MAT4143

From Department of Mathematics at UTSA
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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

Differentiation of integrals with respect to parameter

  • 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