Book: G. Sobczyk, New Foundations in Mathematics


Garret Sobczyk, New Foundations in Mathematics – The Geometric Concept of Number, Birkhauser/Springer, New York, 2013, ISBN978-0-8176-8384-9, 370 pp., Price: 53.95 Euros (Amazon.de), $63.96 (Amazon.com), 39.99 Euros (Springer Yellow Sale until 31 July 2014).

Springer Yellow Sale until 31 July 2014: http://www.springer.com/birkhauser/mathematics/book/978-0-8176-8384-9

The first book of its kind, New Foundations in Mathematics: The Geometric Concept of Number uses geometric algebra to present an innovative approach to elementary and advanced mathematics. Geometric algebra offers a simple and robust means of expressing a wide range of ideas in mathematics, physics, and engineering. In particular, geometric algebra extends the real number system to include the concept of direction, which underpins much of modern mathematics and physics. Much of the material presented has been developed from undergraduate courses taught by the author over the years in linear algebra, theory of numbers, advanced calculus and vector calculus, numerical analysis, modern abstract algebra, and differential geometry. The principal aim of this book is to present these ideas in a freshly coherent and accessible manner.

New Foundations in Mathematics will be of interest to undergraduate and graduate students of mathematics and physics who are looking for a unified treatment of many important geometric ideas arising in these subjects at all levels. The material can also serve as a supplemental textbook in some or all of the areas mentioned above and as a reference book for professionals who apply mathematics to engineering and computational areas of mathematics and physics.

Back cover: The first book of its kind, New Foundations in Mathematics: The Geometric Concept of Number uses geometric algebra to present an innovative approach to elementary and advanced mathematics. Geometric algebra offers a simple and robust means of expressing a wide range of ideas in mathematics, physics, and engineering. In particular, geometric algebra extends the real number system to include the concept of direction, which underpins much of modern mathematics and physics. Much of the material presented has been developed from undergraduate courses taught by the author over the years in linear algebra, theory of numbers, advanced calculus and vector calculus, numerical analysis, modern abstract algebra, and differential geometry. The principal aim of this book is to present these ideas in a freshly coherent and accessible manner.

The book begins with a discussion of modular numbers (clock arithmetic) and modular polynomials. This leads to the idea of a spectral basis, the complex and hyperbolic numbers, and finally to geometric algebra, which lays the groundwork for the remainder of the text. Many topics are presented in a new
light, including:

* vector spaces and matrices;
* structure of linear operators and quadratic forms;
* Hermitian inner product spaces;
* geometry of moving planes;
* spacetime of special relativity;
* classical integration theorems;
* differential geometry of curves and smooth surfaces;
* projective geometry;
* Lie groups and Lie algebras.

Exercises with selected solutions are provided, and chapter summaries are included to reinforce concepts as they are covered. Links to relevant websites are often given, and supplementary material is available on the author’s website.

New Foundations in Mathematics will be of interest to undergraduate and graduate students of mathematics and physics who are looking for a unified treatment of many important geometric ideas arising in these subjects at all levels. The material can also serve as a supplemental textbook in some or all of the areas mentioned above and as a reference book for professionals who apply mathematics to engineering and computational areas of mathematics and physics.

Contents:

1 Modular Number Systems. . . . . . . . . . . . . . . . . . . .1
1.1 Beginnings . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Modular Numbers. . . . . . . . . . . . . . . . . . . . . . 2
1.3 Modular Polynomials . . . . . . . . . . . . . . . . . . . .8
1.4 Interpolation Polynomials . . . . . . . . . . . . . . . . 14

*1.5 Generalized Taylor’s Theorem . . . . . . . . . . . . . .17
1.5.1 Approximation Theorems. . . . . . . . . .  . . . . . . .18
1.5.2 Hermite–Pade Approximation .. . . . . . . . . .. . .    20

2 Complex and Hyperbolic Numbers. . . . . . . . .. . . . . . .23
2.1 The Hyperbolic Numbers.. . . . . .. . . . . . . . . . . . 24
2.2 Hyperbolic Polar Form . . . . . . . . . . . . . . . . . . 26
2.3 Inner and Outer Products . . . . . . . . . . . . . . . . .30
2.4 Idempotent Basis . . . . . . . . . . . . . . . . . . . . .33
2.5 The Cubic Equation.. . . . . . . . . . . . . . . . . . . .35
2.6 Special Relativity and Lorentzian Geometry . . . . . . . .37

3 Geometric Algebra . . . . . . . . . . . . . . . . . . . . . 43
3.1 Geometric Numbers of the Plane. . . . . . . . . . . . . . 45
3.2 The Geometric Algebra G3 of Space . . . . . . . . . . . . 50
3.3 Orthogonal Transformations . . . . . . . . . . . . . . . .54
3.4 Geometric Algebra of Rn . . . . . . . . . . . . . . . . . 57
3.5 Vector Derivative in Rn . . . . . . . . . . . . . . . . . 63

4 Vector Spaces and Matrices . . . . . . . . . . . . . . . . .67
4.1 Definitions .. . . . . . . . . . . . . . . . . . . . . . .67
4.2 Matrix Algebra . . . . . . . . . . . . . . . . . . . . . .70
4.3 Matrix Multiplication . . . . . . . . . . . . . . . . . . 73
4.4 Examples of Matrix Multiplication . . . . . . . . . . . . 75
4.5 Rules of Matrix Algebra . . . . . . . . . . . . . . . . . 78
4.6 The Matrices of G2 and G3 . . . . . . . . . . . . . . . . 79

5 Outer Product and Determinants . . . . . . . . . . . . . . .85
5.1 The Outer Product . . . . . . . . . . . . . . . . . . . . 85
5.2 Applications to Matrices. . . . . . . . . . . . . . . . . 92

6 Systems of Linear Equations . . . . . . . . . . . . . . . . 95
6.1 Elementary Operations and Matrices . . . . . . . . . . . .95
6.2 Gauss–Jordan Elimination .. . . . . . . . . . . . . . . .100
6.3 LU Decomposition .. . . . . . . . . . . . . . . . . . . .103

7 Linear Transformations on Rn . . . . . . . . . . . . . . . 107
7.1 Definition of a Linear Transformation . . . . . . . . . .107
7.2 The Adjoint Transformation .. . . . . . . . . . . . . . .113

8 Structure of a Linear Operator . . . . . . . . . . . . . . 117
8.1 Rank of a Linear Operator .. . . . . . . . . . . . . . . 117
8.2 Characteristic Polynomial . . . . . . . . . . . . . . .  120
8.3 Minimal Polynomial of f . . . . . . . . . . . . . . . .  122
8.4 Spectral Decomposition . . . . . . . . . . . . . . . . . 125
*8.5 Jordan Normal Form . . . . . . . . . . . . . . . . . .  130

9 Linear and Bilinear Forms . . . . . . . . . . . . . . . .  137
9.1 The Dual Space . . . . . . . . . . . . . . . . . . . . . 137
9.2 Bilinear Forms . . . . . . . . . . . . . . . . . . . . . 142
9.3 Quadratic Forms . . . . . . . . . . . . . . . . . . . .  144
9.4 The Normal Form .. . . . . . . . . . . . . . . . . . . . 145

10 Hermitian Inner Product Spaces . . . . . . . . . . . . .  153
10.1 Fundamental Concepts. . . . . . . . . . . . . . . . . . 154
10.2 Orthogonality Relationships in Pseudo-Euclidean Space . 157
10.3 Unitary Geometric Algebra of Pseudo-Euclidean Space. . .161
10.4 Hermitian Orthogonality.. . . . . . . . . . . . . . . . 166
10.5 Hermitian, Normal, and Unitary Operators . . . . . . . .172
*10.6 Principal Correlation .. . . . . . . . . . . . . . . . 175
*10.7 Polar and Singular Value Decomposition .. . . . . . . .178

11 Geometry of Moving Planes . . . . . . . . . . . . . . . . 181
11.1 Geometry of Space–Time . . . . . . . . . . . . . . . . .181
11.2 Relative Orthonormal Basis . . . . . . . . . . . . . . .186
11.3 Relative Geometric Algebras . . . . . . . . . . . . . . 189
11.4 Moving Planes . . . . . . . . . . . . . . . . . . . . . 191
*11.5 Splitting the Plane . . . . . . . . . . . . . . . . .  194

12 Representation of the Symmetric Group. . . . . . . . . .  201
12.1 The Twisted Product .. . . . . . . . . . . . . . . . .  201
12.1.1 Special Properties . . . . . . . . . . . . . . . . . .203
12.1.2 Basic Relationships . . . . . . . . . . . . . . . . . 204
12.2 Geometric Numbers in Gn,n . . . . . . . . . . . . . . . 205
12.3 The Twisted Product of Geometric Numbers . . . . . . . .207
12.4 Symmetric Groups in Geometric Algebras . . . . . . . . .210
12.4.1 The Symmetric Group S4 in G4,4 . . . . . . . . . . . .211
12.4.2 The Geometric Algebra G4,4 . . . . . . . . . . . . . .214
12.4.3 The General Construction in Gn,n . . . . . . . . . . .217
*12.5 The Heart of the Matter . . . . . . . . . . . . . . . .218

13 Calculus on m-Surfaces . . . . . . . . . . . . . . . . . .223
13.1 Rectangular Patches on a Surface . . . . . . . . . . . .223
13.2 The Vector Derivative and the Directed Integral . . . . 229
13.3 Classical Theorems of Integration . . . . . . . . . . . 236

14 Differential Geometry of Curves . . . . . . . . . . . . . 243
14.1 Definition of a Curve . . . . . . . . . . . . . . . . . 243
14.2 Formulas of Frenet–Serret .. . . . . . . . . . . . . . .245
14.3 Special Curves . . . . . . . . . . . . . . . . . . . .  248
14.4 Uniqueness Theorem for Curves . . . . . . . . . . . . . 249

15 Differential Geometry of k-Surfaces. . . . . . . . . . . .253
15.1 The Definition of a k-SurfaceM in Rn . . . . . . . . . .254
15.2 The Shape Operator .. . . . . . . . . . . . . . . . . . 261
15.3 Geodesic Curvature and Normal Curvature . . . . . . . . 267
15.4 Gaussian, Mean, and Principal Curvatures ofM . . . . . .270
15.5 The Curvature Bivector of a k-SurfaceM . . . . . . . .  271

16 Mappings Between Surfaces. . . . . . . . . . . . . . . . .275
16.1 Mappings Between Surfaces . . . . . . . . . . . . . . . 275
16.2 Projectively Related Surfaces . . . . . . . . . . . . . 279
16.3 Conformally Related Surfaces . . . . . . . . . . . . . .282
16.4 ConformalMapping in Rp,q . . . . . . . . . . . . . . . .286
16.5 M¨obius Transformations and Ahlfors–VahlenMatrices . . 287
*16.6 Affine Connections . . . . . . . . . . . . . . . . . . 291

17 Non-euclidean and Projective Geometries . . . . . . . . . 297
17.1 The Affine n-Plane A n
17.2 The Meet and Joint Operations . . . . . . . . . . . . . 299
17.3 Projective Geometry . . . . . . . . . . . . . . . . . . 304
17.4 Conics . . . . . . . . . . . . . . . . . . . . . . . . .312
17.5 Projective Geometry Is All of Geometry . . . . . . . . .319
17.6 The HorosphereH p,q . . . . . . . . . . . . . . . . . . 321

18 Lie Groups and Lie Algebras. . . . . . . . . . . . . . . .329
18.1 Bivector Representation . . . . . . . . . . . . . . . . 329
18.2 The General Linear Group.. . . . . . . . . . . . . . . .333
18.3 The Algebra Ωn,n . . . . . . . . . . . . . . . . . . . 337
18.4 Orthogonal Lie Groups and Their Lie Algebras .. . . . . 339
18.5 Semisimple Lie Algebras . . . . . . . . . . . . . . . . 345
18.6 The Lie Algebras An . . . . . . . . . . . . . . . . . . 348

 

See also previous announcement of incomplete online version (no longer available) on author homepage: https://gaupdate.wordpress.com/2011/03/04/book-g-sobczyk-new-foundations-in-mathematics-the-geometric-concept-of-number/

Source: Email by springer_AT_news.springer.com, 28 Apr. 2014,  http://www.amazon.de/New-Foundations-Mathematics-Garret-Sobczyk-ebook/dp/B00DGEQUYC/ref=sr_1_1?ie=UTF8&qid=1398664087&sr=8-1&keywords=New+Foundations+in+Mathematics, http://www.springer.com/birkhauser/mathematics/book/978-0-8176-8384-9, http://www.amazon.com/New-Foundations-Mathematics-Geometric-Concept/dp/0817683844/ref=sr_1_1?ie=UTF8&qid=1398664426&sr=8-1&keywords=New+Foundations+in+Mathematics

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