Differential Geometry

Front Cover
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

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About the author (1989)

James J Stoker was an American applied mathematician and engineer. He was director of the Courant Institute of Mathematical Sciences and is considered one of the founders of the institute, Courant and Friedrichs being the others. Stoker is known for his work in differential geometry and theory of water waves.

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