Mathematics 2

Module aims

This module builds on first year Mathematics and introduces more advanced concepts including Vector Calculus, Partial Differential Equations, Complex Analysis, Probability, and the theory of Signals and Systems. 

Learning outcomes

On successfully completing this module, you should be able to:
1. Evaluate line, surface and volume integrals.
2. Apply Green’s, Gauss’ and Stokes’ theorems to simple geometries.
3. Analyse simple first and second order systems using linear and quasi-linear first-order, and linear second-order partial differential equations.
4. Differentiate complex valued functions and identify harmonic functions.
5. Demonstrate understanding of fundamentals of probability theory and their application to reliability analysis.
6. Identify a linear system and analyse its properties.
7. Apply Fourier series and Fourier transforms to analyse periodic and aperiodic signals, including the response of linear systems to such signals. 

Module syllabus

1) Vector Calculus: double integrals; inversion of order of integration; mappings; Jacobian; change of variables. Line integrals in the plane and Green’s theorem. Vector operators: grad, div and curl.  Existence of a potential function. Surface and volume integrals; Gauss’s and Stokes’s theorems.  
2) Partial Differential Equations: First order PDEs: linear and quasi-linear; characteristics. Second order PDE’s: classification and reduction to canonical form; identification of elliptic, parabolic and hyperbolic equations; d’Alembert’s solution and separation of variables for the one-dimensional wave equation; separation of variables of two-dimensional Laplace equation in Cartesian and polar coordinates. Separation of variables and similarity solutions.
3) Complex Analysis: Differentiability; holomorphic functions Cauchy-Riemann equations; conformal mapping; 
4) Probability and Statistics: Laws of probability; combining probabilities; conditional probabilities and independence. Random variables: discrete and continuous; probability density functions; cumulative distribution functions; binomial and Poisson random variables; uniform, exponential, normal random variables; normal approximations to the binomial and Poisson random variables; Reliability Analysis: hazard rate and hazard function; Modelling failure times of components and systems. 
5) Signals and Systems: Basic concept of a signal; Linear Time Invariant Systems; Step Response and Impulse Response for continuous and discrete time systems; Properties of the convolution in continuous and discrete time; Fourier transform.

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.  

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 at the end of the module.
Assessment type Assessment description Weighting Pass mark
Examination Written examination 100% 40%

You will receive feedback on examinations 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