# Mechanical Engineering BEng/MEng

## Engineering Mathematics 2 ENG2086

**Academic Session:**2024-25**School:**School of Engineering**Credits:**20**Level:**Level 2 (SCQF level 8)**Typically Offered:**Semester 1**Available to Visiting Students:**No**Collaborative Online International Learning:**No

### Short Description

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.

### Timetable

4 lectures per week

### Excluded Courses

None

### Co-requisites

None

### Assessment

100% exam

**Main Assessment In:** December

### Course Aims

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

### Intended Learning Outcomes of Course

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).

### Minimum Requirement for Award of Credits

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.