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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.
1. Coordinate Manifolds
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
- No. of pages:
- © Academic Press 1961
- 1st January 1961
- Academic Press
- eBook ISBN:
Graduate Institute for Mathematics and Mechanics Indiana University, Bloomington, Indiana
Departments of Mathematics, Electrical Engineering, and Medicine University of Southern California Los Angeles, California
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