Concepts from Tensor Analysis and Differential Geometry

✍ Scribed by Tracy Y. Thomas (Eds.)

• Publisher: Elsevier Science
• Year: 1961
• Tongue: English
• Leaves: 122
• Series: Mathematics in Science and Engineering 1
• Category: Library

✦ Synopsis

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; andmethods 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 particularbranches, 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

✦ Table of Contents

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