Foundations of the theory of algebraic invariants

Cover......Page 1
Title Page......Page 2
Copyright......Page 3
Foreword......Page 4
Contents......Page 5
1 The invariants of transformations of Cartesian coordinates......Page 7
2 The group of motions......Page 19
3 The group of motions and reflections; the principal group......Page 22
4 Affine transformations of the plane......Page 27
5 Projective transformations of a straight line......Page 45
6 Projective transformations of the plane......Page 57
7 The reduction of affine invariants to projective invariants and of metric invariants to affine and projective invariants......Page 74
8 Multi-dimensional space......Page 91
9 Tensors and operations on tensors......Page 105
10 The geometric meaning of tensors; the projective point of view......Page 121
11 The geometric meaning of tensors; the affine point of view......Page 132
12 Relative tensors and their operations......Page 139
13 Invariants and concomitants of tensors......Page 145
14 Examples of invariants and concomitants......Page 149
15 The simplest properties of invariants......Page 165
16 Tensors with constant components......Page 176
17 Proof of the fundamental theorem......Page 188
18 The symbolic method in invariant theory......Page 195
19 Fundamental identities......Page 215
20 Invariant processes......Page 229
21 Complete systems of invariants......Page 241
22 Linear and quadratic binary forms......Page 252
23 Cubic binary forms......Page 267
24 Gordan's method for finding complete systems of concomitants of binary forms. Gordan-C'lebsch series......Page 278
25 Binary forms of fourth order......Page 291
26 Differential equations for invariants and semi-invariants of binary forms......Page 300
27 The quadratic ternary form......Page 314
28 Geometric interpretations of the simplest concomitants of ternary forms of order r......Page 323
29 The quadratic n-ary form......Page 332
30 Mixed second order tensors (affinors)......Page 344
31 Certain general properties of polyvectors......Page 359
32 Geometric interpretation of polyvectors. Weitzenbock's complex symbolic notation......Page 369
33 Condition for divisibility of a polyvector by one or several vectors......Page 380
34 The bi-vector......Page 387
35 The tri-vector......Page 396
Answers and hints to exercises......Page 402
Literature......Page 424
Index......Page 425