This course provides the essential mathematics needed throughout all engineering disciplines. Topics covered include: Functions of several variables; Partial differentiation; Line integrals and multidimensional integrals; Ordinary Differential Equations; Laplace Transforms; Fourier Series.

4 lectures per week

None

None

100% exam

This course aims to ensure that students are competent in the essential mathematics required for engineering programmes.

By the end of this course students will be able to:

Functions of several variables (MEM Sections 9.5 to 9.7)

■ use contour plots to visualise functions of more than one variable;

■ calculate partial derivatives, including chain rule, product rule, quotient rule, etc.;

■ evaluate the total derivative and use it to estimate experimental errors;

■ establish whether a differential is "exact" and determine its parent function;

■ apply Taylor's theorem for functions of many variables;

■ find the stationary points of a function of two variables and determine their nature;

Line integrals and multidimensional integrals (AMEM Section 3.4)

■ define and evaluate integrals along a contour in the plane;

■ evaluate double and triple integrals by repeated integration;

Ordinary Differential Equations (MEM Chapter 10)

■ derive differential equations for simple engineering systems and be able to derive appropriate boundary or initial conditions;

■ classify differential equations as to order and degree, ordinary or partial, homogeneous or inhomogeneous, linear or nonlinear;

■ recognise separable differential equations and solve by integration of each side;

■ recognise first-order linear differential equations and solve by the integrating factor method;

■ recognise the form of the solution of higher-order differential equations: linear independence of solutions, general solution with arbitrary coefficients, complementary function and particular integral;

■ obtain the general solution for second-order, ordinary, differential equations by the trial function method;

■ solve second order ODEs using the auxiliary equation, complementary function and particular integral;

■ solve differential equations numerically;

■ derive and solve some examples of second order partial differential equations;

Fourier Series (MEM Chapter 12, AMEM Chapter 7)

■ determine the Fourier series representation of simple periodic functions using the trigonometric and complex exponential forms, using the symmetry properties of the function as appropriate;

■ apply Fourier series to solve engineering problems with a periodic input.

Introduction to Vector calculus (AMEM Sections 3.2 and 3.3)

■ define and calculate the gradient vector of a scalar function, and explain its magnitude and direction;

■ define and apply the test for conservative vector fields and potential functions, and calculate a potential function for a conservative vector field;

■ state the divergence and curl of a vector field in Cartesian coordinates;

■ apply the test for whether a vector field can be represented by a vector potential (solenoidal);

Laplace Transforms (MEM Chapter 11)

■ derive the Laplace Transform for simple functions, and of more general functions and derivatives using known properties of the transform and tables of transforms;

■ derive the inverse Laplace transforms of standard functions, and of more general functions using known properties of the inverse transform and tables of transforms;

■ solve ordinary differential equations up to second order with initial conditions using the method of Laplace transforms;

■ use the techniques for differential equations to solve engineering problems;

This course follows the syllabus in Modern Engineering Mathematics (MEM) and Advanced Modern Engineering Mathematics (AMEM) by Glyn James (Pearson).

Students must attend the degree examination and submit at least 75% by weight of the other components of the course's summative assessment.

Students should attend at least 75% of the timetabled classes of the course.

Note that these are minimum requirements: good students will achieve far higher participation/submission rates.  Any student who misses an assessment or a significant number of classes because of illness or other good cause should report this by completing a MyCampus absence report.