This book (published in two volumes) provides a comprehensive and coherent account of Tensor theory. It shows that the three principal formulations of tensors—through transformation laws, as multilinear mappings, and as elements of tensor product spaces—are mathematically equivalent formulations of the same underlying multilinear structure.

Rather than treating these viewpoints separately, the book reveals their deep unity within a single structural framework that connects multilinear algebra, symmetry, geometry, physics, and computation. From fundamentals to advanced applications, it guides the reader from elementary concepts to sophisticated topics while developing the structural insight that establishes tensor theory as a unifying language of modern science.

Some Salient Features of the Book:

• Unifying the Principal Definitions of Tensors — Systematically develops and reconciles tensors defined through transformation laws, multilinear mappings, and tensor product spaces.
• One Mathematical Language—Many Disciplines — Presents tensor theory as a common language connecting linear algebra, differential geometry, continuum mechanics, relativity, representation theory, machine learning, and data science.
• From Foundations to Modern Tensor Theory — Builds from first principles in multilinear algebra, manifolds, and tensor fields to advanced topics including Christoffel symbols, Lie derivatives, differential forms, curvature, Young tableaux, and tensor decompositions.
• Learning Through Theory and Computation — Combines theoretical development with worked examples, conceptual discussions, review questions, exercises, including applications of modern tensor software.
• Coherent, Rigorous, and Comprehensive — Integrates algebraic, geometric, analytical, physical, and computational perspectives within a single structural framework.