Mathematics 1

Module aims

This module introduces mathematics as a logical and structured discipline, to ensure that all students acquire the mathematical knowledge and skills required for the first year aeronautical course, and to provide a basis for the more advanced mathematical techniques which are required in later years of the course.

Learning outcomes

On successfully completing this module, you should be able to:

1. Demonstrate mastery of basic mathematical concepts associated with functions, differential calculus, integral calculus, and series;
2. Evaluate partial derivatives and understand their practical importance;
3. Employ complex numbers and Fourier series, and appreciate their practical relevance;
4. Derive analytical solutions of certain first and second order ordinary differential equations and understand their practical importance;
5. Employ linear algebra methods and understand their application in engineering;
6. Carry out a range of matrix manipulations, and calculate matrix eigenvalues and eigenvectors;
7. Demonstrate understanding of the properties of the inner product and apply Gram-Schmidt orthogonalisation;
8. Reduce systems of linear ODEs with constant coefficients to matrix form and determine solutions in terms of eigenvalues and eigenvectors.

Module syllabus

Differential Calculus: basic rules for derivatives, Parametric representation, Implicit differentiation, higher order derivatives, Leibniz’s rule, relative extremes and inflexion points, monotonicity, asymptotes, convexity and concavity
Integral Calculus: integration techniques for rational and trigonometric functions, integrals to evaluate areas and volumes
Series: tests of convergence, Taylor and Maclaurin series
Limits: Advanced techniques for calculating limits
Partial Differentiation: Total and partial derivatives of a function of two variables, change of variables, identification and classification of stationary points, Taylor’s theorem for functions of two variables, contours
Complex Numbers: Complex arithmetic, complex plane, polar form, modulus and argument, Euler’s formula, Geometric properties, De Moivre’s theorem and application to trigonometric formulae, powers of complex numbers, logarithm of a complex number, hyperbolic functions
Fourier Series: Periodic functions, Fourier series representation of continuous functions, odd/even functions, sine/cosine Fourier series, exponential formulation of Fourier series, convergence of Fourier series
Ordinary Differential Equations: solving first-order and second-order equations, particular integrals, special equations and techniques, Laplace transforms, application of Laplace transforms to solve ordinary differential equations
Linear Algebra: Matrix operations, linear systems, Gaussian elimination, LU decomposition, vector spaces and subspaces, column/row/null space, determinant of a matrix, Cramer's rule, eigenvalues and eigenvectors, link between matrices and systems of ordinary differential equations; linear independence of eigenvectors; diagonalisation; powers of a matrix; Cayley-Hamilton theorem for distinct eigenvalues case; introduction to inner product space; Gram-Schmidt orthogonalisation, adjoint operator

Teaching methods

The module will be delivered primarily through large-class lectures introducing the key concepts and methods, supported by a variety of delivery methods combining the traditional and the technological. The content is presented via a combination of slides, whiteboard and visualiser.

Learning will be reinforced through tutorial question sheets and laboratory exercises, featuring analytical, computational and experimental tasks representative of those carried out by practising engineers.

Assessments

This module presents opportunities for both formative and summative assessment. 
You will be formatively assessed through progress tests and tutorial sessions.
You will have additional opportunities to self-assess your learning via tutorial problem sheets.
You will be summatively assessed by a written examination.  

Your prior mathematical ability will be assessed through an introductory maths test at the start of session based on the core Mathematics A-level syllabus, with the aim of bringing the ability of all students up to the minimum required for the module. Should you fail the introductory test, a resit will be available within three weeks with additional support offered in the form of optional lectures and tutorials.

 
Assessment type Assessment description Weighting Pass mark
Examination Written examination 100% 40%
Examination Introductory mathematics test 0% 40%

You will receive feedback on the examination in the form of an examination feedback report on the performance of the entire cohort.
You will receive feedback on your performance whilst undertaking tutorial exercises, during which you will also receive instruction on the correct solution to tutorial problems.
Further individual feedback will be available to you on request via this module’s online feedback forum, through staff office hours and discussions with tutors.

Reading list