Differentiable Manifolds and Tensor Calculus: Foundations, Differential Forms, Curvature, and Applications in Mechanics, Relativity, and Optimization

Differentiable Manifolds and Tensor Calculus — Volume VIII provides a rigorous and progressive introduction to modern differential geometry and tensor calculus, developing the mathematical foundations that support numerous areas of pure mathematics, theoretical physics, engineering, and optimization.

This volume presents a systematic treatment of differentiable manifolds, smooth mappings, tangent and cotangent spaces, differential forms, exterior algebra, tensor fields, orientation, and integration on manifolds. It also explores affine connections, covariant differentiation, parallel transport, Christoffel symbols, curvature, and the Generalized Stokes Theorem, providing a solid foundation for Riemannian geometry and its applications.

Throughout the book, formal definitions, rigorous proofs, worked examples, and geometric interpretations are combined to help readers understand advanced concepts while connecting theory with practical applications.

Inside this volume, you will learn:

• Foundations of differentiable manifolds.

• Tangent and cotangent spaces.

• Differential forms and exterior algebra.

• Tensor fields and tensor calculus.

• Integration on manifolds and the Generalized Stokes Theorem.

• Affine connections and covariant differentiation.

• Parallel transport and curvature.

• Fundamentals of Riemannian geometry.

• Applications to classical mechanics, general relativity, mathematical physics, and optimization on manifolds.

Designed for advanced undergraduate and graduate students, researchers, instructors, mathematicians, physicists, and engineers, this book offers a modern, systematic, and mathematically rigorous treatment of differential geometry and tensor calculus.

Part of the Advanced Topics in Applied Mathematics series, this volume continues the collection’s mission of providing comprehensive and rigorous references on contemporary mathematics and its scientific applications.