Appreciated for its balance between rigor and physical application

New chapters on algebras, representation of Clifford algebras and spinors, fiber bundles, and gauge theories

Includes historical notes and over 900 worked-out examples.

Includes supplementary material: [...]

Mathematical Preliminaries.- I Finite-Dimensional Vector Spaces.- 1 Vectors and Linear Maps.- 2 Algebras.- 3 Operator Algebra.- 4 Matrices.- 5 Spectral Decomposition.- II Infinite-Dimensional Vector Spaces.- 6 Hilbert Spaces.- 7 Classical Orthogonal Polynomials.- 8 Fourier Analysis.- III Complex Analysis.- 9 Complex Calculus.- 10 Calculus of Residues.- 11 Advanced Topics.- IV Differential Equations.- 12 Separation of Variables in Spherical Coordinates.- 13 Second-Order Linear Differential Equations.- 14 Complex Analysis of SOLDEs.- 15 Integral Transforms and Differential Equations.- V Operators on Hilbert Spaces.- 16 Introductory Operator Theory.- 17 Integral Equations.- 18 Sturm-Liouville Systems.- VI Green's Functions.- 19 Green's Functions in One Dimension.- 20 Multidimensional Green's Functions: Formalism.- 21 Multidimensional Green's Functions: Applications.- VII Groups and Their Representations.- 22 Group Theory.- 23 Representation of Groups.- 24 Representations of the Symmetric Group.- VIII Tensors and Manifolds.- 25 Tensors.- 26 Clifford Algebras.- 27 Analysis of Tensors.- IX Lie Groups and Their Applications.- 28 Lie Groups and Lie Algebras.- 28.2 An Outline of Lie Algebra Theory.- 29 Representation of Lie Groups and Lie Algebras.- 30 Representation of Clifford Algebras.- 31 Lie Groups and Differential Equations.- 32 Calculus of Variations, Symmetries, and Conservation Laws.- X Fiber Bundles.- 33 Fiber Bundles and Connections.- 34 Gauge Theories.- 35 Differential Geometry.- 36 Riemannian Geometry.