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New Foundations in Mathematics : The Geometric Concept of Number / by Garret Sobczyk
Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: SpringerLink BücherPublisher: Boston : Birkhäuser Boston, 2013Description: Online-Ressource (XIV, 370 p. 55 illus., 32 illus. in color, digital)ISBN:- 9780817683856
- 512.5
- QA184-205
Contents:
Summary: 1 Modular Number Systems -- 2 Complex and Hyperbolic Numbers -- 3 Geometric Algebra -- 4 Vector Spaces and Matrices -- 5 Outer Product and Determinants -- 6 Systems of Linear Equations -- 7 Linear Transformations on R^n -- 8 Structure of a Linear Operator -- 9 Linear and Bilinear Forms -- 10 Hermitian Inner Product Spaces -- 11 Geometry of Moving Planes -- 12 Representations of the Symmetric Group -- 13 Calculus on m-Surfaces -- 14 Differential Geometry of Curves -- 15 Differential Geometry of k-Surfaces -- 16 Mappings Between Surfaces -- 17 Non-Euclidean and Projective Geometries -- 18 Lie Groups and Lie Algebras -- References -- Symbols.Summary: 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.PPN: PPN: 1651882436Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
New Foundations in Mathematics; Preface; Contents; Chapter 1 Modular Number Systems; 1.1 Beginnings; 1.2 Modular Numbers; 1.3 Modular Polynomials; 1.4 Interpolation Polynomials; *1.5 Generalized Taylor's Theorem; 1.5.1 Approximation Theorems; 1.5.2 Hermite-Pade Approximation; Chapter 2 Complex and Hyperbolic Numbers; 2.1 The Hyperbolic Numbers; 2.2 Hyperbolic Polar Form; 2.3 Inner and Outer Products; 2.4 Idempotent Basis; 2.5 The Cubic Equation; 2.6 Special Relativity and Lorentzian Geometry; Chapter 3 Geometric Algebra; 3.1 Geometric Numbers of the Plane
3.2 The Geometric Algebra G3 of Space3.3 Orthogonal Transformations; 3.4 Geometric Algebra of Rn; 3.5 Vector Derivative in Rn; Chapter 4 Vector Spaces and Matrices; 4.1 Definitions; 4.2 Matrix Algebra; 4.3 Matrix Multiplication; 4.4 Examples of Matrix Multiplication; 4.5 Rules of Matrix Algebra; 4.6 The Matrices of G2 and G3; Chapter 5 Outer Product and Determinants; 5.1 The Outer Product; 5.2 Applications to Matrices; Chapter 6 Systems of Linear Equations; 6.1 Elementary Operations and Matrices; 6.2 Gauss-Jordan Elimination; 6.3 LU Decomposition; Chapter 7 Linear Transformations on Rn
7.1 Definition of a Linear Transformation7.2 The Adjoint Transformation; Chapter 8 Structure of a Linear Operator; 8.1 Rank of a Linear Operator; 8.2 Characteristic Polynomial; 8.3 Minimal Polynomial of f; 8.4 Spectral Decomposition; *8.5 Jordan Normal Form; Chapter 9 Linear and Bilinear Forms; 9.1 The Dual Space; 9.2 Bilinear Forms; 9.3 Quadratic Forms; 9.4 The Normal Form; Chapter 10 Hermitian Inner Product Spaces; 10.1 Fundamental Concepts; 10.2 Orthogonality Relationships in Pseudo-Euclidean Space; 10.3 Unitary Geometric Algebra of Pseudo-Euclidean Space; 10.4 Hermitian Orthogonality
10.5 Hermitian, Normal, and Unitary Operators*10.6 Principal Correlation; *10.7 Polar and Singular Value Decomposition; Chapter 11 Geometry of Moving Planes; 11.1 Geometry of Space-Time; 11.2 Relative Orthonormal Basis; 11.3 Relative Geometric Algebras; 11.4 Moving Planes; *11.5 Splitting the Plane; Chapter 12 Representation of the Symmetric Group; 12.1 The Twisted Product; 12.1.1 Special Properties; 12.1.2 Basic Relationships; 12.2 Geometric Numbers in Gn,n; 12.3 The Twisted Product of Geometric Numbers; 12.4 Symmetric Groups in Geometric Algebras; 12.4.1 The Symmetric Group S4 in G4,4
12.4.2 The Geometric Algebra G4,412.4.3 The General Construction in Gn,n; *12.5 The Heart of the Matter; Chapter 13 Calculus on m-Surfaces; 13.1 Rectangular Patches on a Surface; 13.2 The Vector Derivative and the Directed Integral; 13.3 Classical Theorems of Integration; Chapter 14 Differential Geometry of Curves; 14.1 Definition of a Curve; 14.2 Formulas of Frenet-Serret; 14.3 Special Curves; 14.4 Uniqueness Theorem for Curves; Chapter 15 Differential Geometry of k-Surfaces; 15.1 The Definition of a k-Surface M in Rn; 15.2 The Shape Operator; 15.3 Geodesic Curvature and Normal Curvature
15.4 Gaussian, Mean, and Principal Curvatures of M
1 Modular Number Systems -- 2 Complex and Hyperbolic Numbers -- 3 Geometric Algebra -- 4 Vector Spaces and Matrices -- 5 Outer Product and Determinants -- 6 Systems of Linear Equations -- 7 Linear Transformations on R^n -- 8 Structure of a Linear Operator -- 9 Linear and Bilinear Forms -- 10 Hermitian Inner Product Spaces -- 11 Geometry of Moving Planes -- 12 Representations of the Symmetric Group -- 13 Calculus on m-Surfaces -- 14 Differential Geometry of Curves -- 15 Differential Geometry of k-Surfaces -- 16 Mappings Between Surfaces -- 17 Non-Euclidean and Projective Geometries -- 18 Lie Groups and Lie Algebras -- References -- Symbols. .
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