Mathematical Techniques 1 (MT1 | SPA4121)

Year: 1 | Semester: A | Level: 4 | Credits: 15

Course organiser: Dr Jeanne Wilson | Course deputy: Prof David Arrowsmith

Synopsis:

Techniques of mathematics, mostly calculus, required in the study of the physical sciences. Complex numbers and functions, differentiation, partial differentiation, series, integration, polar coordinates and multiple integration. The course structure includes both lectures and self-paced programmed learning, with assessment by coursework and an end of year examination.

Aims:

This is a first course in applications of calculus; it also introduces complex quantities. Its main purpose is to prepare the students for the mathematical manipulations, which they will encounter in all subsequent Physics courses. The types of examples used are, consequently, heavily Physics based.

Outcomes:

At a basic level, students should be able to: find the first and second derivatives of functions of a single variable; apply calculus methods to curve sketching; calculate the first and second derivatives of quantities defined parametrically; calculate the first and second partial derivatives of functions of many variables; use partial derivative methods to solve small change problems and rate of change problems for functions of many variables and parametric situations; establish Maclaurin, Taylor, Binomial, and Fourier series expansions for (relatively) simple functions; Series expansion for two dimensional problems, proof by induction, manipulate complex numbers in cartesian and exponential form (including converting to and from both forms); find roots of complex numbers; hyperbolic functions; integrate functions of a single variable by various methods (substitution, parts, partial fractions etc.); integrate functions defined parametrically; establish reduction formulae for given integration problems; use integration methods to evaluate areas and averages of functions (including rms values); use integration methods to calculate volumes of revolution and centroids of figures; express simple multiple integrals; apply multiple integration methods for evaluating areas bound by curves and volumes bound by planes as well as other Physics based problems (e.g. moments of inertia); Compute Fourier transforms; understand the meaning of an analytic function, and integrate and differentiate complex functions; be able to solve problems related to conformal mapping.

Recommended books:

Stroud, K.A.
Engineering Mathematics 7th Ed.
Palgrave, (2007)
ISBN 978-1-4039-4246-3
&
Riley, K.F.; Hobson, M.P.; Bence, S.J.
Mathematical Methods for Physics and Engineering
Cambridge University Press (2006)
ISBN 0-521-67971-0

Other books to consult: 

M. L. Boas
Mathematical Methods for the Physical Sciences
Wiley (2005)
ISBN-13: 978-0471365808