A Brief on Tensor Analysis [electronic resource] by J.G. Simmonds.

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When I was an undergraduate, working as a co-op student at North American Aviation, I tried to learn something about tensors. In the Aeronautical En­ gineering Department at MIT, I had just finished an introductory course in classical mechanics that so impressed me that to this day I cannot watch a...

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Bibliographic Details
Uniform Title:Undergraduate Texts in Mathematics, 2197-5604
Main Author: Simmonds, J.G (Author)
Corporate Author: SpringerLink (Online service)
Language:English
Published: New York, NY : Springer New York : Imprint: Springer, 1982.
Edition:1st ed. 1982.
Series:Undergraduate Texts in Mathematics,
Subjects:
Mathematical analysis.
Online Access:
Springer Book Archive - Mathematics: 1982 (Springer Link)
Format: Electronic eBook
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
  • Alternate 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 Alternate Formula for Computing the Christoffel Symbols
  • A Change of Coordinates
  • Transformation of the Christoffel Symbols
  • Exercises
  • IV The Gradient Operator, Covariant Differentiation, and the Divergence Theorem
  • The Gradient
  • Linear and Nonlinear Eigenvalue Problems
  • The Del or Gradient 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
  • Exercises.