A Gentle Introduction to Tensors: From Linear Algebra to Tensor Calculus, Geometry, and Physics

A Gentle Introduction to Tensors is a clear, example-driven guide to tensor analysis and tensor calculus. It is written for readers who want to understand not only the formulas, but also the geometric meaning behind them.

The book begins with the foundations: vectors, covectors, bases, dual spaces, metrics, tensor products, tensor types, and index notation. It then develops tensor fields, coordinate transformations, manifolds, pullbacks, pushforwards, covariant derivatives, curvature, Lie derivatives, embedded curves and surfaces, differential forms, integration, and applications in mechanics and physics.

Designed for self-study, the book emphasizes step-by-step derivations, consistent notation, carefully placed examples, and remarks that address common sources of confusion. Vectors and vector-like objects are written in bold, upper and lower indices are used consistently, and geometric ideas are connected to their coordinate formulas throughout.

Topics include:

Vector spaces, dual spaces, covectors, and metrics Tensor products, tensor types, and transformation laws Scalar, vector, covector, and tensor fields Manifolds, tangent spaces, and tensor bundles Pullbacks, pushforwards, and coordinate transformations Connections, Christoffel symbols, covariant derivatives, and geodesics Curvature, torsion, Lie derivatives, and differential forms Surface geometry, stress, strain, elasticity, fluids, relativity, and gravitation Whether you are studying mathematics, physics, engineering, continuum mechanics, relativity, or differential geometry, this book provides a gentle but systematic path from linear algebra to modern tensor calculus.