Author | Simmonds, James G. author |
---|---|
Title | A Brief on Tensor Analysis [electronic resource] / by James G. Simmonds |
Imprint | New York, NY : Springer US, 1982 |
Connect to | http://dx.doi.org/10.1007/978-1-4684-0141-7 |
Descript | online resource |
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 Newtonโs Law and Tensor Calculus -- Rigid Bodies -- New Conservation Laws -- Nomenclature -- Newtonโs Law in Cartesian Components -- Newtonโs Law in Plane Polar Coordinates -- The Physical Components of a Vector -- The Christoffel Symbols -- General Three-Dimensional Coordinates -- Newtonโs 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 ? ยท v, ? ร v, and ?v -- The Covariant Derivative -- The Component Forms of ? ยท v, ? ร v, and ?v -- The Kinematics of Continuum Mechanics -- The Divergence Theorem -- Exercises