Course description
Mathematical Physics tentative topics list. This course is aimed at physics majors who wish to deepen their understanding or mathematical methods used in physics.
Topics List
| Date |
Sections |
Topics |
Prerequisite Skills |
Student Learning Outcomes
|
| Week 1
|
|
Complex Analysis Part I
|
|
- Definition and algebraic properties of complex numbers, Riemann Sphere, Holomorphic functions and conformal mappings
|
| Week 2
|
|
Complex Analysis Part II
|
|
- Integrals in the Complex Plane, Cauchy's theorem, Calculus of Residues
|
| Week 3
|
|
Complex Analysis Part III
|
|
- Harmonic functions and Poisson's formula
|
| Week 4
|
|
Tensor Calculus Basics I
|
|
- Using indices in three-dimensional cartesian vector analysis, deriving vector identities using index calculus, divergence, grad and curl in index notation, divergence and Stokes' theorem.
|
| Week 5
|
|
Tensor Caluclus Basics II
|
|
- Manifolds and coordinate transformations, vector fields, Riemannian geometry, covariant derivatives and Christoffel symbols
|
| Week 6
|
|
Applied Functional Analysis Part I
|
|
- Hilbert spaces and inner products, orthogonality and completeness.
|
| Week 7
|
|
Applied Functional Analysis Part II
|
|
- Operators in Hilbert spaces, eigenvalue problem, self-adjointness and spectral properties
|
| Week 8
|
|
Applied Functional Analysis Part III
|
|
- Examples of Hilbert spaces in quantum mechanics, standard examples such as potential wells and harmonic oscillator
|
| Week 9
|
|
Overview about ordinary differential equations I
|
|
- Systems of nonlinear/linear equations, basic existence and uniqueness theorems
|
| Week 10
|
|
Overview about ordinary differential equations II
|
|
- Phase-plane, linearization, stability, chaos
|
| Week 11
|
|
PDE's of Mathematical Physics
|
|
- Standard examples, qualitative properties, conservation laws
|
| Week 12
|
|
Introduction to Lie Groups and Symmetries I
|
|
- Definition of a Lie group and examples, commutators and Lie brackets, Lie algebras
|
| Week 13
|
|
Introduction to Lie Groups and Symmetries II
|
|
- Exponential maps, applications of Lie groups to differential equations, Noether's theorem
|
| Week 14
|
|
KdV equation, completely integrable systems
|
|
- Soliton solutions, infinite hierarchy of conservation laws
|
