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1 Mathematical Preliminaries Part I Finite-Dimensional Vector Spaces 2 Vectors and Linear Maps 3 Algebras 4 Operator Algebra 5 Matrices 6 Spectral Decomposition Part II Infinite-Dimensional Vector Spaces 7 Hilbert Spaces 8 Classical Orthogonal Polynomials Fourier Analysis Part III Complex Analysis 10 Complex Calculus 11 Calculus of Residues 12 Advanced Topics Part IV Differential Equations 13 Separation of Variables in Spherical Coordinates 14 Second-Order Linear Differential Equations 15 Complex Analysis of SOLDEs 16 Integral Transforms and Differential Equations Part V Operators on Hilbert Spaces 17 Introductory Operator Theory 18 Integral Equations 19 Sturm-Liouville Systems Part VI Green's Functions 20 Green's Functions in One Dimension 21 Multidimensional Green's Functions: Formalism 22 Multidimensional Green's Functions: Applications Part VII Groups and Their Representations 23 Group Theory 24 Representation of Groups 25 Representations of the Symmetric Group Part VIII Tensors and Manifolds 26 Tensors 27 Clifford Algebras 28 Analysis of Tensors Part IX Lie Groups and Their Applications 29 Lie Groups and Lie Algebras 30 Representation of Lie Groups and Lie Algebras 31 Representation of Clifford Algebras 32 Lie Groups and Differential Equations 33 Calculus of Variations, Symmetries, and Conservation Laws Part X Fiber Bundles 34 Fiber Bundles and Connections 35 Gauge Theories 36 Differential Geometry 37 Riemannian Geometry References Index