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==Course description==
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#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.
 
==Topics List==
 
{| class="wikitable sortable"
 
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
 
|-
 
|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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*
 
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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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|}
 

Latest revision as of 09:14, 24 March 2023

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