Concepts from Tensor Analysis and Differential Geometry - 1st Edition - ISBN: 9781483230184, 9781483263717

Concepts from Tensor Analysis and Differential Geometry

1st Edition

Authors: Tracy Y. Thomas
Editors: Richard Bellman
eBook ISBN: 9781483263717
Imprint: Academic Press
Published Date: 1st January 1961
Page Count: 128
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Description

Concepts from Tensor Analysis and Differential Geometry discusses coordinate manifolds, scalars, vectors, and tensors. The book explains some interesting formal properties of a skew-symmetric tensor and the curl of a vector in a coordinate manifold of three dimensions. It also explains Riemann spaces, affinely connected spaces, normal coordinates, and the general theory of extension. The book explores differential invariants, transformation groups, Euclidean metric space, and the Frenet formulae. The text describes curves in space, surfaces in space, mixed surfaces, space tensors, including the formulae of Gaus and Weingarten. It presents the equations of two scalars K and Q which can be defined over a regular surface S in a three dimensional Riemannian space R. In the equation, the scalar K, which is an intrinsic differential invariant of the surface S, is known as the total or Gaussian curvature and the scalar U is the mean curvature of the surface. The book also tackles families of parallel surfaces, developable surfaces, asymptotic lines, and orthogonal ennuples. The text is intended for a one-semester course for graduate students of pure mathematics, of applied mathematics covering subjects such as the theory of relativity, fluid mechanics, elasticity, and plasticity theory.

Table of Contents


Preface

1. Coordinate Manifolds

2. Scalars

3. Vectors and Tensors

4. A Special Skew-Symmetric Tensor

5. The Vector Product. Curl of a Vector

6. Riemann Spaces

7. Affinely Connected Spaces

8. Normal Coordinates

9. General Theory of Extension

10. Absolute Differentiation

11. Differential Invariants

12. Transformation Groups

13. Euclidean Metric Space

14. Homogeneous and Isotropic Tensors

15. Curves in Space. Frenet Formulae

16. Surfaces in Space

17. Mixed Surface and Space Tensors. Coordinate Extension and Absolute Differentiation

18. Formulae of Gauss and Weingarten

19. Gaussian and Mean Curvature of a Surface

20. Equations of Gauss and Codazzi

21. Principal Curvatures and Principal Directions

22. Asymptotic Lines

23. Orthogonal Ennuples and Normal Congruences

24. Families of Parallel Surfaces

25. Developable Surfaces. Minimal Surfaces

General References

Subject Index

Details

No. of pages:
128
Language:
English
Copyright:
© Academic Press 1961
Published:
Imprint:
Academic Press
eBook ISBN:
9781483263717

About the Author

Tracy Y. Thomas

Affiliations and Expertise

Graduate Institute for Mathematics and Mechanics Indiana University, Bloomington, Indiana

About the Editor

Richard Bellman

Affiliations and Expertise

Departments of Mathematics, Electrical Engineering, and Medicine University of Southern California Los Angeles, California