Simmonds A Brief on Tensor Analysis

Erscheinungsjahr 2012
ISBN: 978-1-4684-0141-7
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, eBook

Reihe: Undergraduate Texts in Mathematics

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.

Zielgruppe

Research

Autoren/Hrsg.

Simmonds, J.G.

Inhaltsverzeichnis

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.- TheCovariant Derivative.- The Component Forms of ? · v, ? × v, and ?v.- The Kinematics of Continuum Mechanics.- The Divergence Theorem.- Exercises.