Chapter 12. Geometrical Methods in Physics

  1. Prof. Dr. Charlie Harper

Published Online: 28 JAN 2005

DOI: 10.1002/3527603077.ch12

Analytic Methods in Physics

Analytic Methods in Physics

How to Cite

Harper, C. (1999) Geometrical Methods in Physics, in Analytic Methods in Physics, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, FRG. doi: 10.1002/3527603077.ch12

Author Information

  1. California State University, Hayward, USA

Publication History

  1. Published Online: 28 JAN 2005
  2. Published Print: 12 OCT 1999

ISBN Information

Print ISBN: 9783527402168

Online ISBN: 9783527603077

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Keywords:

  • geometrical methods;
  • contravariant and covariant tensors;
  • tensor algebra;
  • line element;
  • metric tensor;
  • tensor calculus;
  • covariant differentiation;
  • geodesic line;
  • curvature tensor;
  • exterior differential forms

Summary

This chapter contains sections titled:

  • Introduction

  • Transformation of Coordinates in Linear Spaces

  • Contravariant and Covariant Tensors

    • Tensors of Rank One

    • Higher-Rank Tensors

    • Symmetric and Antisymmetric Tensors

    • Polar and Axial Vectors

  • Tensor Algebra

    • Addition (Subtraction)

    • Multiplication (Outer Product)

    • Contraction

    • Inner Product

    • The Quotient Law

  • The Line Element

    • The Fundamental Metric Tensor

    • Associate Tensors

  • Tensor Calculus

    • Introduction

    • Christoffel Symbols

    • Covariant Differentiation of Tensors

  • The Equation of the Geodesic Line

  • Special Equations Involving the Metric Tensor

    • The Riemann-Christoffel Tensor

    • The Curvature Tensor

    • The Ricci Tensor

    • The Einstein Tensor and Equations of General Relativity

  • Exterior Differential Forms

    • Introduction

    • Exterior Product

    • Exterior Derivative

    • The Exterior Product and Exterior Derivative in ℜ3

    • The Generalized Stokes Theorem

  • Problems