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.
Summary:
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 plane in flight-especially in a tum-without imaging it bristling with vec tors. Near the end of the course the professor showed that, if an airplane is treated as a rigid body, there arises a mysterious collection of rather simple looking integrals called the components of the moment of inertia tensor. Tensor-what power those two syllables seemed to resonate. I had heard the word once before, in an aside by a graduate instructor to the cognoscenti in the front row of a course in strength of materials. "What the book calls stress is actually a tensor. . . ." With my interest twice piqued and with time off from fighting the brush fires of a demanding curriculum, I was ready for my first serious effort at self instruction. In Los Angeles, after several tries, I found a store with a book on tensor analysis. In my mind I had rehearsed the scene in which a graduate stu dent or professor, spying me there, would shout, "You're an undergraduate.
