Difference between revisions of "MAT4143"

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(Created page with "==Course description== Mathematical Physics tentative topics list. This course is aimed at physics majors who wish to deepen their understanding or mathematical methods used i...")
 
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Introduction and classification of PDE, Calculus review
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Complex Analysis Part I
 
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Multivariable Calculus, Chain Rule
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* Definition of a PDE as a relation between partial derivatives of an unknown function. Classification of PDE according to order - linear/nonlinear/quasilinear
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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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Applied examples of PDE
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Complex Analysis Part II
 
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Multivariable Calculus, Chain Rule
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* Origin and background of common PDE's: heat equation, wave equation, transport equation, etc.
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Integrals in the Complex Plane, Cauchy's theorem, Calculus of Residues
 
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|Week 3
 
|Week 3
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The method of characteristics for first-order quasilinear equations
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Complex Analysis Part III
 
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Multivariable Calculus, Chain Rule
 
Multivariable Calculus, Chain Rule
 
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* Solving quasilinear first-order equations using the method of characteristics
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* Harmonic functions and Poisson's formula
 
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|Week 4
 
|Week 4
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The method of characteristics for first-order fully nonlinear equations
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Tensor Calculus Basics I
 
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Multivariable Calculus, Chain Rule
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* Solving fully nonlinear first-order equations (e.g. the Eikonal equation) using the method of characteristics
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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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Heat and wave equation on the whole real line
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Tensor Caluclus Basics II
 
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Differentiation of integrals with respect to a parameter, integration by parts
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* Fundamental solution of the heat equation, D'Alembert's formula for the wave equation
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* Manifolds and coordinate transformations, vector fields, Riemannian geometry, covariant derivatives and Christoffel symbols
 
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|Week 6
 
|Week 6
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Initial-boundary value problem for heat and wave equation I
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Applied Functional Analysis Part I
 
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Partial derivatives, chain rule
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* Separation of variables method for heat and wave equation
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* Hilbert spaces and inner products, orthogonality and completeness.
 
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|Week 7
 
|Week 7
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Initial-boundary value problem for heat and wave equation II, introduction to Fourier series
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Applied Functional Analysis Part II
 
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Partial derivatives, chain rule
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* Forming more general solutions out of infinite superposition of basic solutions
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* Operators in Hilbert spaces, eigenvalue problem, self-adjointness and spectral properties
 
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|Week 8
 
|Week 8
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Introduction to Fourier series
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Applied Functional Analysis Part III
 
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Infinite series
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* Orthonormal systems of functions, spectral method for the wave and heat equation
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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
 
|Week 9
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Schroedinger equation
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Overview about ordinary differential equations I
 
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Complex numbers
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* Basic properties of Schroedinger equation, particle in a potential well 
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* systems of nonlinear/linear equations, basic existence and uniqueness theorems 
 
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|Week 10
 
|Week 10
 
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Complex numbers
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Qualitative properties of PDE's
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Overview about ordinary differential equations II
 
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Differentiation of integrals with respect to parameter  
 
Differentiation of integrals with respect to parameter  
 
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* Uniqueness of solutions, finite and infinite propagation speed for wave and heat equation
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* phase-plane, linearization, stability, chaos
 
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|Week 11
 
|Week 11
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Introduction to numerical methods for PDE (optional)
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PDE's of Mathematical Physics
 
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Derivatives, Calculus
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* Basic finite difference schemes for first-order quasilinear equations, CFL condition
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* standard examples, qualitative properties, conservation laws
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|Week 12
 
|Week 12
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Matrices, Linear Algebra
 
Matrices, Linear Algebra
 
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Introduction to the Laplace and Poisson equation
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Introduction to Lie Groups and Symmetries I
 
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* Solving the Laplace equation on the whole space and on a simple bounded region (square, disc)
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* Definition of a Lie group and examples, commutators and Lie brackets, Lie algebras
 
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Introduction to the Calculus of Variations
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Introduction to Lie Groups and Symmetries II
 
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Differentiation of an integral with respect to a parameter, parametric surfaces
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* Compute the variational derivative of a functional 
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* Exponential maps, applications of Lie groups to differential equations, Noether's theorem
 
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|Week 14
 
|Week 14
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Review, advanced topics
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KdV equation, completely integrable systems
 
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Revision as of 07:35, 23 March 2023

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

Multivariable Calculus, Chain Rule

  • 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

Matrices, Linear Algebra

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

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