From Vectors to Tensors

It is true that there exist many books dedicated to linear algebra and some­ what fewer to multilinear algebra, written in several languages, and perhaps one can think that no more books are needed. However, it is also true that in algebra many new results are continuously appearing, different points of view can be used to see the mathematical objects and their associated structures, and different orientations can be selected to present the material, and all of them deserve publication. Under the leadership of Juan Ramon Ruiz-Tolosa, Professor of multilin­ ear algebra, and the collaboration of Enrique Castillo, Professor of applied mathematics, both teaching at an engineering school in Santander, a tensor textbook has been born, written from a practical point of view and free from the esoteric language typical of treatises written by algebraists, who are not interested in descending to numerical details. The balance between follow­ ing this line and keeping the rigor of classical theoretical treatises has been maintained throughout this book. The book assumes a certain knowledge of linear algebra, and is intended as a textbook for graduate and postgraduate students and also as a consultation book. It is addressed to mathematicians, physicists, engineers, and applied scientists with a practical orientation who are looking for powerful tensor tools to solve their problems.

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.

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.