# PX149 Mathematics for Physicists

## Lecturers: Bogdan Hnat (term 1) and Nicholas d'Ambrumenil (term 2)

## Weighting: 24 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 analytically 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 proficient at evaluating a definite integral that describes the area under a curve
- 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 the basic ideas from linear algebra: matrices, determinants, condition for a unique solution to a set of linear equations, eigenvalues and eigenvectors and the diagonalisation of 2x2 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.

Introduction to line, surface and volume integrals in physics

Fourier series: Computation of Fourier coefficients by using orthogonality of sin mx and cos mx. Examples: sawtooth and square wave. Half range sin and cos expansions. Arbitrary period. Applications to solution of wave equation.

Linear algebra: Generalisation of the algebra of vectors to n dimensions. Linear independence, bases and dimension. Linear maps and matrices. Kernel, image, rank and nullity. Application to simultaneous linear equations. Computation of rank (and inversion of matrices) by row and column operations.

Determinants: Elementary properties with emphasis on the 2 x 2 and 3 x 3 cases (general proofs not required). Rank and inverse of matrix, and solution of simultaneous equations ( Cramer's rule) using determinants (only useful for n < 3: if n > 3 use row and column operations). Eigenvectors and eigenvalues, characteristic equation.

**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%).

This module has a home page.

**Recommended Text:**

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

**Other Texts:** JB Staunton (Ed.), *Maths for Scientists* , Pearson/Addison Wesley

G.Stephenson, *Mathematical Methods for Science Students,* Longman,

If you have forgotten your A-level Mathematics the following is useful

R. Lambourne and M. Tinker, *Basic Mathematics for the Physical Sciences,* Wiley

and the same authors have also written a book that covers the first half of the module

R. Lambourne and M. Tinker, *Further Mathematics for the Physical Sciences,* Wiley.

**Leads from:** A level Mathematics