Keele University Mathematics Undergraduate Programmes 2020 2020 Entry | Page 12

12 | Mathematics
Year 2 Modules
Differential Equations Probability Analysis II
This module builds on Calculus.
It aims to develop skills in mathematical
techniques, focusing on methods for
solving ordinary differential equations.
The topics covered include: solutions
to first-order differential equations,
unforced and forced linear differential
equations with constant coefficients,
the harmonic oscillator, power series
methods of solution and graphical
aspects of differential equations. Probability is the mathematics of
uncertainty. Motivated by a discussion
of classical probability based on coins,
cards, dice etc., an axiomatic approach
to probability theory is developed.
This is a sound framework in which
to handle independence of several
events, conditional probability and
Baye’s Theorem. Thereafter the module
focuses on discrete and continuous
random variables and their probability
distributions, concluding with the Central
Limit Theorem. Finally confidence
intervals and hypothesis testing are
introduced with some applications. This module builds upon the introduction
to formal analysis that appeared in
Exploring Algebra and Analysis by
studying limits of functions, continuity,
and differentiation from a rigorous point
of view. This allows students to prove that
the methods of calculus that appear in
other modules are valid and consistent,
and paves the way for more abstract and
powerful versions of analysis.
Exploring Algebra
and Analysis
Algebra and Analysis are two of the main
branches of Pure Mathematics. In this
module, students will explore these topics
via a combination of guided investigations
and lectures. The concepts and styles
of thinking developed are important for
other modules in Years 2 and 3. Linear
Algebra generalizes the vector and
matrix algebra encountered at level 4
by studying abstract vector spaces and
maps between them. Analysis arose out
of the need to place Calculus on a secure
theoretical footing. In particular, the
module will develop rigorous proofs
of a number of results appearing in level
4 Calculus.
Abstract Algebra
This module introduces and studies the
abstract algebraic structure known as
a group. Beginning with the axiomatic
and theoretical foundations, the
module progresses, through a study
of subgroups, to the proof of one of
the most important theorems in Group
Theory, namely Lagrange’s Theorem.
The module also examines applications
of group theory, and the often beautiful
way in which it interacts with geometry
and number theory. It concludes with a
preliminary exploration of other closely
related algebraic structures, namely rings
and fields.
keele.ac.uk/scm
Computational Mathematics
This module covers aspects of
computational mathematics by
introducing computer algebra systems
(CAS). This application of technology
is intended to enrich the students’
undergraduate studies and furnish them
with valuable transferable skills.
Complex Variable I
and Vector Calculus
The first half of the module is concerned
with functions of a complex variable.
Starting with a quick revision of complex
numbers from Algebra I, students will next
study elementary functions of complex
variables before examining analytic
functions and the Cauchy-Riemann
equations. The module then moves on
to the fundamental results embodied in
Cauchy’s Theorem, Cauchy’s Integral
Formula and Cauchy’s Residue Theorem,
providing some elementary applications
along the way. The second half of the
module is an introduction to vector
calculus, which provides a framework for
solving physical and geometric problems.
First developing familiarity with the basic
ideas, language and operators of vector
calculus, the module then progresses to
the classical integral theorems of Green,
Gauss and Stokes
Mathematical Modelling
The aim of this module is to demonstrate
how real-world problems can be modelled
mathematically. The mathematical
modelling process is introduced through
a six-step, problem-solving approach.
Mathematical tools used in the model
construction and solution process will
include ordinary differential equations
and their solution methods (including
phase-plane analysis), dimensional
analysis and difference equations.
The modelling ideas will be developed
through novel and innovative case
studies of real-world scenarios and
through individual/group projects.
Introduction to Mathematics
Education
This module introduces students to
principles of reflective learning and
critical pedagogy linked to the history
and theory of Mathematics education.
Through a mixture of individual and
group work, students will investigate
and explore aspects of mathematical
learning and critically analyse these
with reference to their effectiveness
from a personal perspective.