Mathematics

Learning Outcomes

Subject Specific Intellectual and Research Skills

Having successfully completed this module you will be able to:

• Manipulate algebraic expressions of real and complex numbers, vectors, and matrices.
• Critically analyse and solve mathematical problems.

Subject Specific Practical Skills

Having successfully completed this module you will be able to:

• Show logical thinking in problem solving.
• Perform calculations of simple problems and work through longer examples.

Knowledge and Understanding

Having successfully completed this module, you will be able to demonstrate knowledge and understanding of:

• Mathematical methods both in the abstract and in relation to specific example problems from technical engineering subjects.

Transferable and Generic Skills

Having successfully completed this module you will be able to:

• Plan learning and revision activities in a self-study environment.
• Demonstrate organisational and time-management skills.

Numbers and functions: • Complex numbers: graphical representation; algebra; polar form; Euler's formula and exponential form. • Complex numbers: trigonometric and hyperbolic functions; logarithm of a complex number; De Moivre's theorem; roots; simple loci in the complex plane • Functions: inverse; trigonometric; exponential, logarithmic and hyperbolic. • Oscillating function; Wave function Calculus • Differentiation: basic rules; differentiation of standard functions; Newton's method for finding roots; partial differentiation. • Differentiation: maxima, minima and points of inflection; curve sketching; parametric, implicit and logarithmic differentiation; Taylor and Maclaurin series. • Integration: definition of integral; standard integrals; substitution; integration by parts; numerical integration. • Integration: substitution; applications to centroids, volumes of revolution, etc. • Integration: rational functions; improper integrals. • Integration: double integrals; polar coordinates; triple integrals. Differential Equations • classification; simple first and second order differential equations. • solution of first order ODEs (separable, homogenous, linear and exact). • linear operators; linear inhomogeneous second order ODEs; free and forced oscillators Vectors: • Vectors: triple products; differentiation and integration of vectors; vector equations of lines and planes • Vectors: basic properties; Cartesian components, scalar and vector products. Vectors in 2-Space and 3-Space. Inner Product (Dot Product) • Vector Product (Cross Product) • Vector and Scalar Functions and Their Fields. Vector Calculus: Derivatives • Curves. Arc Length. Curvature. Torsion • Gradient of a Scalar Field. • Divergence of a Vector Field • Curl of a Vector Field • Directional derivatives. • Alternative coordinate systems. Matrices: • Matrix algebra: terminology; addition, subtraction and multiplication of matrices; determinants. • Matrix algebra: inverse of a matrix using cofactors; systems of linear equations; inverse of a matrix using the elimination method. Statistics: • Statistics: probability; conditional probability; combinations and permutations; discrete and continuous random variables. • Statistics: mean and standard error of sample data; normal distribution; sampling; confidence intervals; hypothesis testing

Taught using a blended approach, incorporating “flipped mode” teaching and traditional problem classes. • Comprehensive explanatory video lectures on each topic • Case study recordings • Weekly discussion tutorials and seminars • Case study based problem sheets • Weekly intensive problem classes – problem sheets worked on • Self-study notes for each topic. • Self-testing on each block • Mini-test at end of each block • Past examination papers and solutions.

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.