Description

This unique book complements traditional textbooks by providing a visual yet rigorous survey of the mathematics used in theoretical physics beyond that typically covered in undergraduate math and physics courses. The exposition is pedagogical but compact, and the emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions, alternative notations and jargon, and relevant facts and theorems. Special attention is given to detailed figures and geometric viewpoints. Certain topics which are well covered in textbooks, such as historical motivations, proofs and derivations, and tools for practical calculations, are avoided. The primary physical models targeted are general relativity, spinors, and gauge theories, with notable chapters on Riemannian geometry, Clifford algebras, and fiber bundles. **Contents:** Mathematical Structures Abstract Algebra Vector Algebras Topological Spaces Algebraic Topology Manifolds Lie Groups Clifford Groups Riemannian Manifolds Fiber Bundles Categories and Functors **Readership:** Students in mathematics and physics who want to explore a level deeper into actual mathematical content. Keywords: Gauge Theory;Spinor;Clifford Algebra;Homology;Homotopy;Differential Geometry;Lie Group;Lie Algebra;Pauli Matrix;Dirac Matrix;Riemannian Geometry;Parallel Transport;Ricci Curvature;Divergence;Ehresmann ConnectionReview: **Key Features:** The approach taken by this book to the subject material differs from other titles in that it is intuitive and visual yet also mathematically rigorous, allowing concise coverage of a large breadth of material and providing a cross-subject synthesis while at the same time serving as a useful reference The book is also unique in that it can be used in three ways: as the basis for a course, as a supporting text for related courses, and as a reference. These uses could apply to both undergraduate and graduate students. In addition, there is a substantial audience for the book among independent researchers, amateur physicists, and readers of popular science who want to explore a level deeper into actual mathematical content The treatments of Riemannian geometry, Clifford algebras, and fiber bundles are particularly notable, including detailed figures and geometric viewpoints that would seem to be novel to the literature