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Semester 2

Methods of Theoretical Physics (PHYS10105)

Subject

Physics and Astronomy

College

SCE

Credits

20

Normal Year Taken

3

Delivery Session Year

2023/2024

Pre-requisites

Course Summary

The first part of the course, Complex Calculus, provides an introduction to complex numbers and analytic functions. We discuss the general properties of such functions, and develop the machinery of differential and integral calculus of complex functions. A special emphasis is put on using the Residue Theorem to evaluate real integrals. We introduce the Fourier and Laplace transforms and demonstrate how they can be used in solving linear PDE's.The second part of the course, Vectors, Tensors, and Continuum Mechanics, provides an introduction to continuum mechanics. It is used as a natural way to formally introduce the following concepts: vectors, bases, matrices determinants and the index notation, the general theory of Cartesian tensors, and rotation and reflection symmetries. These concepts are then applied in formulating a theory of elastic solids (linear elasticity) and viscous fluids (the Navier-Stokes equation) in terms of strain and stress tensors. These demonstrate how the resulting equations can be solved in various cases. We briefly introduce the concept of turbulence and describe its phenomenology and mechanism.

Course Description

Complex calculus* Complex numbers review (operations, properties and inequalities)* Functions: a brief introduction to mapping. Limits & Continuity. Differentiation. Cauchy-Riemann relations* Analytic Functions: definition & properties. Introduction to conformal mapping. Branchs in z, z 2 1.* Dirichlet problem using conformal mapping* Complex integration: contours. Examples (1/z). Some theorems.* Antiderivatives. Complex fundamental theorem of calculus. Closed contours.* Cauchy-Goursat theorem. Deformation theorem. Applications of CauchyGoursat theorem: Cauchy's Integral Formula. Liouville's Theorem. Fundamental theorem of Algebra. Examples.* Morera's theorem. Complex series. Taylor series. Analytic continuation.* Laurent Series. Examples. Zeroes & Singularities. Residues. Examples.* Residue theorem. Jordan's Lemma, Indentation Lemma. Examples.* Branches and branch cuts. Improper integrals. Kramers-Kronig relations.* Fourier and Laplace transforms. Solving ODE's with Fourier and Laplace transforms. Vectors, Tensors, and Continuum Mechanics.* Vectors, bases, Einstein summation convention, the delta and epsilon symbols, matrices, determinants.* Rotations of bases, composition of two rotations, reflections, projection operators, passive and active transformations, the SO(3) symmetry group.* Cartesian tensors: - definition/transformation properties and rank - quotient theorem, pseudotensors, the delta and epsilon symbols as tensors - isotropic (pseudo)tensors* Taylor's theorem: the one- and three-dimensional cases* Linear Elasticity - the strain tensor, stretching and shear - the stress tensor, and some properties - elastic deformations of solid bodies, generalised Hooke's Law, isotropic media (and the various parameterisations for constants, ie Lam´e constants; Young's modulus and Poisson's ratio; bulk and shear modulus)* Fluid Mechanics - The Navier-Stokes equation. Incompressibility. The Reynolds number. Simple exact solutions. - Stokes equation. Drag on a sphere. Method of singularities. Multipole expansion. - Advection-diffusion equation. - Phenomenon of the transition to turbulence. Kolmogorov-Richardson cascade (qualitatively!). - Linear viscoelasticity

Assessment Information

Written Exam 80%, Coursework 20%, Practical Exam 0%

Additional Assessment Information

Coursework: 20%Examination: 80%

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