© 2005

From Vectors to Tensors


Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages I-XVI
  2. Basic Tensor Algebra

  3. Special Tensors

  4. Exterior Algebras

  5. Tensors over Linear Spaces with Inner Product

  6. Classic Tensors in Geometry and Mechanics

    1. Pages 581-658
  7. Back Matter
    Pages 659-674

About this book


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.


Mathematica Matrix Permutation Transformation algebra computer mechanics

Authors and affiliations

  1. 1.Depto. Matemática AplicadaUniversidad CantabriaSantanderSpain

About the authors

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:

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.

Bibliographic information

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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)