### Welcome to the MT3 (SPA5218) - Mathematical Techniques III - Home Page

Module Organiser : Prof. Carl Murray

Office hours: 12:30-13:30 Monday, 10:30-11:30 Thursday

Deputy Module Organiser : Dr. Sanjaye Ramgoolam

Lectures: 1200-1300 Tuesday, 1000-1200 Wednesday (David Sizer LT)

Exercise Classes: 1000-1200 Friday (BR:3.01), 1400-1600 Friday (QB:EB2)

### Learning Outcomes

On completing this course students should be able to: Use Index notation as a powerful tool for manipulating matrices and vectors. Appreciate the concept of a vector space, basis vectors and linear independence, with examples using coordinate vectors and functions. Manipulate matrix and differential operators; Solve a variety of 1st and 2nd order differential equations both ordinary and partial for physical problems Use various methods, including separation of variables and Green function techniques. Understand the basics of variational calculus Understand contour integration, the residue theorem, special functions and some physical applications.

### Weekly Outline

### Week 1 :

#### Vector spaces, Linear Operators

Vectors - Examples, Definitions of Abstract Vector Spaces, Linear independence

Matrices - Matrix Multiplication, Index notation

Reading : Riley, Hobson, Bence (RHB) Chapter 8 ; Arfken Chapter 3

### Week 2

#### Vector spaces, Linear Operators

Definition of Basis of a vector space, Dimension, Linear operators and their relation to Matrices, Rotations as examples of linear operators. Definition of inner products for Vector spaces over real or complex numbers. The Kronecker Delta symbol.

Reading : Riley Hobson and Bence Chapter 8, Arfken Chapter 3

### Week 3

#### Vector spaces, Linear Operators

Rotation operators in any dimension and orthogonal matrices. Change of basis, Similarity transformation, Gramm-Schmidt. Characteristic Polynomial, Eigenvectors and eigenvalues of Hermitian matrices. Diagonalization of a Hermitian matrix.

Reading : Riley Hobson and Bence Chapter 8, Arfken Chapter 3

### Week 4

#### Vector spaces, Linear Operators

Function spaces as Vector Spaces. Inner products for Function Spaces. Fourier Series.

Reading : Riley Hobson and Bence Chapter 12 and 17 ( intro and 17.1-17.3)

### Week 5

#### Ordinary differential equations

First order ordinary Differential Equations. Differential operators, Hermitian condition, First Order ODE, Separation of variables, Exact equations, Integrating Factors, Integrating factor for general linear first order.

Reading : RHB Chapter 14, Arfken (4th edition) Chapter 8, Sec. 8.2 , pgs 463-471

### Week 6

#### Ordinary Differential Equations

Review first order : complementary function and particular integral ;Second order linear equations, quick review of case of constant coefficients (MT2) ; More general case of variable coefficients, Homogeneous and Inhomogeneous, Series solution, Indicial Equation. Examples of Hermite and Legendre equations, Truncations to Polynomials.

**Reading** : RHB - parts of chapter 15, notably 15-introduction, 15.1.1-15.1.3, 15.2.6 and most of 16.Arfken -- Chapter 8.

**ENJOY READING WEEK **!! I recommend going through all the Exercise sheets and Homeworks 1-5 again and checking your answers against the solutions provided. This will be excellent preparation for the final exam, and will improve your ability to use mathematical techniques/concepts effectively in understanding the math of subsequent physics courses. And if you have questions about any of this, come talk to lecturer or instructors after reading week. ( office hours Wed. 12-1, Fri 12-1)

### Week 7

#### Inhomogeneous Equations, Dirac Delta Functions, Green's Functions for solving ordinary differential equations.

#### Reading
- Some methods for inhomogenous equations-- particular integrals in cases with constant coefficients -- RHB page 494-495
- For Dirac Delta functions. See
- RHB pages 439-443 ;
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### Week 8

#### Partial Differential Equations, Separation of Variables, Heat Equation, Laplace's Equation

#### Reading :

RHB Chapter 21,

### Week 9

#### Green's function for Poisson's equation in 2D and 3D, Variational Principle.

#### Reading :

Arfken Chapter 17

### Week 10

#### Functions of complex variables, Analytic functions, Cauchy-Riemann equations, Taylor Expansion, Cauchy theorem, Cauchy integral formula

**Reading** : Arfken Chapter 6, 7 ; RHB Chapter 24

**NOTE :** The final exam will not test the material from Week 10 and 11 on Complex variables.

### Week 11

#### Laurent series, Residue theorem, Examples (Lecture 1). Review of Index Notation from Weeks 1-4 and some further applications (Lecture 2-3)

**Reading :** Arfken Chapter 6, 7 ; RHB Chapter 24

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### Deadlines

Homeworks are due at 4 p.m on Wednesdays.

### Marking

Exam 90%, Homeworks 10%, Exercise class work is unassessed but attendance is compulsory.

### Lecture notes

Lecture notes will be distributed via the module's QMPlus pages

### Exercises Classes and Homework

Exercise classes start on Friday 2 October. They are normally held in Bancroft Road 3.01 (10:00-12:00) for Group A and Queens' Building EB2 (14:00-16:00) for Group B. However, for the first week only (i.e. Friday 2 October) both classes take place 9:00-11:00 with Group A in the Bancroft Building 1.02.6 and Group B in Laws 1.02

Homeworks and their solutions will be distributed via the module's QMPLus pages. The first homework is due at 4pm on Wednesday 7 October.