Mathematics

Learning Outcomes

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.