Quantum Physics --
Analytical mechanics --
Newtonian mechanics --
Lagrangian formalism --
Hamiltonian formalism --
Canonical quantization --
Hilbert space, bras and kets --
Axioms of canonical quantization --
Heisenberg equation, Heisenberg picture and Schrodinger picture --
Wavefunction --
Harmonic oscillator --
Path integral quantization of a Bose particle --
Path integral quantization --
Imaginary time and partition function --
Time-ordered product and generating functional --
Harmonic oscillator --
Transition amplitude --
Partition function --
Path integral quantization of a Fermi particle --
Fermionic harmonic oscillator --
Calculus of Grassmann numbers --
Differentiation --
Integration --
Delta-function --
Gaussian integral --
Functional derivative --
Complex conjugation --
Coherent states and completeness relation --
Partition function of a fermionic oscillator --
Quantization of a scalar field --
Free scalar field --
Interacting scalar field --
Quantization of a Dirac field --
Gauge theories --
Abelian gauge theories --
Non-Abelian gauge theories --
Higgs fields --
Magnetic monopoles --
Dirac monopole --
The Wu-Yang monopole --
Charge quantization --
Instantons --
The (anti- )self-dual solution --
Mathematical Preliminaries --
Equivalence relation and equivalence class --
Vector spaces --
Vectors and vector spaces --
Linear maps, images and kernels --
Dual vector space --
Inner product and adjoint --
Tensors --
Topological spaces --
Continuous maps --
Neighbourhoods and Hausdorff spaces --
Closed set.

This book introduces several current mathematical methods to postgraduate students of theoretical physics. This is achieved by presenting applications of the mathematics to physics, high-energy physics, general relativity and condensed matter physics.