Differential Geometry

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John Wiley & Sons, Jan 18, 1989 - Mathematics - 404 pages
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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 The vector notation
1
Addition of vectors
2
Multiplication by scalars
3
Vector product
5
Scalar triple product
6
Invariance under orthogonal transformations
7
Vector calculus
11
Plane Carves 1 Introduction
12
The Gauss equations
134
The Christoffel symbols evaluated
135
The Weingarten equations
136
Uniqueness of a surface for given gk and Lk
138
The theorema egregium of Gauss
139
How Gauss may have hit upon his theorem
141
Compatibility conditions in general
143
CodazziMainardi equations
144

Change of parameters
14
Invariance under changes of parameter
16
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 r
23
Formulas for c
26
Existence of a plane curve for given curvature k
27
Frenet equations for plane curves
28
Evolute and involute of a plane curve
29
Envelopes of families of curves
31
The Jordan theorem as a problem in differential geometry in the large
34
Additional properties of Jordan curves
41
The total curvature of a regular Jordan curve
45
Simple closed curves with 0 as boundaries of convex point sets
46
Four vertex theorem
48
Chapter in Space Carres 1 Regular curves
53
Length of a curve
54
Principal normal and osculating plane
55
Binomial vector
57
The Frenet equations for space curves
58
The Darboux vector
62
Formulas for and r
63
Canonical representation of a curve
64
Existence and uniqueness of a space curve for given s tj
65
What about 0?
67
Another way to define space curves
68
Some special curves
70
The Basic Elements of Surface Theory 1 Regular surfaces in Euclidean space
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
78
Angle measurement on surfaces
80
Area of a surface
82
A few examples
83
Second fundamental form of a surface
85
Osculating paraboloid
86
Curvature of curves on a surface
88
Principal directions and principal curvatures
91
Mean curvature H and Gaussian curvature K
92
Another definition of the Gaussian curvature K
93
Lines of curvature
95
Third fundamental form
98
Characterization of the sphere as a locus of umbilical points
99
Asymptotic lines
100
Introduction of special parameter curves
101
Asymptotic lines and lines of curvature as parameter curves
103
Analogues of polar coordinates on a surface
104
Some Special Surfaces 1 Surfaces of revolution
109
Developable surfaces in the small made up of parabolic points
114
Edge of regression of a developable
118
Why the name developable?
120
Developable surfaces in the large
121
Developables as envelopes of planes
129
The Partial Differential Equations of Surface Theory 1 Introduction
133
Existence of a surface with given gu and Llk
146
An application of the general theory to a problem in the large
148
Inner Differential Geometry in the Small from the Extrinsic Point of View 1 Introduction Motivations for the basic concepts
151
Approximate local parallelism of vectors in a surface
155
Parallel transport of vectors along curves in the sense of LeviChrita
157
Properties of parallel fields of vectors along curves
160
Parallel transport is independent of the path only for surfaces hav ing K m 0
162
the geodetic curvature
163
lines with 0
165
Geodetic lines as candidates for shortest arcs
167
Straight lines as shortest arcs in the Euclidean plane
168
A general necessary condition for a shortest arc
171
Geodesies in the small and geodetic coordinate systems
174
Geodesies as shortest arcs in the small
178
Further developments relating to geodetic coordinate systems
179
Surfaces of constant Gaussian curvature
183
Parallel fields from a new point of view
184
Models provided by differential geometry for nonEuclidean geometries
185
Parallel transport of a vector around a simple closed curve
191
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
227
Geodetically convex domains
231
The GaussBonnet formula applied to closed surfaces
237
Vector fields on surfaces and their singularities
239
Poincares 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 0
261
Hilberts theorem on surfaces in E3 with K 1
265
The form of complete surfaces of positive curvature in threedimen sional space
272
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
Parallel displacement Necessary condition for Euclidean metrics
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
Special theory of relativity
318
Relativistic dynamics
323
The general theory of relativity
326
The Wedge Product and die Exterior Demati?e of Differential
335
Appendix A Tensor Algebra in Affine Euclidean and Minkowski Spaces
371
Appendix B Differential Equations
388
Bibliography
396
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