Mathematics (Extended Degree) BSc (Hons)

Entry Requirements 2026/27

Standard Entry

64 to 80 Tariff Points

From a combination of acceptable Level 3 qualifications which may include: A-level, T Level, BTEC Diplomas/Extended Diplomas, Scottish and Irish Highers, Access to HE Diplomas, or the International Baccalaureate.

Find out how many points your qualifications are worth by using the UCAS Tariff calculator:

��ϲ�� is committed to supporting all individuals to achieve their ambitions and we understand that every applicant’s circumstances can be different, which is why we take a flexible approach when making offers for this course. We have a range of schemes and alternative offers to make sure as many individuals as possible are given an opportunity to study at our university regardless of personal circumstances or background. Typically, offers range from 64 to 80 UCAS tariff points, but we’ll assess your individual circumstances and potential when reviewing your application

To find out more, review our Northumbria Entry Requirement Essential Information page for further details

Subject Requirements:

There are no specific subject requirements for this course.

GCSE Requirements:

Applicants will need Maths and English Language at minimum grade 4/C, or an equivalent.

Additional Requirements:

There are no additional requirements for this course.

International Qualifications:

We welcome applicants with a range of qualifications which may not match those shown above.

If you have qualifications from outside the UK, find out what you need by visiting

English Language Requirements:

International applicants should have a minimum overall IELTS (Academic) score of 6.0 with 5.5 in each component (or an approved equivalent*).

*The university accepts a large number of UK and International Qualifications in place of IELTS. You can find details of acceptable tests and the required grades in our English Language section:

Fees and Funding 2026/27 Entry

UK Fee in Year 1*: TBC

* Government has yet to announce 26/27 tuition fee levels.  As a guide, 25/26 fees were £9,535 per year.

International Fee in Year 1: TBC

Scholarships for 26/27 have not yet been announced. Please see the main Funding Pages for 25/26 scholarship information.

ADDITIONAL COSTS

TBC

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Modules Overview 2026/27

Modules

Module information is indicative and is reviewed annually therefore may be subject to change. Applicants will be informed if there are any changes.

Foundation year Year one Year two Year three Year four

Foundation year

KL3004 -

Applications of Physics (Core,20 Credits)

This module will introduce you to key physics topics through a range of industrial and practical contexts. The contexts will include space technology and astrophysics, medical applications including bioengineering and non-invasive testing, music, renewable energy technologies, geophysics and the history of physics.
The physics topics studied in this module are:
- waves and oscillations including reflection, refraction, polarisation, stationary waves, simple harmonic motion and damping.
- material properties including flow of liquids and viscosity, properties of materials and Young modulus
- particle and nuclear physics including the structure of the atom, radioactive decay, nuclear fusion and fission, luminosity of stars and blackbody radiators.
- nature of light including wave-particle duality, photoelectric effect, photovoltaic cells, absorption and emission spectra, and energy levels

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KL3005 -

Introduction to Logic and Algorithms (Core,20 Credits)

In this module, you will explore the principles of mathematical logic, proof techniques, and basic algorithms of discrete mathematics and number theory. You will learn about such concepts as:
• Fundamental proof techniques, including induction, contradiction, and contrapositive.
• A formal language for mathematical statements, enabling compact, precise notation and its translation to natural language.
• How to interpret mathematical statements unambiguously and implement them as algorithms.
• Practical application of these concepts using Python and the interactive Jupyter Notebook interface.
• Techniques for numerical and symbolic computations, as well as data analysis and visualization in Python.
By completing this module, you will develop foundational knowledge that is valuable in itself and will serve as the foundation in mathematics and physics. This knowledge will also underpin applications in fields such as software and electrical engineering, where mathematical logic is crucial. For applied problem-solving, researchers often transition between descriptive, mathematical, numerical, and programming languages. Mastering mathematical logic and algorithms provides an essential framework for these translations, serving as a powerful tool for problem-solving and proof composition.

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KL3008 -

Foundation Trigonometry and Calculus (Core,20 Credits)

You will learn about basic calculus such as graphs and equations of common functions, including trigonometric functions. The module further shows you how these mathematical concepts can be used in a variety of ways to help you in understanding and solving physical problems.

You will learn the below highlighted topics which is indicative rather than prescriptive:

1. You will learn to use of Pythagoras’ Theorem. Calculation of areas and volumes of common solids
2. You will learn to use Cartesian coordinates and simple coordinate geometry. Linear graphs.
3. You will learn to use the derivative related to the rate of change. Differentiation of standard functions. The second derivative. Maximum and minimum values of a function.
4. You will learn integration as the inverse of differentiation. Integration of standard functions. Area and definite integrals.
5. You will learn measurement of angles: degrees and radians. Right angled triangles: Pythagoras’ Theorem. Definition of sine, cosine, and tangent in a right angled triangle. Application problems. Graphs of sine and cosine functions over one cycle. Sine and cosine rules.
6. You will learn the definition of secant, cosecant, cotangent. Arc length and sector area. Definition of sine and cosine as co-ordinates on a unit circle (of angles in all quadrants). Graphs of sine, cosine, and tangent to emphasise their periodic nature. General solution to elementary trigonometric equations. Phase angle. Use of standard trigonometrical identities (Pythagorean, compound angle, double angle), in manipulation of expressions and their use in solving trigonometrical equations.

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KL3011 -

Foundation Algebra and Statistics (Core,20 Credits)

You will learn about elementary algebra and statistics necessary to embark on a mathematics or physics degree. You will learn the below highlighted topics which is indicative rather than prescriptive:

ALGEBRA:
You will learn:
• arithmetical operations on signed numbers: addition, subtraction, multiplication, division and exponentiation.
• square roots. Fractions. Removing brackets. The order of operations.
• substitution and evaluation of algebraic expressions.
• plotting linear and quadratic functions.
• simplification of algebraic expressions, removing brackets.
• multiplication of polynomials.
• exponents - including negative and fraction exponents.
• factorising expressions including quadratics, difference of squares.
• addition, subtraction, multiplication and division of algebraic fractions.
• solution of linear and quadratic equations - including simultaneous equations.
• the exponential and log functions.
• arithmetic and geometric sequences and series.
• the binomial series.

STATISTICS:
You will learn
• representation of various types of data.
• frequency distributions: histogram, frequency polygon.
• cumulative frequency, cumulative frequency polygon.
• the mean, the mode, median.
• range, mean deviation, variance, standard deviation.

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KL3013 -

Introduction to Mechanics and Energy (Core,20 Credits)

You will learn about the principles of mechanics such as the concepts of rigid body, degrees of freedom, displacement, velocity, acceleration, linear momentum, standard constant acceleration equations, motion under gravity, vectors motion of a projectile, moments, couples, addition of forces, Newton’s Theory of Gravitation, angular momentum, equilibrium, free body diagrams and friction. You will explore the principles of energy transfer, including work and power (including electrical energy), conservation of energy; and thermal physics including ideal gases, specific heat capacity and latent heat capacity, conduction, convection and radiation. The module will provide an opportunity for enhancing your study skills: you will learn how to achieve effective learning, working in groups and managing time. You will also be engaging with the theoretical knowledge through experimental work by learning to record observations concisely, produce diagrammatic information and graphical data representation as part of this module.

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KL3014 -

Modelling (Core,20 Credits)

This module provides a first course in applying mathematical tools to solve realistic problems, a different skill to the ability to understanding abstract mathematics. The module aims to develop your problem-solving skills by building mathematical intuition, practicing creativity and demonstrating how to transfer theoretical knowledge to structured and unstructured problems. You will solve a variety of real-life problems using a wide range of mathematical and statistical techniques. You will gain experience in tackling real world problems 'from scratch’, working individually as well as within a group. You will use MATLAB to solve a variety of challenging problems.

Topics and skills include:
• The theory behind, and application in MATLAB, of various mathematical methods including:
o Data analysis exploration (beyond the linear best fit; fundamental principles of AI and optimisation);
o Difference equations;
o Differential equations (first and second order equations; dimensional analysis; analytic vs numerical solutions);
o Randomness and uncertainty.
• Develop and answer problem statements for real-life scenarios.
• Formulate and refine models; adapt the modelling lifecycle (the modelling methodology) for a variety of situations.
• Assess validity of models against the underlying mathematical assumptions and the real-life context.
• Critical analysis of model and of interpretations.
• Tailor modelling, interpretation and communication of results to specific audiences and formats.
• Context for many of the topics covered in a mathematics degree.

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Year one

KC4009 -

Calculus (Core,20 Credits)

The module is designed to introduce you to the principles, techniques, and applications of Calculus. The fundamentals of differentiation and integration are extended to include differential equations and multivariable calculus. On this module you will learn:
• Differentiation: derivative as slope and its relation to limits; standard derivatives; product, quotient, and chain rules; implicit, parametric, and logarithmic differentiation; maxima / minima, curve sketching; Taylor and Maclaurin series; L’Hopital’s rule.
• Integration: standard integrals, definite integrals, area under a curve; integration using substitutions, partial fractions decomposition and integration by parts; calculation of solid volumes.
• Functions of several variables: partial differentiation and gradients; change of coordinate systems; stationary points, maxima / minima / saddle points of functions of two variables; method of Lagrange multipliers (constrained maxima / minima).
• Double integrals: standard integrals, change of order of integration.
• Ordinary differential equations: First-order differential equations solved by direct integration, separation of variables, and integrating factor. Second-order differential equations with constant coefficients solved by the method of undetermined coefficients.

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KC4012 -

Computational Mathematics (Core,20 Credits)

Mathematics students require knowledge of a range of computational tools to complement their mathematical skills. You will be using MATLAB, an interactive programming environment that uses high-level language to solve mathematics and visualise data. In addition, you will be investigating the development of algorithms through a selection of mathematical problems. Elements of the MATLAB language will be integrated throughout with various methods and techniques from numerical mathematics such as interpolation, numerical solution of differential equations, numerical solution of non-linear equations and numerical integration.

The computer skills you will become conversant with include programming concepts such as the use of variables, assignments, expressions, script files, functions, conditionals, loops, input and output. You will be applying MATLAB to solve mathematical problems and display results appropriately.

The range of numerical techniques that will be covered will include a selection from the following topics:
• Solution of non-linear equations by bisection, fixed-point iteration and Newton-Raphson methods.
• Interpolation using linear, least squares and Lagrange polynomial methods.
• Numerical differentiation.
• Numerical integration using trapezoidal and Simpson quadrature formulae.
• Numerical solution of Ordinary Differential Equations using Euler and Taylor methods for first-order initial value problems.
• Numerical solution of systems of linear equations using elementary methods.

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KC4014 -

Dynamics (Core,20 Credits)

This module is designed to provide you with knowledge in a special topic in Applied Mathematics. This module introduces Newtonian mechanics developing your skills in investigating and building mathematical models and in interpreting the results. The following topics will be covered:

Mathematics Review
Euclidean geometry. Vector functions. Position vector, velocity, acceleration.
Cartesian representation in 3D-space. Scalar and vector products, triple scalar product.

Newton’s Laws
Inertial frames of reference. Newton's Laws of Motion. Mathematical models of forces (gravity, air resistance, reaction, elastic force).

Rectilinear and uniformly accelerated motion
Problems involving constant acceleration (e.g., skidding car), projectiles with/without drag force (e.g., parabolic trajectory, parachutist). Variable mass. Launch and landing of rockets.
Linear elasticity. Ideal spring, simple harmonic motion. Two-spring problems. Free/forced vibration with/without damping. Resonance. Real spring, seismograph.

Rotational motion and central forces
Angular speed, angular velocity. Rotating frames of reference.
Simple pendulum (radial and transverse acceleration). Equations of motion, inertial, Coriolis, centrifugal effects. Effects of Earth rotation on dynamical problems (e.g. projectile motion).
Principle of angular momentum, kinetic and potential energy. Motion under a central force. Kepler’s Laws. Geostationary satellite.

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KC4020 -

Probability and Statistics (Core,20 Credits)

This module is designed to introduce you to the important areas of probability and statistics. In this module, you will learn about data collection methods, probability theory and random variables, hypothesis testing and simple linear regression. Real-life examples will be used to demonstrate the applications of these statistical techniques. You will learn how to use R to analyse data in various practical applications.

Outline Syllabus
Data collection: questionnaire design, methods of sampling - simple random, stratified, quota, cluster and systematic. Sampling and non-sampling errors. Random number generation using tables or calculator.

Population and sample, types of data, data collection, frequency distributions, statistical charts and graphs, summary measures, analysis of data using R.

Probability: sample space, types of events, definition of probability, addition and multiplication laws, conditional probability. Discrete probability distributions including Binomial, Poisson. Continuous probability distributions including the Normal. Central Limit Theorem. Mean and variance of linear combination of random variables. Use of Statistics tables.

Hypothesis tests on one sample mean and variance, confidence intervals using the normal and Student t distributions.

Correlation and simple linear regression.

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KL4001 -

Real Analysis (Core,20 Credits)

The module is designed to i) introduce you to the notion of convergence as this applies to sequences, series and functions of one variable; ii) to provide a firm basis for future modules in which the idea of convergence is used; iii) to help you recognize the necessity and power of rigorous argument.

Outline Syllabus:

1) Introduction to propositional logic and sets.
2) Real numbers: equations, inequalities, modulus, bounded sets, maximum, minimum, supremum and infimum.
3) Sequences: convergence, boundedness, limit theorems; standard sequences and rate of convergence, monotone sequences, Cauchy sequences.
4) Series: standard series (geometric, harmonic series, alternating harmonic series, etc ); absolute and conditional convergence; convergence tests.
5) Power Series.
6) Functions: continuity, the intermediate value theorem, the extreme value theorem.
7) Differentiability: basic differentiability theorems, differentiability and continuity, Rolle’s theorem, Lagrange theorem, Taylor’s theorem.
8) Riemann’s Integrability: properties of integrable functions, modulus and integrals, The fundamental theorem of Calculus.

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KL4002 -

Linear Algebra and Geometry (Core,20 Credits)

The module is designed to introduce you to the concepts, definitions and methods linear algebra, coordinate transformations and geometry of curves and surfaces.

Outline Syllabus:

1. Sets, Rings, Groups (basic definitions)
2. Vector Spaces
3. Linear maps (basis expansions, rank, kernel)
4. Matrices (determinants, systems of linear equations, eigenvalues and eigenvectors, similarity transformations)
5. Quadratic forms
6. Euclidean vector spaces
7. Affine spaces
8. Projective spaces
9. Conics
10. Curves in the plane (length of a curve and natural parametrisation, tangent vector, normal vector and curvature)
11. Quadrics
12. Surfaces.

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KL5001 -

Academic Language Skills for Mathematics, Physics and Electrical Engineering (Core – for International and EU students only,0 Credits)

Academic skills when studying away from your home country can differ due to cultural and language differences in teaching and assessment practices. This module is designed to support your transition in the use and practice of technical language and subject specific skills around assessments and teaching provision in your chosen subject. The overall aim of this module is to develop your abilities to read and study effectively for academic purposes; to develop your skills in analysing and using source material in seminars and academic writing and to develop your use and application of language and communications skills to a higher level.

The topics you will cover on the module include:

• Understanding assignment briefs and exam questions.
• Developing academic writing skills, including citation, paraphrasing, and summarising.
• Practising ‘critical reading’ and ‘critical writing’
• Planning and structuring academic assignments (e.g. essays, reports and presentations).
• Avoiding academic misconduct and gaining credit by using academic sources and referencing effectively.
• Listening skills for lectures.
• Speaking in seminar presentations.
• Presenting your ideas
• Giving discipline-related academic presentations, experiencing peer observation, and receiving formative feedback.
• Speed reading techniques.
• Developing self-reflection skills.

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Year two

KC5000 -

Further Computational Mathematics (Core,20 Credits)

This module continues the numerical methods and computational mathematics thread established with KC4012: Computational Mathematics. The module aims to present an introduction to advanced numerical mathematics, in particular multivariable problems, and associated transferable skills. Numerical methods are applied to the solution of several classes of problems, such as: systems of linear and nonlinear equations, eigensystems, optimisation, ordinary and partial differential equations. Theoretical aspects are illustrated and discussed at the lectures, and computational implementation developed at the computer-lab workshops, using appropriate software (e.g. MATLAB).

Topics may include (note this is indicative rather than prescriptive):
1. Vector and matrix spaces: normed spaces; vector norms; matrix norms; compatible norms; spectral radius; condition number.
2. Systems of linear equations: direct and iterative methods.
3. Matrix eigensystems: iterative methods for eigenvalues and eigenvectors.
4. Systems of nonlinear equations: multidimensional Newton method; fixed-point iterations method.
5. Numerical optimization: pattern search methods; descent methods.
6. Ordinary differential equations (ODEs): forward and backward Euler methods; Crank-Nicolson method; convergence, consistency and stability of a method; conditional stability; simple adaptive-step methods; Runge-Kutta methods; predictor-corrector methods; Heun method; systems of ODEs; stiff problems.
7. Numerical approximation of initial, boundary value problems (IBVP) for ordinary and partial differential equations (PDEs): finite difference method for the (Dirichlet) IBVP for the one- and two-dimensional Poisson equations; finite difference method for the (Dirichlet) IBVP for the one-dimensional heat equation; finite-difference method for the (Dirichlet) IBVP for the one-dimensional wave equation.

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KC5001 -

Applied Statistical Methods (Core,20 Credits)

The aim of the module is to enhance your hands-on statistical modelling expertise. The module considers important continuous probability distributions leading on to parameter estimation and goodness of fit. Hypothesis testing for both parametric and non-parametric situations are introduced for each of one and two – possibly paired – samples. This is extended to design, and analysis, of experiments. You will also study residual analysis for model assessment and goodness-of-fit with examples based on the classic simple linear regression model.

The module will cover topics such as:
Probability distributions including standard continuous distributions.
Central Limit Theorem.
Mean and variance of a linear combination of random variables.
Principles of estimation and estimation via the method of moments.
Maximum likelihood estimation. Goodness-of-fit test and contingency tables.
Tests for variances and proportions. Test and confidence intervals using F- and chi-squared distributions.

Nonparametric statistics
Sign test; Wilcoxon signed rank test; Mann-Whitney U-test; Wald-Wolfowitz runs test; Spearman’s rank correlation coefficient.

Regression Analysis
(Pearson’s) correlation coefficient; simple linear regression. Transformations of variables. Residual Analysis.

Design and Analysis of Experiments
Completely randomised, randomised block, Latin square and missing values.

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KC5008 -

Ordinary & Partial Differential Equations (Core,20 Credits)

The module is designed to introduce you to a first mathematical treatment of ordinary and partial differential equations. You will learn fundamental techniques for solving first- and second-order equations as well as approximation methods. These are used to explore the question of the existence of solutions and provide a qualitative behaviour of the solutions. Examples are drawn from applications to physics, engineering, biology, economics and finance and modelling of complex systems.
The module will cover topics such as:

Ordinary Differential Equations (ODEs)

1. First-order ODEs: Classification of ODEs, separable, Bernoulli, Riccati and exact equations as well as integrating factors. Picard iterations and existence of solutions.
2. Second-order ODEs: Solutions of linear equations, independence of solutions, linear stability, initial and boundary value problems, series solutions about ordinary and singular points, special functions

Partial Differential Equations (PDEs)

1. Introduction and classification of PDEs.
2. Method of characteristics for first order linear PDEs.
3. The method of separation of variables and Fourier series.
4. Solutions of Laplace, diffusion/heat and wave equations.
5. Applications

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KC5009 -

Vector Calculus & Further Dynamics (Core,20 Credits)

You will learn about vector calculus and tensor analysis and their applications in ‘Vector Calculus and Further Dynamics’. These powerful mathematical methods provide convenient tools for the description and analysis of the physical world. You will be introduced to the fundamentals of vector calculus and Cartesian tensors, as well as their application to the development and analytical solution of problems in rigid body dynamics. Throughout, the real-world motivation for the techniques chosen and the interpretation of the solutions will be emphasised.

You will learn about mathematical concepts such as:
• Line, surface and volume integrals;
• Vector fields and operators, including Gauss' (Divergence) Theorem, Stokes' Theorem and the Transport Theorem;
• Introduction to Cartesian tensors.

You will be applying these powerful mathematical techniques to planetary motion and rigid body dynamics in ‘Vector Calculus and Further Dynamics’. By studying point particle motion you will become acquainted with the fundamental concepts of central forces and through the application of the principles of linear and angular momentum you will be investigating the dynamics of rigid bodies.

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KL5001 -

Academic Language Skills for Mathematics, Physics and Electrical Engineering (Core – for International and EU students only,0 Credits)

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KL5004 -

Complex Variables (Core,20 Credits)

The module is designed for you to develop your understanding of the principles, techniques and applications of complex variables.

The module will cover topics such as:

Complex numbers: Basic algebraic properties and operations, Trigonometric and exponential forms; Products and powers; Stereographic projection.

Functions of one complex variable: Limits, continuity and mappings; Differentiability, analyticity of a function and Cauchy-Riemann equations; Exponential and trigonometric functions; branches and derivatives of logarithms.

Integrals: Contours, contour integrals, Cauchy’s integral theorem and formula; Liouville’s theorem; Fundamental theorem of algebra.

Series: Convergence of sequences and series; Laurent series; Integration and differentiation of power series.

Residues and poles: Types of isolated singular points; Cauchy’s residue theorem; Meromorphic functions; Applications of residues.

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KL5005 -

Statistical Modelling and Data Visualisation (Core,20 Credits)

This module will provide you with the fundamental tools to identify appropriate exploratory analysis techniques to uncover hidden patterns and unknown correlations in large data sets. You will be able to assess the strength of statistical evidence of the revealed patterns/correlations. You will also develop appropriate technique to visualise data/outputs, implement suitable analytical methods for big data and critically assess the suitability of the chosen analytical technique.
You will have the opportunity to analyse and visualise data for tackling real-life problems. You will work individually and in group and have the opportunity to critically appraise both your own work and the work of others.

The module will include topics such as:
? Exploratory analysis of big data;
? Data visualisation;
? Data manipulation (e.g. dealing with missing values, detecting outliers values, data transformation);
? Univariate statistical methods (e.g. simple linear regression, residual analysis);
? Techniques for predictive data mining (e.g. methods for binary/logistic classification);
? A suite of appropriate computer packages (including R) will be used.

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Year three

KL5001 -

Academic Language Skills for Mathematics, Physics and Electrical Engineering (Core – for International and EU students only,0 Credits)

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KL5006 -

Work placement year (Optional,120 Credits)

This module is designed for all standard full-time undergraduate programmes within the Faculty of Engineering and Environment to provide you with the option to take a one year work placement as part of your programme.

You will be able to use the placement experience to develop and enhance appropriate areas of your knowledge and understanding, your intellectual and professional skills, and your personal value attributes, relevant to your programme of study, as well as accreditation bodies such as BCS, IET, IMechE, RICS, CIOB and CIBSE within the appropriate working environments. Due to its overall positive impact on employability, degree classification and graduate starting salaries, the University strongly encourages you to pursue a work placement as part of your degree programme.

This module is a Pass/Fail module so does not contribute to the classification of your degree. When taken and passed, however, the Placement Year is recognised both in your transcript as a 120 credit Work Placement Module and on your degree certificate.

Your placement period will normally be full-time and must total a minimum of 40 weeks.

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KL5007 -

Study abroad year (Optional,120 Credits)

This module is designed for all standard full-time undergraduate programmes within the Faculty of Engineering and Environment and provides you with the option to study abroad for one full year as part of your programme.

This is a 120 credit module which is available between Levels 5 and 6. You will undertake a year of study abroad at an approved partner University where you will have access to modules from your discipline, but taught in a different learning culture. This gives you the opportunity to broaden your overall experience of learning. The structure of study will be dependent on the partner and will be recorded for an individual student on the learning agreement signed by the host University, the student, and the home University (Northumbria).

Your study abroad year will be assessed on a pass/fail basis. It will not count towards your final degree classification but, it is recognised in your transcript as a 120 credit Study Abroad module and on your degree certificate in the format – “Degree title (with Study Abroad Year)”.

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KL5008 -

Work placement semester (Optional,60 Credits)

This module operates within a partnership between the University, employer and yourself, and provides you with the opportunity to develop core competencies and employability skills relevant to your programme of study in a work based environment.

You will be able to use the placement experience to develop and enhance appropriate areas of your knowledge and understanding, your intellectual and professional skills, and your personal value attributes, relevant to your programme of study, within the appropriate working environments.

This module is a Pass/Fail module so does not contribute to the classification of your degree. When taken and passed, however, the placement is recognised both in your transcript as a 60 credit Work Placement Module and on your degree certificate.

Due to its overall positive impact on employability, degree classification and graduate starting salaries, the University strongly encourages you to pursue a work placement as part of your degree programme.

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KL5009 -

MPEE - Study Abroad Semester (Optional,60 Credits)

This module is designed for all standard full-time undergraduate programmes within the Faculty of Engineering and Environment and provides you with the option to study abroad for one semester as part of your programme.

This is a 60 credit module which is available between Levels 5 and 6. You will undertake a semester of study abroad at an approved partner University where you will have access to modules from your discipline, but taught in a different learning culture. This gives you the opportunity to broaden your overall experience of learning. The structure of study will be dependent on the partner and will be recorded for an individual student on the learning agreement signed by the host University, the student, and the home University (Northumbria).

Your study abroad semester will be assessed on a pass/fail basis. It will not count towards your final degree classification but, if you pass, it is recognised in your transcript as an additional 60 credits for Engineering and Environment Study Abroad Semester.

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Year four

KC6001 -

Financial Mathematics (Optional,20 Credits)

The module introduces the concepts and terminology of financial mathematics and modelling in finance. You will learn about
the properties of interest rates and the key tools of compound interest functions for modelling a range of annuity schemes. The module develops models for life insurance and endowment schemes and enables the students to analyse the behaviour of share prices. The generalised cash-flow model is introduced to describe financial transactions. The student learns how to develop simple models of financial instruments such as bonds and shares.

Outline Syllabus

Interest: Simple and compound interest. Effective and nominal interest rates. Force of interest. Interest paid monthly. Present values. Cash flows and equations of value.
Annuities: Annuities with annual payments, and payments more regularly. Payments in arrear and in advance. Deferred and varying annuities, annuities payable continuously. Loans, loan structure and equal payments.
Discounted cash flow: Generalised cash flow model. Project appraisal at fixed interest rates. Comparison of two investment projects. Different interest rates for lending and borrowing. Payback periods. Measurement of investment performance.
Investments: Types of investments. Valuation of fixed interest securities and uncertain income securities. Real rates of interest. Effects of inflation. Capital gains tax.
Arbitrage in financial mathematics: Forward contracts. Calculating delivery price and delivery value of forward contracts using arbitrage-free pricing methods. Discrete and continuous time rates.
Life Insurance: Term insurance and whole life insurance. Curtate future lifetime. Life tables, expectation of life. Annual and monthly premium. Endowments. Payment at death.
Stochastic Interest Rates: Varying interest rates. Independent rates of return. Expected values. Application of the lognormal distribution. Brownian motion.

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KC6027 -

Fluid Dynamics (Optional,20 Credits)

This module is designed to introduce fundamental concepts in the mathematical area of Fluid Dynamics. You will analyse the equations of continuity and momentum, and will investigate key concepts in this area. We will introduce the Navier-Stokes equations, and case studies will be used to visualise and analyse real-world problems (using appropriate software) as appropriate to delivery of the module. Initially, we will use the inviscid approximation and then utilise analytical and computational techniques to investigate flows. The second half of the module is a specialist course in laminar incompressible viscous flows, encompassing background mathematical theory allied to a case study approach, with solution to problems by both analytical and computational means.

Assessment of the module is by one individual assignment (30%) and one formal examination (70%).

The module is designed to provide you with a useful preparation for employment in an applied mathematical environment, physics environment or engineering environment.

Outline Syllabus
• Introduction of fluid dynamics, Navier-Stokes equations, equations of continuity and momentum for inviscid flow, unsteady one-dimensional flow along a pipe, irrotational flow, Bernouilli's equation, stream function formulation, flow past a cylinder, velocity potential.

• Low Reynolds Number Flow including: (i) lubrication theory, slider bearing, cylinder-plane, journal bearing, Reynolds equation, short bearing approximation; (ii) Flow in a corner, stream function formulation, solution of the biharmonic equation by separation of variables.

• High Reynolds Number Flow including boundary layer equations, skin friction, displacement and momentum thickness, similarity solutions, momentum integral equation, approximate solutions.

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KC6028 -

Dynamical Systems (Optional,20 Credits)

The module aims to present an introduction to Dynamical Systems and associated transferable skills, providing the students with tools and techniques needed to understand the dynamics of those systems. You will analyse non-linear ordinary differential equations and maps, focusing on autonomous systems, and will learn analytical and computational methods to solve them. This module offers the additional opportunity of research-orientated learning through a hands-on approach to selected research-based problems.

Topics may include (note this is indicative rather than prescriptive):
1. Autonomous linear systems, fixed points and their classification.
2. 1-dimensional non-linear systems: critical points; local linear approximations; qualitative analysis; linear stability analysis; bifurcations.
3. Multi-dimensional non-linear systems: linearisation about critical points, limit cycles, bifurcations.
4. Discrete systems: maps (such as tent map, logistic map, Henon map, standard map).
5. Numerical schemes for ordinary differential equations, such as the embedded Runge-Kutta method.
6. Numerical applications and programming: generation of the orbit of a map, Lorenz map for a dynamical system, orbit diagrams, cobwebs, simple fractals.
7. Elements of Chaos theory: Lyapunov exponents, sensitive dependence on initial conditions, strange attractors, Hausdorff dimension, self-similarity, fractals.

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KC6029 -

Advanced Statistical Methods (Optional,20 Credits)

This module covers the three important areas of experimental design, multivariate techniques and regression. Experimental design will be developed using analysis of variance techniques to compare treatments meaningfully using replication, factorial experiments and balanced incomplete block designs. You will then move on to multivariate techniques including multivariate inference, data reduction using principal component analysis and classification with linear discriminant analysis. You will also learn how to extend regression models to the case where there are several explanatory variables including indicator variables. The models will subsequently be scrutinised using variable selection criteria and regression diagnostics to improve the model. Curvilinear and non-linear regression models cover the important aspect where different types of curves are appropriate for the data. The generalised linear model will be introduced, and the specific case of a count response variable is developed.

Outline Syllabus
Experimental Design: design and analysis of 2n factorial experiments with replication, a full replicate and balanced
incomplete block designs.
Multivariate techniques: the multivariate normal distribution and its properties. Hotellings T2 test for one, two and paired
samples. Manova, linear discriminant analysis and principal component analysis.
Multiple linear regression: least squares estimation of the parameters of the model and their properties. The analysis of variance
and the extra sum of squares method. Variable selection techniques and regression diagnostics.
Non-linear and generalised linear models: Non-linear regression models, estimation of parameters and testing the model. Analysis of deviance and the Poisson regression model.

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KC6030 -

Medical Statistics (Optional,20 Credits)

You will learn about a range of appropriate statistical techniques that are used to analyse medical data. You will be introduced to the design and analysis of clinical trials and learn how to design the statistics of clinical trials for a variety of scenarios. These trials are the scientific tests that all medical advances need to go through to assess whether they have merit. You will learn techniques that can be used to handle various types of medical data found in epidemiology and learn when to apply them. You will investigate some of the statistical models used in survival data analysis for the analysis of time to failure data such as transplant data.

By the end of the module, you should have developed an ability to design clinical trials that are scientifically sound and be able to select and apply the appropriate statistical techniques to analyse medical data in a variety of forms.

Outline Syllabus

Design and analysis of Clinical Trials including the four main phases, estimation of sample size and power of a test. Parallel group and cross-over trials.

Categorical data analysis using contingency tables, McNemar's test, Fishers Exact test and test for trend.
Epidemiology: Prospective, retrospective and cross-sectional studies. Analysis of trials including dichotomous response and dichotomous risk factors. Study bias and reliability of a trial. Observer bias and diagnostic tests
Mortality statistics. Survival data analysis

Analysis of covariance, logistic regression

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KC6031 -

Project (Core,40 Credits)

This module is designed specifically to enhance your graduate skills that are essential to your future career and/or postgraduate study. This is achieved by an individual, research-based project work in an area appropriate to your degree.

You will develop the ability to undertake independent research in an area of interest, requiring a survey of current literature, synthesis of ideas, find solutions where required and drawing a coherent appraisal of conclusions. In this process you will learn how to defining clearly a mathematical and/or statistical problem to be investigated/solved, research and appraise current thinking as regards the subject, select methodologies, include appropriate mathematical exemplars to justify your argument and present a well-integrated set of conclusions.

You will also develop the ability to critically appraise both your own work and the work of others in the field.

You will be research-tutored through the module, and you will be assessed by a written project proposal, a poster presentation and a final written report.

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KE6030 -

Geophysical Dynamics (Optional,20 Credits)

In this module you will learn basics concepts of continuum mechanics i.e. kinematics (material derivatives, kinematic boundary conditions), kinetics (stresses, traction), balance laws (mass, momentum, energy) and constitutive relationships (stress strain relationships for fluids).

You will learn the effects of rotation on the fluid flow and how these affect the large-scale flow of the ocean and the atmosphere. The focus is on understanding how the large-scale flow regime of the ocean, atmosphere, glaciers and the mantle can be described mathematically.

Your learning will be set within the context of global environmental changes. The module will provide you with the tool required to understand the physical principles of global circulations models used to describe the movement of the atmosphere, oceans, earth’s mantle and large ice sheets.

The course consists of lectures and exercises using a state-of-the art numerical ice sheet model. Assessment of the module is by one individual numerical modelling assignment (30%) and one formal examination (70%).

On completion of the module you will have developed an improved understanding of the earth system and the principles of climate change.
The module is designed to provide you with a useful preparation for employment in earth sciences with aim at pursuing graduate studies in environmental modelling.

Outline Syllabus

• Introduction and review of continuum mechanics, balance laws (mass, conservation, energy), kinematics of deforming bodies, constitutive laws (non-Newtonian rheology).

• Rotating shallow-water models. Large-scale flow approximations used in earth sciences. Large-scale features of ocean circulation and atmospheric circulation. Ocean gyres, boundary currents, eddy transport. Variation in flow with depth/height. Geostrophic balance. Ekman spiral.

• Hydrostasy in the ocean and atmosphere.

• Mechanics of glaciers and ice sheets. Commonly used flow approximations in glaciology. Ice-sheet instabilities. Grounding-line dynamics and instabilities.

• Scaling of flow equations and linearization. Systematic reduction of equations.

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KL5001 -

Academic Language Skills for Mathematics, Physics and Electrical Engineering (Core – for International and EU students only,0 Credits)

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KL6000 -

Data Science (Optional,20 Credits)

Data Science concerns extracting information from data – in other words giving a voice to the data. Different analysts may have different purposes when analysing data – the intention may be to describe the information in the data, explain the relationships between parts of the data or use a subset of the data to predict the outcome of a variable of interest. For example, that variable could be whether a customer with a particular profile may buy an item of interest. Most companies collect data on their customers and are interested in how this data can be used to improve customer experience as well as profits. Depending on the intention, the approach taken by the analyst will differ and this module will cover the main tools for classification, clustering, association mining and outlier detection allowing you to analyse data with confidence.

By the end of the module, you should have developed an awareness of different approaches to analysing various forms of data and should have an ability to appraise which analytical techniques are appropriate. You will be able to perform the analysis and interpret the results correctly.

Outline Syllabus

Classification techniques that may include decision trees, support vector machines, linear discriminant techniques and logistic regression.

Clustering techniques including k-means clustering, apriori association mining, naïve Bayes and dimensionality reduction.

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KL6002 -

Methods of Applied Mathematics (Optional,20 Credits)

With this module you will learn advanced methods and technical skills to find exact and approximate solutions to complex problems inspired by the real world. Examples of applications include traffic flow, waves in the ocean, optical telecommunications systems, models for climate and biological systems and magnetohydrodynamics.

Outline syllabus will develop the following three areas:

Exact methods

1) Integral transforms:
- Laplace transform,
- Fourier transform

2) Applications of integral transforms to linear differential equations.

3) Theory of quasilinear partial differential equations

4) Method of characteristics.

5) Conservation laws and shock waves.

6) Applications of exact methods to

- traffic flows
- water waves
- magnetohydrodynamics.

Asymptotic methods:

1) Asymptotic methods for algebraic equations
2) Regular and singular perturbation methods for ordinary differential equations;
3) Asymptotic methods for evaluations of integrals

Applications

1) Boundary layers
2) Linear and Nonlinear dispersive waves
3) Solitons.

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To start your application, simply select the month you would like to start your course.