Vector analysis (Part 1):
Concepts within vector analysis and their physical applications: the nabla operator, integral theorems, and potential theory. Tensors with applications from, e.g., electrodynamics and continuum mechanics. Special vector fields and their importance within physical modelling. Modelling using vector analysis. Symmetry concepts with relation to basic group theory and their significance in physics.
Partial differential equations (Part 2):
Physical problems leading to different types of differential equations, e.g., the wave equation, Laplace’ equation, and Poisson’s equation. Numerical solutions to physical problems. Separation of variables in cartesian, cylinder, and spherical coordinates, resulting in special functions, e.g., Bessel functions, Legendre polynomials, and spherical harmonics. Basic theory and application of Green’s functions in physics. Variational calculus and physical modelling using energy principles. Relation between analytical methods and finite difference/element methods.
After completing the course, a student should be able to
Part 1 (TENA):
- Use vector analysis to describe and analyse physical systems
- Model and formulate basic problems in physics within, e.g., electromagnetism and fluid mechanics, using vector analysis
- Describe different physical situations where singular vector fields appear and use these to describe physical systems
- Apply tensor analysis on basic problems in physics within, e.g., solid mechanics
- Use symmetries and basic group theory to draw conclusions regarding physical systems
Part 2 (TENB+INLA):
- Formulate problems in terms of partial differential equations starting from basic physical questions
- Numerically model and solve physical problems described by partial differential equations
- Use expansion in eigenfunctions as a tool for solving problems that appear in, e.g., quantum mechanics and electromagnetism
- Define and in basic cases apply Green’s functions to physical problems, e.g., diffusion and wave propagation
- Analyse physical problems using variational principles and energy arguments
