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Учебно-образовательная физико-математическая библиотека
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Книга
Ninul A.S. Tensor Trigonometry. Moscow: FIZMATLIT. (ISBN 9785891554290)
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| Автор(ы): | Ninul A.S. |
| Название: | Tensor Trigonometry |
| Издательство: | Moscow: FIZMATLIT |
| ISBN: | 9785891554290 |
| Аннотация: | The Tensor Trigonometry, with revealing a tensor nature of the angles and their functions and added by differential trigonometry, is developed for wide applications in various fields. Planimetry includes metric part and trigonometry. In geometries of metric spaces from the end of XIX age their tensor forms are widely used. Trigonometry was remaining in its scalar flat forms. Tensor Trigonometry is its development from Leonard Euler classic forms into spatial k-dimensional (at k>=2) tensor forms with vector and scalar orthoprojections, with step by step increasing a complexity and opportunities. Described in the book are fundamentals of this new mathematical subject with many initial examples of applications. In theoretic plan, Tensor Trigonometry complements naturally Analytic Geometry and Linear Algebra. In practical plan, it gives the clear tools for analysis and solutions of various geometric and physical problems in homogeneous isotropic spaces, as Euclidean, quasi- and pseudo-Euclidean ones, on perfect surfaces of constant radius embedded into them with n-D non-Euclidean Geometries, and in Theory of Relativity. So, it gives classic projective models of non-Euclidean Geometries as trigonometric ones, general laws of summing two-steps and polysteps motions in complete differential and integral forms with polar decomposition of the sum into principal and induced orthospherical motions. The applications were developed till the differential tensor trigonometry of world lines and curves in 3D and 4D pseudo- and quasi-Euclidean spaces, in addition, to the classic Frenet-Serret theory, with absolute and relative differential-geometric parameters of curves, main kinematic and dynamical characteristics of a body moving in space-time along a world line with 4-velocity of Poincare. Due to our tensor trigonometric approach, clear explanations of all well-known and new STR and GR relativistic effects are given with physical interpretations in full agreement with the Law of Energy-Momentum conservation, Quantum Mechanics, Noether Theorem and Higgs Theory. The Tensor Trigonometry can be useful in various domains of mathematics and physics. It is intended to researchers in the felds of analytic geometry of any dimension, linear algebra with matrix theory, non-Euclidean geometries, theory of relativity, quantum mechanics and to all those who is interested in new knowledges and applications, given by exact sciences. It may be useful for educational purposes with this new math subject in the university and graduate schools departments of algebra, geometry and physics - relativistic and classical. |
| Оглавление: | Abstract.....2 To the readers.....3 Contents.....4 Introduction.....7 Notations.....11 Part I. Theory of Exact Matrices: some of general questions as fundamentals of the Tensor trigonometry.....16 Chapter 1. Coefficients of characteristic polynomials 1.1. Simultaneous definition of scalar and matrix coefficients.....17 1.2. The general inequality of means (average values).....18 1.3. The serial method of solving algebraic equations with real roots.....24 1.4. Structures of scalar and matrix characteristic coefficients.....28 1.5. The minimal annulling polynomial of a matrix in explicit form.....34 1.6. Null-prime and null-defective singular matrices.....37 1.7. The reduced form of characteristic coefficients.....40 Chapter 2. Affine and orthogonal eigenprojectors 2.1. Affine (oblique) projectors and quasi-inverse matrix.....40 2.2. Spectral presentation of n × n-matrix with basic canonical form.....41 2.3. Transforming a null-prime matrix in its null-cell form.....43 2.4. Null-normal singular matrices.....44 2.5. Spherically orthogonal projectors and quasi-inverse matrices.....46 Chapter 3. Main scalar invariants of singular matrices 3.1. The minorant of a matrix and its applications.....49 3.2. Sine characteristics of matrices.....52 3.3. Cosine characteristics of matrices.....53 3.4. Limit evaluation of eigenprojectors and quasi-inverse matrices.....54 Chapter 4. Main alternative variants of complexification 4.1. Comparing alternative variants of complexification.....40 4.2. Examples of adequate and pseudoized complexifications.....59 4.3. Examples of Hermitian and symbiotic complexification.....61 Part II. Tensor trigonometry: fundamental contents.....63 Chapter 5. Euclidean and Quasi-Euclidean tensor trigonometry 5.1. Objects of tensor trigonometry and their spatial relations.....64 5.2. Projective tensor sine, cosine and spherically orthogonal reflectors...66 5.3. Projective tensor secant, tangent and a ne (oblique) reflectors.....70 5.4. Comparison of two ways for defining projective tensor angles.....72 5.5. Cell-forms of tensor trigonometric functions and reflectors.....74 5.6. The tensor trigonometric theory of prime roots I.....79 5.7. Rotational functions of motive tensor spherical angles.....81 5.8. Motive-type tensor sine, cosine, secant and tangent.....86 5.9. Relations between projective and motive angles and functions.....88 5.10. Deformational functions of motive tensor spherical angles.....91 5.11. Transformations of orthogonal and oblique eigenprojectors.....94 5.12. Elementary tensor spherical functions with frame axes.....96 Chapter 6. Pseudo-Euclidean tensor and scalar trigonometry as a basis 6.1. Hyperbolic tensor angles, trigonometric functions, and reflectors...100 6.2. Covariant and countervariant spherical-hyperbolic analogies.....102 6.3. Reflector tensor in quasi- and pseudo-Euclidean interpretations....106 6.4. Scalar trigonometry in a pseudoplane with main relations.....108 6.5. Elementary tensor hyperbolic functions with frame axes.....112 Chapter 7. Tensor trigonometric interpretation of prime matrices commutativity and anticommutativity 7.1. Commutativity of prime matrices.....113 7.2. Anticommutativity of prime matrices pairs.....114 Chapter 8. Tensor trigonometric spectra with general inequalities 8.1. Trigonometric spectra of a null-prime matrix.....119 8.2. The general Cosine inequality.....120 8.3. Spectral-cell presentations of tensor trigonometric functions.....123 8.4. The general Sine inequality.....124 Chapter 9. Geometric norms of varied orders for matrix objects 9.1. Quadratic and hierarchical norms.....128 9.2. Absolute and relative norms.....130 9.3. Geometric interpretation of particular norms.....131 9.4. Lineors of special kinds and some figures formed by lineors.....132 Chapter 10. Complexification of tensor trigonometry 10.1. Adequate complexification.....134 10.2. Hermitean complexification.....135 10.3. Pseudoization in binary complex spaces.....1366 Chapter 11. Tensor trigonometry of general pseudo-Euclidean spaces 11.1. Realification of complex quasi-Euclidean spaces.....139 11.2. The general Lorentzian group of pseudo-Euclidean rotations.....140 11.3. Polar representation of general pseudo-Euclidean rotations.....144 11.4. Polysteps hyperbolic rotations with polar decomposition.....147 Chapter 12. Tensor trigonometry of Minkowski pseudo-Euclidean space with geometries of two embedded hyperboloids 12.1. Trigonometric models of bi-associated hyperbolic geometries.....150 12.2. Rotations and deformations in their elementary tensor forms.....157 12.3. The mathematical principle of relativity.....160 Appendix Trigonometric models of motions in non-Euclidean Geometries and STR Preface with additional notations.....163 Chapter 1A. Space-time of Lagrange and space-time of Poincare and of Minkowski as mathematical abstractions and physical reality...168 Chapter 2A. Tensor trigonometric model of Lorentzian homogeneous principal transformations.....177 Chapter 3A. Minkowskian real kinematic dilation of time as a consequence of the time-arrow hyperbolic rotation.....180 Chapter 4A. Lorentzian seeming contraction of moving object extent as a consequence of moving Euclidean subspace hyperbolic deformation 184 Chapter 5A. Trigonometric models of two-steps, polysteps and integral collinear motions in STR and in two hyperbolic geometries.....191 Chapter 6A. Isomorphic mapping of pseudo-Euclidean space of index 1 into Special quasi-Euclidean space with Beltrami pseudosphere.....207 Chapter 7A. Trigonometric models of two-steps, polysteps and integral non-collinear motions in STR and in two hyperbolic geometries.....213 Chapter 8A. Trigonometric models of two-steps and polysteps non-collinear motions in quasi-Euclidean and spherical geometries.....243 Chapter 9A. Real and observable by us space-time in the gravity field. 254 Chapter 10A. Differential tensor trigonometry of world lines and curves 268 Mathematical-Physical Kunstkammer.....305 Literature.....309 Name Index.....313 Subject Index.....316 Exit data.....320 |
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