1b45intro

   PART 49

---------------------------------------------------------

 Please note: This site is NOT an official course Website

----------------------------------------------------------

  1B45 MATHEMATICAL METHODS 1

  Introduction 

The 1B45 Mathematics course attempts to accommodate the changes in mathematical preparation afforded by A levels. These changes have become evident in the difficulties some of our students encounter in our physics and astronomy courses. Many of these difficulties stem from a loss of fluency in mathematics and associated manipulative skills. In physics and astronomy laws are expressed mathematically, and mathematics is also needed to derive the consequences and interrelationships of these laws.

I (Prof. Tegid Wyn Jones) urge you to accept the challenge of mastering this course because effort now will make it much easier to cope with the physics and astronomy courses. The challenge is as much to me as it is to you!!!

How are we to meet this challenge?

Come to all of the 40 lecture periods and take a good set of notes. You must participate in the lectures and taking good notes is the start of this. I intend to develop the course in a reasonably logical and rigorous manner and to do many, many examples. Over the years I have taught eight of the first and second year core courses and a good fraction of the examples are drawn from these courses. The emphasis will be on the mathematics you will need for your physics and astronomy courses, and on acquiring fluency in mathematical manipulation. In four weeks or so the first tranche of my printed lecture notes will become available. In no way, however, can these be a substitute for you writing down your own set of notes as we go along. You really should go through your notes after every lecture, rewriting them as necessary, and going through everything I wrote down on the board.

Do the weekly problem sheets, normally given out in the Thursday lectures and collected a week later. Three of these problem sheets will be devoted to simple manipulative skills in differentiation, integration and vectors. You will be required to achieve more than $80 \%$ in each of the three, repeating them as necessary before you can complete the course. Solutions to the other problem sheets are posted on the web a few days after your solutions have been collected.

Starting in four weeks time there will be six weekly two-hourly problem classes where groups of 10 - 12 students attempt a problem sheet. Demonstrators will circulate around to give help as needed. Again, don't be afraid to ask for help.

In the last week of this term there will be a class exam to tell me and you how well the course has gone. There is, of course, the final 1B45 exmination at the end of the academic year.

The book Mathematical Methods for Physics and Engineering , by K. F. Riley, M. P. Hobson and S. J. Bence, (Second Edition - Oxford), available from Waterstones, is a good text book for 1B45 and the follow on course 1B46. However it is not an essential purchase.

Finally, do not underestimate this course. If you think you are familiar with most of the material, particularly early on in the course, persevere, there will be new material, and I will have some quite challenging problems to keep the most able of you happy!! This course has been extremely successful with a good response from the students and a most satisfactory examination performance. Remember that a good effort now will make your physics and astronomy courses so much easier and much more fun!!

Prof. Tegid Wyn Jones. D27 physics Department - twj@hep.ucl.ac.uk

Aims

• provide the mathematical foundations required for all the first semester and some of the second semester courses in the first year of the physics and astronomy programmes;
• prepare students for the second semester follow-on mathematics course PHYS1B46;
• give students practice in mathematical manipulation and problem solving.

Objectives

After completing this half-unit course, the student should be able to:

• solve simultaneous and quadratic equations with examples taken from physical situations;
• sum arithmetic, geometric and other simple series;
• appreciate the relation between powers, exponentials and logarithms and the more general concept of the inverse function in terms of a graphical approach;
• derive the values of the trigonometric functions for special angles;
• understand the relation between the hyperbolic and exponential functions;
• differentiate simple functions and apply the product and chain rules to evaluate the differentials of more complicated functions;
• find the positions of the stationary points of a function of a single variable and determine their nature;
• understand integration as the reverse of differentiation and to use this to evaluate integrals almost 'by inspection';
• evaluate integrals by using substitutions and integration by parts;
• understand a definite integral as an area under a curve;
• evaluate the Gaussian, Feynman, Gamma and Breit Wigner (Lorentzian) integrals and generate further definite integrals by differentiation w. r. t. a parameter;
• differentiate up to second order a function of 2 or 3 variables and be able to test when an expression is a perfect differential;
• change the independent variables by using the chain rule and, in particular, work with polar coordinates;
• find the stationary points of a function of two independent variables and to determine their nature;
• find the stationary points of function of two or more variables subject to constraints (Lagrange multipliers);
• manipulate real three-dimensional vectors, evaluate scalar and vector products, find the angle between two vectors in terms of components;
• construct vector equations for lines and planes and find the angles between them;
• express vectors, including velocity and acceleration, in terms of basis vectors in polar coordinate systems;
• understand the concept of convergence for an infinite series, be able to apply simple tests to investigate it;
• expand an arbitrary function of a single variable as a power series (Maclaurin and Taylor), make numerical estimates, and be able to apply L'Hopitals rule to evaluate the ratio of two singular expressions;
• represent complex numbers in Cartesian and polar form on an Argand diagram.
• perform algebraic manipulations with complex numbers, including finding powers and roots;
• apply de Moivres theorem to derive trigonometric identities and understand the relation between trigonometric and hyperbolic functions through the use of complex arguments. components;

Course Outline / Syllabus

40 lectures plus 6 problems classes.

• Elementary Functions (mainly revision)
  Manipulation of algebraic equations, powers, exponentials and logarithms, inverse functions, trigonometric functions, sine, cosine and tangent for special angles, hyperbolic functions.
• Vectors
  Definition, addition, subtraction, scalar and vector multiplication. Vector and scalar triple products, vector equations (Third order determinants only very briefly).
• Differentiation (mainly revision) Definition, product rule, function of a function rule, implicit functions, logarithmic derivative, parametric differentiation, maxima and minima.
• Integration (mainly revision)
  Integration as converse of differentiation, changing variables, integration by parts, partial fractions, trigonometric and other substitutions, definite integral, integral as the area under a curve, trapezium rule, integral of odd and even functions. The Gaussian, Feynman, Breit Wigner (Lorentzian) integrals. Further definite integrals obtained by differentiation w. r. t. a parameter.
• Partial Differentiation
  definition, surface representation of functions of two variables, total differentials, chain rule, change of variables, second order derivatives. Maxima, minima and saddle points for functions of two variables. Stationary values of functions subject to constraints.
• More Vectors
  Vector geometry - straight lines and planes. Vector differentiation, vectors in plane polar, cylindrical, and spherical polar coordinates.
• Series
  Summation of arithmetic, geometric and other simple series. Sequences and series, convergence of infinite series. Power series, radius of convergence, simple examples including the binomial series. Taylor and Maclaurin series, L'Hopital's rule.

• Complex Numbers
  Representation, addition, subtraction, multiplication, division, Cartesian, polar exponential forms, De Moivre's theorem, powers and roots, complex equations.

Problem Classes/Exams

Starting in three weeks time there will be six weekly two-hourly problem classes where groups of 10 - 12 students attempt a problem sheet. Demonstrators will circulate around to give help as needed. Again, don't be afraid to ask for help.

In the last week of this term there will be a class exam to tell me and you how well the course has gone. There is, of course, the final 1B45 exmination at the end of the academic year.

---------------------------------------------------------

   PHYS2B21 - Mathematical Methods in Physics and Astronomy

Prerequisites

In order to take this course, students should have studied the material in the precursor PHYS1B21 mathematics course.

Aims

This course aims to:

• provide the remaining mathematical foundations for all the second and third year compulsory Physics and Astronomy courses;
• prepare students for the second semester Mathematics option MATHB8;
• give students practice in mathematical manipulation and problem solving at second-year level.

Objectives

The PHYS1B21 and PHYS2B21 syllabuses together cover all the mathematical requirements of the compulsory Physics and Astronomy courses. The five major areas treated in 2B21 are of special relevance to Quantum Mechanics and Electromagnetism, and the applications of these subjects to many other topics, including plasma, condensed matter, atomic, molecular, and particle physics. At the end of each section of the course, students should be able to appreciate when to use a particular technique to solve a given problem and be able to carry out the relevant calculations. Specifically,

For Matrices, students should be able to:

• manipulate vectors in a complex n-dimensional space and represent linear transformations in this space by matrices;
• perform matrix algebra, including multiplication and inversion, using a wide variety of matrices including unitary, Hermitian, and orthogonal matrices;
• solve linear simultaneous equations through the use of matrices and determinants;
• find the eigenvalues and eigenvectors of a matrix up to 3x3;
• diagonalise a matrix up to 3x3 and apply the technique to physical and mathematical problems.

For Differential Equations, students should be able to:

• solve a variety of second order linear partial differential equations, including the Laplace and wave equations, by the method of separation of variables, using both Cartesian and polar coordinates, and impose boundary conditions;
• know what type of series solution to seek when solving an ordinary second-order linear and homogeneous differential equation by examining its singularity structure and derive a solution consistent with the boundary conditions;
• investigate the convergence of the series solution and understand how and why polynomial solutions arise.

For Legendre Functions, students should be able to:

• solve the Legendre differential equation by series method and find the conditions necessary for a polynomial solution;
• derive and apply the generating function and recurrence relations for Legendre polynomials;
• employ the orthogonality relation of Legendre polynomials to develop functions as series of such polynomials;
• manipulate spherical harmonics up to l=2.

In Fourier Analysis, students should be able to:

• derive the formulae for the expansion coefficients for real and complex Fourier series;
• make analyses using sinusoidal and complex functions for both periodic and non-periodic functions and be aware of possible convergence problems;
• understand the possible complications when differentiating or integrating Fourier series;
• use Parseval's identity to deduce the values of some infinite series;
• derive the formulae for the expansion coefficients for real and complex Fourier transforms;
• perform Fourier transforms of a variety of functions and derive and use Dirac delta functions;
• apply the convolution theorem to physical problems.

In Vector Calculus, students should be able to:

• understand the concepts of scalar and vector fields;
• carry out algebraic manipulations with the div, grad, curl, and Laplacian operators in Cartesian coordinates;
• derive and apply the divergence and Stokes' theorems in physical situations, and deduce coordinate-independent expressions for the vector operators;
• derive and use expressions for the vector operators in cylindrical and spherical polar coordinates.

Methodology and Assessment

The 33 lectures in this half-unit course are reinforced by approximately 11 discussion periods, where the lecturer goes over examples of relevant problems without introducing any new examinable material. In addition there are 2 revision lectures in Term-3. The end-of-session written examination counts for 90% of the assessment. The 10% continuous assessment component is based primarily on the best eight out of ten homework sheets (8%). Up to 2% credit will be derived from the results of the mid-sessional test examination held just before the Christmas break.

Textbooks

A book which covers essentially everything in both this and the first-year 1B21 mathematics course is Mathematical Methods in the Physical Sciences, by Mary Boas (Wiley). This book will also be of use in the B8 option given in the second semester. A more specialised second-year book is Mathematical Methods for Physics and Engineering, by K.F. Riley, M.P. Hobson and S.J. Bence (Cambridge University Press). Mathematical Methods for Physicists, by G.B. Arfken and H.-J. Weber (Academic Press), is more challenging but is rewarding for the mathematically-inclined students.

  Syllabus

   (The approximate allocation of lectures to topics is shown in brackets below.):

Linear Vector Spaces and Matrices [12.5]

Definition and properties of determinants, especially 3 x 3. [1.5]

Revision of real 3-dimensional vectors, Complex linear vector spaces, Linear transformations and their representation in terms of matrices, Multiple transformations and matrix multiplication. [3]

Properties of matrices, Special matrices, Matrix inversion, Solution of linear simultaneous equations. [4]

Eigenvalues and eigenvectors, Eigenvalues of unitary and Hermitian matrices, Real quadratic forms, Normal modes of oscillation. [4]

Partial Differential Equations [3.5]

Superposition principle for linear homogeneous partial differential equations, Separation of variables in Cartesian coordinates, Boundary conditions, One-dimensional wave equation, Derivation of Laplace's equation in spherical polar coordinates, Separation of variables in spherical polar coordinates, the Legendre differential equation, Solutions of degree zero.

Series Solution of Ordinary Differential Equations [2]

Derivation of the Frobenius method, Application to linear first order equations, Singular points and convergence, Application to second order equations.

Legendre Functions [4]

Application of the Frobenius method to the Legendre equation, Range of convergence, Quantisation of the l index, Generating function for Legendre polynomials, Recurrence relations, Orthogonality of Legendre functions, Expansion in series of Legendre polynomials, Solution of Laplace's equation for a conducting sphere, Associated Legendre functions, Spherical harmonics.

Fourier Analysis [5]

Fourier series, Periodic functions, Derivation of basic formulae, Simple applications, Gibbs phenomenon (empirical), Differentiation and integration of Fourier series, Parseval's identity, Complex Fourier series. [2.5]

Fourier transforms, Derivation of basic formulae and simple application, Dirac delta function, Convolution theorem. [2.5]

Vector Operators [6]

Gradient, divergence, curl and Laplacian operators in Cartesian coordinates, Flux of a vector field, Divergence theorem, Stokes' theorem, Coordinate-independent definitions of vector operators. Derivation of vector operators in spherical and cylindrical polar coordinates.

----------------------------------------

  Relativity and Gravitation

This is an advanced course in General Relativity; students enrolling on SPA-7019 are strongly recommended to have taken Spacetime & Gravity (STG | SPA-6308) or equivalent. The assessment for SPA-7019 is as follows:

• Assessed homework problems 10%
• Final examination (two and a half hours) 90%

SYNOPSIS and Syllabus

Einstein’s theory of relativity is one of the pillars of modern physics, and is currently enjoying a renaissance due to recent progress in cosmology and gravitational wave detection. This course is aimed at providing sufficient tools to understand the deep physics that underpins these advances, and to provide the foundational mathematics and physics required for more advanced study. This will begin with an introduction to differential geometry, before moving on to Einstein’s gravitational field equations and their solutions. It will include the study of black hole physics and gravitational wave emission. In particular, you will be presented with:

• An introduction to differential geometry, including an explanation of how physics should be understood in curved spaces.
• A presentation of Einstein’s theory of general relativity, including some exact solutions to the field equations of the theory.
• The formalism used for studying perturbative relativistic gravity, for use in the Solar System and for calculating the gravitational wave signals from inspiralling binaries.
• Some of the modern developments in general relativity, including the LIGO detection of gravitational waves.

   Outcomes

By the end of this course students should have a detailed knowledge of how to describe physics in curved space-times, as well as how to calculate observables within these space-times. This will include the use of orthonormal frames, and the development of the ideas of differential geometry from first principles. The student will have an understanding of the different vacuum solutions to Einstein's equations, including the properties and global structure of rotating and static black holes. The student will be able to perform calculations in perturbation theory, which will include both gravitational waves and Newtonian-like gravitational potentials. This will include an understanding of why gravity propagates at the speed of light in Einstein's theory, and will allow the student to calculate the amount of gravitational radiation that is emitted from multiple bodies in orbit around each other. The student will understand why observations of the Hulse-Taylor binary pulsar system allowed the existence of gravitational waves to be inferred, and be able to reproduce the gravitational wave signals from inspiralling binary systems (as recently detected with LIGO).