A Brief on Tensor Analysis

Overview

In this text which gradually develops the tools for formulating and manipulating the field equations of Continuum Mechanics, the mathematics of tensor analysis is introduced in four, well-separated stages, and the physical interpretation and application of vectors and tensors are stressed throughout. This new edition contains more exercises. In addition, the author has appended a section on Differential Geometry.

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Paperback (Softcover reprint of the original 2nd ed. 1994)
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Overview

In this text which gradually develops the tools for formulating and manipulating the field equations of Continuum Mechanics, the mathematics of tensor analysis is introduced in four, well-separated stages, and the physical interpretation and application of vectors and tensors are stressed throughout. This new edition contains more exercises. In addition, the author has appended a section on Differential Geometry.

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Editorial Reviews

Booknews
Intended for advanced undergraduates in engineering, physics, mathematics, and applied sciences, this material can serve as a springboard for studies in continuum mechanics and general relativity. Topics include an introduction to vectors and tensors, general bases and tensor notation, Newton's law and tensor calculus, the gradient, del operator, covariant differentiation, and the divergence theorem. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9781461264248
  • Publisher: Springer New York
  • Publication date: 4/30/2013
  • Series: Undergraduate Texts in Mathematics Series
  • Edition description: Softcover reprint of the original 2nd ed. 1994
  • Edition number: 2
  • Pages: 114
  • Product dimensions: 6.14 (w) x 9.21 (h) x 0.28 (d)

Table of Contents

I Introduction: Vectors and Tensors.- Three-Dimensional Euclidean Space.- Directed Line Segments.- Addition of Two Vectors.- Multiplication of a Vector v by a Scalar—.- Things That Vectors May Represent.- Cartesian Coordinates.- The Dot Product.- Cartesian Base Vectors.- The Interpretation of Vector Addition.- The Cross Product.- Alternative Interpretation of the Dot and Cross Product. Tensors.- Definitions.- The Cartesian Components of a Second Order Tensor.- The Cartesian Basis for Second Order Tensors.- Exercises.- II General Bases and Tensor Notation.- General Bases.- The Jacobian of a Basis Is Nonzero.- The Summation Convention.- Computing the Dot Product in a General Basis.- Reciprocal Base Vectors.- The Roof (Contravariant) and Cellar (Covariant) Components of a Vector.- Simplification of the Component Form of the Dot Product in a General Basis.- Computing the Cross Product in a General Basis.- A Second Order Tensor Has Four Sets of Components in General.- Change of Basis.- Exercises.- III Newton’s Law and Tensor Calculus.- Rigid Bodies.- New Conservation Laws.- Nomenclature.- Newton’s Law in Cartesian Components.- Newton’s Law in Plane Polar Coordinates.- The Physical Components of a Vector.- The Christoffel Symbols.- General Three-Dimensional Coordinates.- Newton’s Law in General Coordinates.- Computation of the Christoffel Symbols.- An Alternative Formula for Computing the Christoffel Symbols.- A Change of Coordinates.- Transformation of the Christoffel Symbols.- Exercises.- IV The Gradient, the Del Operator, Covariant Differentiation, and the Divergence Theorem.- The Gradient.- Linear and Nonlinear Eigenvalue Problems.- The Del Operator.- The Divergence, Curl, and Gradient of a Vector Field.- The Invariance of— · v,— × v, and—v.- The Covariant Derivative.- The Component Forms of— · v,— × v, and—v.- The Kinematics of Continuum Mechanics.- The Divergence Theorem.- Differential Geometry.- Exercises.

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