MAT5143

Mathematical Physics - MAT4143/5153

Course description

Mathematical Physics tentative topics list. This course is aimed at physics majors who wish to deepen their understanding of 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.

Catalog entry

Prerequisite: Calculus III MAT2214 and Differential Equations II MAT3623 with a letter grade of C- or better, or successful completion of at least three credits of equivalent courses.

Content: 1. Topics in Complex Analysis. 2. Differential Equations: Dynamical systems, nonlinearity & chaos. 3. Nonlinear Waves in PDEs: Continuous Systems, Hamiltonian formulation of plasmas and fluids, KdV eq., Nonlinear Schroedinger Eq., Sine/Klein-Gordon Equation(s). 4. Asymptotic Analysis methods, time-dependent/independent perturbation theory. 5. Functional Analysis in Mathematical Physics 6. Mathematical Formalism of PDEs for Physicists 7. Group Theory in Physics, Lie Algebras. 8. Tensor calculus: theory and applications. 3 Credit Hours

Textbooks:

• 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