The book provides a bridge from courses in general physics to the intermediate-level courses in classical mechanics, electrodynamics and quantum mechanics. The author bases the mathematical discussions on specific physical problems to provide a basis for developing mathematical intuition.

1 A Review.- 1.1 Electrostatics.- 1.2 Electric Current.- 1.3 Magnetic Flux.- 1.4 Simple Harmonic Motion.- 1.5 A Rigid Rotator.- 1.6 Exercises.- 2 Vectors.- 2.1 Representations of Vectors.- 2.2 The Scalar Product of Two Vectors.- 2.3 The Vector Product of Two Vectors.- 2.4 Exercises.- 3 Vector Calculus.- 3.1 Partial Derivatives.- 3.2 A Vector Differential Operator.- 3.3 Components of the Gradient.- 3.4 Flux.- 3.5 Exercises.- 4 Complex Numbers.- 4.1 Why Study Complex Numbers?.- 4.2 Roots of a Complex Number.- 4.3 Exercises.- 5 Differential Equations.- 5.1 Infinite Series.- 5.2 Analytic Functions.- 5.3 The Classical Harmonic Oscillator.- 5.4 Boundary Conditions.- 5.5 Polynomial Solutions.- 5.6 Elementary Functions.- 5.7 Singularities.- 5.8 Exercises.- 6 Partial Differential Equations.- 6.1 The Method of Separation of Variables.- 6.2 The Quantum Harmonic Oscillator.- 6.3 A Conducting Sphere in an Electric Field.- 6.4 The Schrödinger Equation for a Central Field.- 6.5 Exercises.- 7 Eigenvalue Problems.- 7.1 Boundary Value Problems.- 7.2 A Vibrating Drumhead.- 7.3 A Particle in a One-Dimensional Box.- 7.4 Exercises.- 8 Orthogonal Functions.- 8.1 The Failure of Classical Physics.- 8.2 Observables and Their Measurement.- 8.3 Mathematical Operators.- 8.4 Eigenvalue Equations.- 8.5 The Quantum Harmonic Oscillator.- 8.6 Sturm-Liouville Theory.- 8.7 The Dirac Delta Function.- 8.8 Fourier Integrals.- 8.9 Fourier Series.- 8.10 Periodic Functions.- 8.11 Exercises.- 9 Matrix Formulation of the Eigenvalue Problem.- 9.1 Reformulating the Eigenvalue Problem.- 9.2 Systems of Linear Equations.- 9.3 Back to the Eigenvalue Problem.- 9.4 Coupled Harmonic Oscillators.- 9.5 A Rotating Rigid Body.- 9.6 Exercises.- 10 Variational Principles.- 10.1 Fermat's Principle.- 10.2 Another VariationalCalculation.- 10.3 The Euler-Lagrange Equation.- 10.4 Exercises.- Appendix A Vector Relations.- A.1 Vector Identities.- A.2 Integral Theorems.- A.3 The Functions of Vector Calculus.- Appendix B Fundamental Equations of Physics.- B.1 Poisson's Equation.- B.2 Laplace's Equation.- B.3 Maxwell's Equations.- B.4 Time-Dependent Schrödinger Equation.- Appendix C Some Useful Integrals and Sums.- C.1 Integrals.- C.2 Sums.- Appendix D Algebraic Equations.- D.1 Quadratic Equation.- D.2 Cubic Equation.- References.