## Differential GeometryThis 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 |

396 | |

401 | |

403 | |

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### Common terms and phrases

angle assumed basis boundary bounded calculation called Chapter circle clear closed coefficients compact complete condition connected Consider const constant containing continuous convex coordinate system corresponding course covering curvature curve defined definition derivatives determined developable differential equations direction discussion distance Euclidean example exists exterior derivative fact field fixed follows formula functions geodetic geometry given hence holds important initial integral interest introduced invariant joining leads length line element linear manifold mapping means metric neighborhood normal noted notion obtained once origin orthogonal pair parallel parameter plane positive possible present problem proof proved quantities reasonable regular relation respect result satisfy scalar seen shortest shown side simply solution space sphere straight surface tangent vectors tensor theorem theory tion transformations triangle turn uniquely vanish zero