvature of plane curves 21 The normal vector and the sign of 23 Formulas for 26 Existence of a plane curve for given curvature 27 Frenet equations for plane curves 28 Evolute and involute of a planc curve 15 Envelopes of families of curves 33 The Jordan theorem as a problem in differential geometry in the large 34 Additional properties of Jordan curves 43 The total curvature of a regular Jordan curve 45 Simple closed curves with 0 as boundaries of convex point sets 46 Four vertex theorem 20 49 21 53 23 54 Principal normal and osculating plane 55 Binormal vector 6 Torsion 7 of a space curve 57 The Frenet equations for space curves 58 Rigid body motions and the rotation vector 59 The Darboux vector 62 Formulas for x and a 11 The sign of 63 Canonical representation of a curve 64 Existence and uniqueness of a space curve for given ks rs 14 What about 67 Another way to define space curves 69 Some special curves 57 57 58 58 62 63 63 64 65 67 68 70 The Basic Elements of Surface Theory 74 75 76 77 78 78 80 82 74 Change of parameters 75 Curvilinear coordinate curves on a surface 76 Tangent plane and normal vector 77 Length of curves and first fundamental form 6 Invariance of the first fundamental form 78 Angle measurement on surfaces 81 Area of a surface 82 A few examples 83 Second fundamental form of a surface 85 Osculating paraboloid 87 Curvature of curves on a surface 89 Principal directions and principal curvatures 91 Mean curvature H and Gaussian curvature 92 26 93 27 94 29 95 31 97
 The Partial Differential Equations of Surface Theory 133 91 146 Inner Differential Geometry in the Small from the Extrinsic 151 Derivation of the GaussBonnet formula 195 Consequences of the GaussBonnet formula 196 Tchebychef nets 198 Differential Geometry in the Large 1 Introduction Definition of ndimensional manifolds 203 Definition of a Riemannian manifold 206 Facts from topology relating to twodimensional manifolds 211 Surfaces in threedimensional space 217 Abstract surfaces as metric spaces 218 Complete surfaces and the existence of shortest arcs 220 Angle comparison theorems for geodetic triangles 226 Geodetically convex domains 231 The GaussBonnet formula applied to closed surfaces 237 Vector fields on surfaces and their singularities 239 Poincarés theorem on the sum of the indices on closed surfaces 244 Conjugate points Jacobis conditions for shortest arcs 247 The theorem of BonnetHopfRinow 254 Synges theorem in two dimensions 255 Covering surfaces of complete surfaces having K sO 259 Hilberts theorem on surfaces in Es with K 1 265 92 271 Intrinsic Differential Geometry of Manifolds Relativity 1 Introduction 282 Tensor Calculus in Affine and Euclidean Spaces 2 Affine geometry in curvilinear coordinates 284 Tensor calculus in Euclidean spaces 287 Tensor calculus in mechanics and physics 292 Tensor Calculus and Differential Geometry in General Manifolds 5 Tensors in a Riemannian space 294 Basic concepts of Riemannian geometry 296 46 300 Normal coordinates Curvature in Riemannian geometry 307 Geodetic lines as shortest connections in the small 310 Geodetic lines as shortest connections in the large 311 Theory of Relativity 11 Special theory of relativity 318 Relativistic dynamics 323 95 325 The general theory of relativity 326 The Wedge Product and the Exterior Derivative of Differential 335 Vector differential forms and surface theory 342 99 347 Scalar and vector products of vector forms on surfaces and their 349 Minimal surfaces 356 Appendix A Tensor Algebra in Affine Enclidean and Minkowski Spaces 371 Appendix B Differential Equations 388 Bibliography 396 103 401 104 403 Copyright

References to this book

 Theory of ShellsPhilippe G. CiarletLimited preview - 2000
 An Introduction to Differentiable Manifolds and Riemannian GeometryLimited preview - 1986
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