مفاهیم تحلیل تنسوری و هندسه دیفرانسیل

In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation; methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; and methods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory. As a result, the book represents a blend of new methods in general computational analysis, and specific, but also generic, techniques for study of systems theory ant its particular branches, such as optimal filtering and information compression. - Best operator approximation, - Non-Lagrange interpolation, - Generic Karhunen-Loeve transform - Generalised low-rank matrix approximation - Optimal data compression - Optimal nonlinear filtering Content: Edited by Page iii Copyright page Page iv Preface Page v T.Y. Thomas 1. Coordinate Manifolds Pages 1-5 2. Scalars Page 6 3. Vectors and Tensors Pages 7-12 4. A Special Skew-symmetric Tensor Pages 13-15 5. The Vector Product. Curl of a Vector Page 16 6. Riemann Spaces Pages 17-28 7. Affinely Connected Spaces Pages 29-31 8. Normal Coordinates Pages 32-38 9. General Theory of Extension Pages 39-44 10. Absolute Differentiation Pages 45-47 11. Differential Invariants Pages 48-53 12. Transformation Groups Pages 54-56 13. Euclidean Metric Space Pages 57-64 14. Homogeneous and Isotropic Tensors Pages 65-69 15. Curves in Space. Frenet Formulae Pages 70-74 16. Surfaces in Space Pages 75-80 17. Mixed Surface and Space Tensors. Coordinate Extension and Absolute Differentiation Pages 81-86 18. Formulae of Gauss and Weingarten Pages 87-89 19. Gaussian and Mean Curvature of a Surface Page 90 20. Equations of Gauss and Codazzi Pages 91-92 21. Principal Curvatures and Principal Directions Pages 93-98 22. Asymptotic Lines Pages 99-100 23. Orthogonal Ennuples and Normal Congruences Pages 101-107 24. Families of Parallel Surfaces Pages 108-113 25. Developable Surfaces. Minimal Surfaces Pages 114-115 Subject index Pages 117-119 Coordinate manifolds Scalars Vectors and tensors Special skew-symmetric tensor Vector product, curl of a vector Riemann spaces Affinely connected spaces Normal coordinates General theory of extension Absolute differentiation Differential invariants Transformation groups Euclidean metric space Homogeneous and isotropic tensors Curves in spsace, Frenet formulae Surfaces in space Mixed surface and space tensors, coordinate extension and absolute differentiation Formulae of Gauss and Weingarten Gaussian and mean curvature of a surface Equations of Gauss and Codazzi Principla curvatures and principal directions Asymptotic lines Orthogonal ennuples and normal congruences Families of parallel surfaces Developable surfaces, minimal surfaces.