A Geometrical Introduction to Tensor Calculus

Jeroen Tromp

• Publisher: Princeton University Press
• Year: 2025
• Print ISBN: 9780691267982, 9780691267975
• eBook ISBN: 9780691269221
• Topic: Physical Sciences
• Subtopic: Calculus

An authoritative, self-contained introduction to geometrical tensor calculus for scientists and engineers

Tensors are widely used in physics and engineering to describe physical properties that have multiple dimensions and magnitudes. In recent years, they have become increasingly important for data analytics and machine learning, allowing for the representation and processing of data in neural networks and the modeling of complex relationships in multidimensional spaces. This incisive book provides a geometrical understanding of tensors and their calculus from the point of view of a physicist. With a wealth of examples presented in visually engaging boxes, it takes readers through all aspects of geometrical continuum mechanics and the field and dynamic equations of Einstein, Einstein-Cartan, and metric-affine theories of general relativity. A Geometrical Introduction to Tensor Calculus gives graduate students, advanced undergraduates, and researchers a powerful and mathematically elegant tool for comprehending the behavior and applications of tensors across an array of fields.

• Offers a physicist’s perspective on geometrical tensor calculus
• Includes dozens of examples that illustrate the geometrical use of tensors in continuum mechanics and general relativity
• Can serve as the basis for a course in tensor calculus for physicists and engineers
• Invaluable as a supplementary guide for anyone studying areas of physics that rely on tensor calculus, such as electrodynamics, geophysics, fluid and continuum mechanics, and general relativity

Table of contents

• Cover
• Contents
• List of Examples
• Preface
• 1. Introduction
• 2. Linear Spaces and Transformations
• 3. Differentiable Manifolds
• 4. Vectors and One-Forms
• 5. Tensors
• 6. Maps between Manifolds
• 7. Differentiation on Manifolds
• 8. Differential Forms
• Glossary
• Bibliography
• Author Index
• General Index