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Starkovich Department of Physics Pacific Lutheran University Tacoma, WA, USA The Structures of Mathematical Physics An Introduction 1.1.1 Set Inclusion, Subsets and Set Equalityġ.1.2 The Algebra of Sets: Union, Intersection and Complementġ.2.2 Equivalence Relations and Quotient Setsġ.3.1 Injective, Surjective and Bijective Mapsġ.4 Cartesian Products of Sets and Projection Mapsġ.5 A Universal Construction for Quotient SetsĢ.4 Morphisms, and a Glance at Algebraic Topology and Categoriesģ.4.2 The Complex Field mathbbC and Hamilton's Search for Number TripletsĤ.2 Linear Independence, Basis Vectors and NormsĤ.2.3 Norms and Distance Functions on Vector SpacesĤ.3.1 Inner Products in mathbbR2 Over mathbbRĤ.3.2 Inner Products in Coordinate SpacesĤ.3.3 Inner Products on Complex and Real Function Spaces-Sesquilinear and Bilinear MapsĤ.4 Orthogonality, Normalization and Complete Sets of VectorsĤ.4.1 Gram-Schmidt Orthogonalization-Coordinate SpaceĤ.4.2 Orthonormalization in Function SpacesĤ.4.3 Gram-Schmidt Orthogonalization-Function SpaceĤ.5 Subspaces, Sums, and Products of Vector SpacesĤ.5.2 Unions, Sums and Direct Sums of Vector Spacesĥ.4 Subalgebras, Quotients and Sums of Algebrasĥ.4.1 Subalgebras, Algebra Ideals and Quotientsĥ.5 Associative Operator Algebras on Inner Product Spacesĥ.5.1 Definitions, Notations and Basic Operations with Matricesĥ.5.2 Linear Transformations, Images and Null Spacesĥ.5.3 Eigenvectors, Similarity Transformations and Diagonalization of Matrices in Real Spacesĥ.7 Unitary, Orthogonal and Hermitian TransformationsĦ Fundamental Concepts of General TopologyĦ.1 General Topology in a Geometric ContextĦ.4.1 Separated and Connected Sets and SpacesĦ.4.2 Separation Axioms and Metric SpacesĦ.5 Compactness, Continuity, Convergence and Completenessħ.1.1 Review of Single-Variable Differentiation and Directional Derivativesħ.1.2 Multi-variable Differentiation and the Jacobianħ.3 Antisymmetric Tensors and p-Forms in mathbbRnħ.5 Correspondences Between Exterior and Vector Calculus in mathbbR3ħ.6 Hamilton's Equations and Differential Formsħ.6.1 Lagrange's Equation, Legendre Transformations and Hamilton's Equationsħ.6.2 Hamiltonian Phase Space as a Symplectic Manifoldħ.7 Transformations of Vectors and Differential FormsĨ Aspects of Integration and Elements of Lie GroupsĨ.2 Line Integrals and the Integration of One-FormsĨ.3 Homotopy and the Cauchy Theorems of Complex AnalysisĨ.4 Integration of p-Forms and the Vector Integral Theorems