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| − | ==Course description==
| + | #REDIRECT[[MAT5143]] |
| − | Mathematical Physics tentative topics list. This course is aimed at physics majors who wish to deepen their understanding or mathematical methods used in physics.
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| − | ==Topics List==
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| − | {| class="wikitable sortable"
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| − | ! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
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| − | |Week 1
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| − | *
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| − | Complex Analysis Part I
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| − | * Definition and algebraic properties of complex numbers, Riemann Sphere, Holomorphic functions and conformal mappings
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| − | |Week 2
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| − | *
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| − | Complex Analysis Part II
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| − | *Integrals in the Complex Plane, Cauchy's theorem, Calculus of Residues
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| − | |Week 3
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| − | *
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| − | Complex Analysis Part III
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| − | * Harmonic functions and Poisson's formula
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| − | |Week 4
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| − | *
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| − | Tensor Calculus Basics I
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| − | * 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.
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| − | |Week 5
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| − | *
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| − | Tensor Caluclus Basics II
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| − | * Manifolds and coordinate transformations, vector fields, Riemannian geometry, covariant derivatives and Christoffel symbols
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| − | |Week 6
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| − | *
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| − | Applied Functional Analysis Part I
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| − | * Hilbert spaces and inner products, orthogonality and completeness.
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| − | |Week 7
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| − | *
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| − | Applied Functional Analysis Part II
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| − | * Operators in Hilbert spaces, eigenvalue problem, self-adjointness and spectral properties
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| − | |Week 8
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| − | *
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| − | Applied Functional Analysis Part III
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| − | * Examples of Hilbert spaces in quantum mechanics, standard examples such as potential wells and harmonic oscillator
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| − | |Week 9
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| − | Overview about ordinary differential equations I
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| − | * Systems of nonlinear/linear equations, basic existence and uniqueness theorems
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| − | |Week 10
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| − | Overview about ordinary differential equations II
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| − | * Phase-plane, linearization, stability, chaos
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| − | |Week 11
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| − | *
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| − | PDE's of Mathematical Physics
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| − | * Standard examples, qualitative properties, conservation laws
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| − | |Week 12
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| − | Introduction to Lie Groups and Symmetries I
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| − | * Definition of a Lie group and examples, commutators and Lie brackets, Lie algebras
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| − | |Week 13
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| − | *
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| − | Introduction to Lie Groups and Symmetries II
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| − | * Exponential maps, applications of Lie groups to differential equations, Noether's theorem
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| − | |Week 14
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| − | *
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| − | KdV equation, completely integrable systems
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| − | *
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| − | * Soliton solutions, infinite hierarchy of conservation laws
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| − | |}
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