Mathematics, Physics, and Electrical Engineering Foundation Year

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

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.

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.

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.

This module will introduce you to the fundamental principles of electricity. You will learn about the application of electrical circuits analysis theory to solve electrical problems, such as resistor networks, voltage dividers, RC circuits and rectifiers. You will consider Ohm’s Law, Joule’s Law and Kirchhoff’s law extensively to assist your understanding of electrical circuit analysis. You will learn about the basic concepts and application principles of DC circuits. Topics include terms and symbols, basic electrical and electronic components/devices, electrical measurement instruments and the laws of DC circuits, including series and parallel connections. You will cover how Ohm’s law and Joule’s law are used to calculate the power supply of batteries and power consumption of resistors. You will also learn about A.C. circuits including topics like safety, terms and symbols, sine waves, AC electrical theory and AC measurement instruments. You will understand about Digital logic circuits and numbering systems: These topics include principles of NOT/AND/OR/XOR gates, terms and symbols, combinational gates, Boolean expression, truth table and circuit schematics. These topics are foundations for the generation and use of electricity in a range of applications.

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.

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.