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Учебно-образовательная физико-математическая библиотека
Библиотека > Книга
Книга
Ninul A.S. Tensor Trigonometry. Moscow: FIZMATLIT, 2021. (ISBN 9785940522782)
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| Автор(ы): | Ninul A.S. |
| Название: | Tensor Trigonometry |
| Издательство: | Moscow: FIZMATLIT |
| Год: | 2021 |
| ISBN: | 9785940522782 |
| Аннотация: | Planimetry includes metric part and trigonometry. In geometries of metric spaces from the end of XIX age
their tensor forms are widely used. However the trigonometry is remained only in its scalar form in a plane.
The tensor trigonometry is development of the flat scalar trigonometry from Leonard Euler classic forms into general multi-dimensional tensor forms with vector and scalar orthoprojections and 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 its applications. In theoretic plan, the tensor trigonometry complements naturally Analytic Geometry and Linear Algebra. In practical plan, it gives the clear instrument for solutions of various geometric and physical problems in homogeneous isotropic spaces, such as Euclidean, quasi- and pseudo-Euclidean ones. So in these spaces, the tensor trigonometry gives very clear general laws of motions in complete forms and with polar decompositions into principal and secondary motions, their descriptive trigonometric vector models, which are applicable also to n-dimensional non-Euclidean geometries in subspaces of constant radius embedded into enveloping metric spaces, and in the theory of relativity. In STR, the applications were considered till a trigonometric pseudo-analog of the classic theory by Frenet–Serret with absolute differentially-geometric, kinematic and dynamic characteristics in the current points of a world line. New methods of the tensor trigonometry can be also useful in other domains of mathematics and physics. The book is intended for researchers in the fields of multi-dimensional spaces, analytic geometry, linear algebra with theory of matrices, non-Euclidean geometries, theory of relativity 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 departments of algebra, geometry and physics. |
| Оглавление: | To the readers.....3 Contents.....5 Introduction.....7 Notations.....11 Part I. Theory of Exact Matrices: some of general questions.....16 Chapter 1. Coefficients of characteristic polynomials 1.1. Simultaneous definition of scalar and matrix coefficients.....17 1.2. The general inequality of means.....18 1.3. The serial method for solving an algebraic equation with real roots.....24 1.4. Structures of scalar and matrix characteristic coefficients.....28 1.5. The minimal annulling polynomial of a matrix in its 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 (oblique) and orthogonal eigenprojectors 2.1. Affine (oblique) eigenprojectors and quasi-inverse matrix.....43 2.2. Spectral representation of an n × n-matrix and its basic canonical form..43 2.3. Transforming a null-prime matrix in the null-cell canonical form.....47 2.4. Null-normal singular matrices.....48 2.5. Spherically orthogonal eigenprojectors and quasi-inverse matrices.....50 Chapter 3. Main scalar invariants of singular matrices 3.1. The minorant of a matrix and its applications.....53 3.2. Sine characteristics of matrices.....57 3.3. Cosine characteristics of matrices.....58 3.4. Limit methods for evaluating projectors and quasi-inverse matrices.....59 Chapter 4. Two alternative complexification variants 4.1. Comparing two variants.....61 4.2. Examples of adequate complexification.....64 4.3. Examples of Hermitian and symbiotic complexification.....67 Part II. Tensor trigonometry: fundamental contents.....68 Chapter 5. Euclidean and quasi-Euclidean tensor trigonometry 5.1. Objects of tensor trigonometry and their space relations.....69 5.2. Projective tensor sine, cosine, and spherically orthogonal reflectors.....71 5.3. Projective tensor secant, tangent, and affine (oblique) reflectors.....75 5.4. Comparison of two ways for defining projective tensor angles.....78 5.5. Canonical cell-forms of trigonometric functions and reflectors.....80 5.6. The trigonometric theory of prime roots I.....85 5.7. Rotational trigonometric functions and motive-type spherical angles...87 5.8. The tensor sine, cosine, secant, and tangent of a motive type angle.....92 5.9. Relations between projective and motive angles and functions.....94 5.10. Deformational trigonometric functions and cross projecting.....97 5.11. Special transformations of orthogonal and oblique eigenprojectors...100 5.12. Elementary tensor spherical trigonometric functions with frame axes...103 Chapter 6. Pseudo-Euclidean tensor and scalar trigonometry as a basis 6.1. Hyperbolic tensor angles, trigonometric functions, and reflectors.....107 6.2. Covariant concrete (or specific) spherical-hyperbolic analogy.....109 6.3. The reflector tensor in quasi-Euclidean and pseudo-Euclidean interpretation 111 6.4. Scalar trigonometry in a pseudoplane.....116 6.5. Elementary tensor hyperbolic trigonometric functions with frame axes..120 Chapter 7. Trigonometric interpretation of matrices commutativity and anticommutativity 7.1. Commutativity of prime matrices.....121 7.2. Anticommutativity of prime matrices pairs.....122 Chapter 8. Trigonometric spectra and trigonometric inequalities 8.1. Trigonometric spectra of a null-prime matrix.....127 8.2. The general cosine inequality.....129 8.3. Spectral-cell representations of tensor trigonometric functions.....132 8.4. The general sine inequality.....133 Chapter 9. Geometric norms of matrix objects 9.1. Quadratic norms of matrix objects in Euclidean and quasi-Euclidean spaces 136 9.2. Absolute and relative norms.....139 9.3. Geometric interpretation of particular quadratic norms.....139 9.4. Lineors of special kinds and simplest figures formed by lineors.....141 Chapter 10. Complexification of tensor trigonometry 10.1. Adequate complexification.....143 10.2. Hermitian complexification.....144 10.3. Pseudoization in binary complex spaces.....146 Chapter 11. Tensor trigonometry of general pseudo-Euclidean spaces 11.1. Realification of complex quasi-Euclidean spaces.....148 11.2. The general Lorentzian group of pseudo-Euclidean rotations.....149 11.3. Polar representation of general pseudo-Euclidean rotations.....154 11.4. Multistep hyperbolic rotations.....157 Chapter 12. Tensor trigonometry of Minkowski pseudo-Euclidean space 12.1. Trigonometric models for two concomitant hyperbolic geometries.....160 12.2. Rotations and deformations in elementary tensor forms.....169 12.3. The special mathematical principle of relativity.....172 Appendix. Trigonometric models of motions in STR and non-Euclidean Geometries. Preface with additional notations.....175 Chapter 1A. Space-time of Lagrange and space-time of Minkowski as mathematical abstractions and physical reality.....180 Chapter 2A. The tensor trigonometric model of Lorentzian homogeneous principal transformations.....190 Chapter 3A. Einsteinian dilation of time as a consequence of the time-arrow hyperbolic rotation.....193 Chapter 4A. Lorentzian seeming contraction of moving object extent as a consequence of the moving Euclidean subspace hyperbolic deformation..197 Chapter 5A. Trigonometric models of two-step, multistep and integral collinear motions in STR and in hyperbolic geometries.....205 Chapter 6A. Isomorphic mapping of a pseudo-Euclidean space into time-like and space-like quasi-Euclidean ones, Beltrami pseudosphere.....222 Chapter 7A. Trigonometric models of two-step, multistep and integral non-collinear motions in STR and in hyperbolic geometries.....228 Chapter 8A. Trigonometric models of two-step and multistep non-collinear motions in quasi-Euclidean space and in spherical geometry.....260 Chapter 9A. Real and observable space-time in the general relativity.....272 Chapter 10A. Motions along world lines in hP 3+1 i and their geometry...285 Mathematical–Physical Kunstkammer.....304 Literature.....308 Name Index.....312 Subject Index.....315 Abstract.....319 Exit data.....320 |
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