From Vectors to Tensors: (Universitext)

This textbook, deals with tensors that are treated as vectors, and has a practical orientation. In addition to dealing with the classical topics of tensor books, new tensor concepts are introduced, such as the rotation of tensors, the transposer tensor, the eigentensors, the permutation tensor structure, etc. The book covers an existing gap between the classic theory of tensors and the possibility of solving tensor problems with a computer. In fact, the computational algebra is formulated in matrix form to facilitate its implementation on computers. For the first time tensor contraction is formulated in terms of matrix operations. A computer package , written in Mathematica, is available through Internet at: http://personales.unican.es/castie/tensors that complements the book. In summary, the book is not a standard book on tensors because of its orientation, the many novel contributions included in it, the careful notation and the stretching-condensing techniques used for most of the transformations used in the book.

Table of Contents:
Basic Tensor Algebra.- Tensor Spaces.- to Tensors.- Homogeneous Tensors.- Change-of-basis in Tensor Spaces.- Homogeneous Tensor Algebra: Tensor Homomorphisms.- Special Tensors.- Symmetric Homogeneous Tensors: Tensor Algebras.- Anti-symmetric Homogeneous Tensors, Tensor and Inner Product Algebras.- Pseudotensors; Modular, Relative or Weighted Tensors.- Exterior Algebras.- Exterior Algebras: Totally Anti-symmetric Homogeneous Tensor Algebras.- Mixed Exterior Algebras.- Tensors over Linear Spaces with Inner Product.- Euclidean Homogeneous Tensors.- Modular Tensors over En (IR) Euclidean Spaces.- Euclidean Exterior Algebra.- Classic Tensors in Geometry and Mechanics.- Affine Tensors.

About the Author :
Enrique Castillo is Professor of Applied Mathematics at the University of Cantabria in Santander (Spain). He is a Mathematician and a Civil engineer and Member of the Spanish Royal Academy of Engineering. He has taught at several other universities in the Europe and America. The author/coauthor of eleven other books in English and fourteen in Spanish, and more than 300 papers in journals and Congresses. More information can be found at his Web site: http://personales.unican.es/castie//. Juan R. Ruiz-Tolosa is an Industrial and Civil Engineer and has been Professor of Algebra, Tensors, Topology, Differential Geometry and Calculus at the Civil Engineering School, University of Cantabria for 30 years. His field of research includes Number Theory, Euclidean Geometry, Elliptic Integrals, Algebraic Roots of Equations, etc.

Review :
From the reviews: "The book bridges a gap between the classical theory of tensors and the possibility of solving tensor problems with a computer. … For the first time, tensor contraction is formulated in terms of matrix operations. … Addressed primarily to students of engineering, applied mathematics and mathematical physics, this unconventional approach to tensors is original for its orientation, its novel concepts, the choice of notation and the stretching-condensing techniques applied to most of the transformations used." (Bulletin Bibliographique, 51:1-2, 2005) "In this textbook, the authors present a new and, to some extent, unconventional approach to tensors … . This volume gives a thorough treatment of the topic of sensors. … the exposition of the book is a very transparent one and a pleasure to read. … Primary addressees of the book are (graduate) students in applied mathematics, mathematical physics, and engineering sciences as well. Moreover, the book can be recommended without reservations to instructors … ." (A. R. Kräuter, Internationale Mathematische Nachrichten, Issue 203, 2006)