# PX153 Mathematics for Physicists

## Lecturers: Neil Wilson (term 1) and Pier-Emmanuel Tremblay (term 2)

## Weighting: 20 CATS

All scientists use mathematics to state the basic laws and to analyze quantitatively and rigorously their consequences. The module introduces the concepts and techniques, which will be assumed by future modules. These include: complex numbers, functions of a continuous real variable, integration, functions of more than one variable and multiple integration.

**Aims:**

To revise relevant parts of the A-level syllabus, to cover the mathematical knowledge to undertake first year physics modules, and to prepare for mathematics and physics modules in subsequent years.

**Objectives:**

*Term 1:*

At the end of the first half of module you should:

- Understand the concept of a vector and be able to carry out vector algebraic manipulations
- Understand the notion of a complex number and be able to manipulate expressions involving complex numbers.
- Have met the general idea of mathematical modelling and how this leads to a description in terms of a differential equation.
- Be able to solve some types of first and second order Ordinary Differential Equations.
- Understand the notion of convergence and be able to obtain the interval of convergence of a series.
- Be familiar with functions of more than one variable and with partial differentiation.
- Be able to determine the gradient vector of a function of 2 and 3 variables.

*Term 2:*

At the end of the second half of the module, you should:

- Be able to deal with multiple integrals and know how to evaluate the volume of a three dimensional object
- Understand the concepts of line and surface integrals
- Know what is meant by a Fourier series and be able to compute the Fourier coefficients for simple functions
- Be familiar with ideas from linear algebra: matrices, determinants, the condition for a unique solution to a set of linear equations, eigenvalues and eigenvectors and the diagonalisation of matrices.

**Syllabus:**

*Term 1*

Vector Algebra: Scalar and vector product.

Complex numbers: Algebra, Argand diagram, exp (iθ), De Moivre's theorem, roots.

Introduction to Differential equations: (ordinary, constant coefficients): Concept of complete solution and boundary or initial conditions. How such equations arise. Second order homogeneous linear differential equations with constant coefficients. Application of complex numbers.

First order differential equations: separation of variables; first order inhomogeneous equations with non-constant coefficients; integrating factor method. Inhomogeneous second order constant coefficient equations; particular integral and complementary function; Physics applications.

Series and Taylor Series: Convergence, ratio test and interval of convergence, Taylor Series and applications.

Functions of two or more variables: Contours and cross-sections. Partial differentiation, Taylor series, Maxima,Minima and saddle points. Introduction to partial differential equations. Change of co-ordinates ((x,y)- (r,θ)). The gradient vector of functions of 2 and 3 variables.

*Term 2*

Riemann Integral: Standard Integrals, use of change of variables.

Multiple Integrals: Double integrals and volume under a surface. Change of variable and change of order of integration.

Introduction to line, surface and volume integrals.

Fourier series: Conditions for convergence, orthogonality relations, computation of Fourier coefficients, examples. Periodic extensions. Symmetric and antisymmetric functions. Sine and cosine series. Arbitrary periods. Parseval's theorem, Gibbs phenomenon.

Linear algebra: Matrices, matrix addition and multiplication. Simultaneous linear equations and their solution by Gaussian elimination, row-reduced echelon form. Idea of a linear map and relation to matrices.

Determinants: Definition using co-factors and properties. Inverse of a square matrix, conditions for its existence, adjugate matrix, relation to solution of simultaneous equations. Efficient algorithms, LU decomposition. Orthogonal, Hermitian and unitary matrices. Eigenvalues and characteristic equation. Eigenvectors, their orthogonality for Hermitian matrices.

Similarity transformations, corresponding invariance of determinant and trace, use to diagonalise matrices.

**Commitment:** 60 Lectures + 20 problems classes

**Assessment:**

Examination (85%):

Part 1 - a 2 hour examination (January); Part 2 - a 2 hour examination (June). Continuous Assessment (15%): Based on answers to weekly problems sheets (7.5%) and computer based assessments (7.5%).

**Recommended Text:**

KF Riley, MP Hobson and SJ Bence, Mathematical Methods for Physics and Engineering: a Comprehensive Guide, CUP

**Leads from:** A level Mathematics