Going beyond standard textbooks, this book presents mathematical topics with basic physics and also selected quantum physics topics with underlying mathematics. Aimed at first-year graduate students, it is concise and discusses selected topics in full without omitting any steps. It covers the mathematical tools needed throughout common graduate level courses in physics and features around 450 end-of-chapter problems, with solutions available to lecturers from the Wiley website. From the contents: Vectors Tensors and Matrices Hamiltonian Mechanics Coupled Oscillator and Normal Modes Stretched String Vector Calculus and the del Operator Electromagnetic Waves Fluid Dynamics Irreversible Processes The Entropy Thermodynamic Inequalities Probability, Statistics and Density Liouville Equation Generalized Vectors and Linear Operators Quantum Mechanics Fourier Series and Transforms Angular Momentum Spin Angular Momentum Time-dependent Perturbation Theory Laplace Transformation Quantum Harmonic Oscillator Permutation Group Quantum Statistics The Free-Electron Model Bose-Einstein Condensation Magnetic Susceptibility Theory of Variations Second Quantization Quantum Statistics of Composites Superconductivity Complex Numbers Analyticity and Cauchy-Reimann Equations Cauchy's Fundamental Theorem Laurent Series Multivalued Functions Residue Theorem and its Applications.