Concepts from Tensor Analysis and Differential Geometry

Concepts from Tensor Analysis and Differential Geometry

1st Edition - January 1, 1961

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  • Author: Tracy Y. Thomas
  • eBook ISBN: 9781483263717

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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.

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

Product details

  • No. of pages: 128
  • Language: English
  • Copyright: © Academic Press 1961
  • Published: January 1, 1961
  • 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

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