Ninul A. Tensor Trigonometry 2021

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Ninul A. Tensor Trigonometry 2021

Textbook in PDF format In the monograph, we undertake constructing general forms of the Tensor Trigonometry in multi-dimensional homogeneous and isotropic spaces with quadratic metrics (as Euclidean, quasi- and pseudo-Euclidean ones). The classic Scalar Trigonometry acts on eigenplanes of the binary trigonometric subspace of a tensor angle. The angle between two lines (or vectors), between two subspaces (or lineors) in multi-dimensional linear spaces has accordingly the nature of bivalent tensors, determined by the set reflector tensor of the binary space. However, its kind is determined by the concrete quadratic metric. In these metric spaces, a tensor angle and its trigonometric functions are respectively either orthogonal, or quasi-orthogonal, or pseudo-orthogonal tensors. (In particular, for Euclidean spaces, the simplest reflector tensor is a unity matrix, and we can deal only with the middle reflector of the concrete tensor angle.) These tensor angles and all their trigonometric functions can be defined in the two forms: projective one by a pair of eigenprojectors or eigenreflectors; motive one by the given rotational or deformational matrix. Projective and motive angles are one-to-one connected. In order to obtain the tensor construction, it was necessary to consider highly thoroughly a number of related questions in the Theory of Exact Matrices, what is a part of Linear Algebra. In addition to this, our efforts were rewarded by attainments of interesting and unexpected results in Algebra, Geometry and Theoretical Physics. Tensor Trigonometry point of view gives such advantages, that some rather difficult and not easily perceivable mathematical or physical theories became quietly transparent and natural for understanding. We exposed this on more elementary examples of trigonometric modelling different motions with the use of their polar representations in quasi-, pseudo-Euclidean and non-Euclidean geometries and in the Theory of Relativity. So, the hyperbolic tensor of motion with the certain scalar multipliers produces all the kinematic and dynamic scalar, vector and tensor physical relativistic characteristics of a moving material body, and gives the general law of summing motions and relativistic velocities. The hyperbolic tensor of deformation produces all the relativistic seeming geometric parameters of a moving object. Main content of the book are at the joint of problems studied in multi-dimensional Geometry and Linear Algebra. Since the exposition of the theory required many of additional notations and terms, the author tried to give them the most convenient and logical forms. So, this relates to the matrix alphabet based on wide-spreading examples. To the readers Introduction Notations Theory of Exact Matrices: some of general questions Tensor trigonometry: fundamental contents Appendix Literature