Advanced Mathematical Techniques of Physics
Full course description
The Italian physicist Galileo already remarked in the 16th century that “the book of nature is written in mathematics”. In the centuries and development of physics since, this has become true to the point that advanced mathematics is inseparably entwined with physics. Indeed, for a professional career in physics research, a rigorous training in advanced mathematical techniques is a necessity. In this course, we will provide a number of the most important topics needed in active research in physics. Topics include integral transforms, techniques of solving partial differential equations, finding particular solutions by Green’s function techniques, complete sets, Fourier analysis and its relationship to data-analysis and quantum mechanics, and variational calculus. In all cases, the mathematics will be practiced in the context of real-life examples of fundamental theories of physics, such as quantum field theory and relativity.
To provide students training and fluency in the following mathematical techniques of physics:
- Fourier series
Theorem of Riemann-Lebesque, Dirichlet, Jordan, theorem of Abel-Poison, Cauchy series.
- Fourier integrals
Banach space, Hilbert space; Schwarz inequality, Parseval, connection to Heisenberg uncertainty relation.
- Bessel and Legendre functions
Complete sets of orthonormal functions, Euler’s constant, Fourier-Bessel series, Hankel transform.
- Laplace transformation
Complex function theory, s-plane, initial value problems for (partial) differential equations.
- Variational methods
First order PDEs, second order PDEs; Lagrangian, Euler-Lagrange equation.
- Green’s functions
Solving of potential equations, Dirichlet and von Neumann problems, Wronskian determinant.
- MAT2004 Linear Algebra
- MAT2009 Multivariable Calculus
- Arfken & Weber & Harris: Mathematical Method for Physicists, 7th edition or higher. ISBN 978-0-12-384654-9.
- lecture notes by Jo van den Brand & Gideon Koekoek.