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Mathematical Physics - MAT4143/5153

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. This course is also suitable for Mathematics majors, who can deepen their understanding of applications and proofs of major theorems in Functional Analysis.


  • Needham, T. (1997). Visual Complex Analysis. United Kingdom: Clarendon Press.
  • Grimshaw, R. (1993). Nonlinear Ordinary Differential Equations (1st ed.). Routledge.
  • R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. II: Partial Differential Equations, Wiley Classics Library, John Wiley & Sons Inc., New York, 1989.
  • Bender, C. and Orszag, S. Advanced Mathematical Methods for Scientists and Engineers. Mc-Graw Hill.
  • Kreyszig, E. (1989). Introductory Functional Analysis with Applications. Wiley.
  • Methods of Applied Mathematics. Todd Arbogast and Jerry L. Bona. Department of Mathematics, and Institute for Computational Engineering and Sciences, University of Texas at Austin, 2008.
  • P. J. Olver , Applications of Lie Groups to Differential Equations, Second Edition, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1993.
  • Rutherford Aris, Vectors, Tensors and the Basic Equations of Fluid Mechanics. Dover Publications

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