Tensor Analysis

Description

This book presents tensors and tensor analysis as primary mathematical tools for engineering and engineering science students and researchers. The discussion is based on the concepts of vectors and vector analysis in three-dimensional Euclidean space, and although it takes the subject matter to an advanced level, the book starts with elementary geometrical vector algebra so that it is suitable as a first introduction to tensors and tensor analysis. Each chapter includes a number of problems for readers to solve, and solutions are provided in an Appendix at the end of the text. Chapter 1 introduces the necessary mathematical foundations for the chapters that follow, while Chapter 2 presents the equations of motions for bodies of continuous material. Chapter 3 offers a general definition of tensors and tensor fields in three-dimensional Euclidean space. Chapter 4 discusses a new family of tensors related to the deformation of continuous material. Chapter 5 then addresses constitutive equations for elastic materials and viscous fluids, which are presented as tensor equations relating the tensor concept of stress to the tensors describing deformation, rate of deformation and rotation. Chapter 6 investigates general coordinate systems in three-dimensional Euclidean space and Chapter 7 shows how the tensor equations discussed in chapters 4 and 5 are presented in general coordinates. Chapter 8 describes surface geometry in three-dimensional Euclidean space, Chapter 9 includes the most common integral theorems in two- and three-dimensional Euclidean space applied in continuum mechanics and mathematical physics. Front Matter ....Pages i-xxi Mathematical Foundation (Fridtjov Irgens)....Pages 1-33 Dynamics. The Cauchy Stress Tensor (Fridtjov Irgens)....Pages 35-74 Tensors (Fridtjov Irgens)....Pages 75-130 Deformation Analysis (Fridtjov Irgens)....Pages 131-162 Constitutive Equations (Fridtjov Irgens)....Pages 163-182 General Coordinates in Euclidean Space E3 (Fridtjov Irgens)....Pages 183-229 Elements of Continuum Mechanics in General Coordinates (Fridtjov Irgens)....Pages 231-269 Surface Geometry. Tensors in Riemannian Space R2 (Fridtjov Irgens)....Pages 271-308 Integral Theorems (Fridtjov Irgens)....Pages 309-338 Back Matter ....Pages 339-385