## MATHEMATICS 2

 SHE Level 2 SCQF Credit Points 20.00 ECTS Credit Points 10.00 Module Code M2G121927 Module Leader Steven Walters School School of Computing, Engineering and Built Environment Subject Mechanical Engineering Trimester A (September start)

### Pre-Requisite Knowledge

Mathematics 1 (M1G108778) or Advanced Higher Mathematics

### Summary of Content

The module is broadly divided into two strands. One strands deals with the solution of ordinary differential equations through a variety of analytical and numerical techniques, one of which is the method of Laplace transforms. The other strand covers a range of applied mathematical topics including double integration, eigenvalues and eigenvectors of square matrices, and Fourier series. The second strand also introduces the student to some basic statistical techniques for the handling of experimental data. Engineering applications of the mathematics and statistics are considered whenever appropriate.

### Syllabus

STATISTICS Determine the mean and standard deviation of ungrouped and grouped data. Classical definition of probability, Normal distribution, Determine confidence limits for the population mean from a sample. One-sample hypothesis testing of the population mean. Univariate regression and correlation. ORDINARY DIFFERENTIAL EQUATIONS Analytical solutions of 1st order, separable ODEs, 1st order linear ODEs and 2nd order, linear ODEs with constant coefficients. Numerical solution of ODEs: Euler's method, Runge-Kutta. LAPLACE TRANSFORMS Definition, properties and applications to 1st and 2nd order ODEs. Unit step function, Dirac function Transforms of the unit step function and the Dirac delta function. Convolution. DOUBLE INTEGRATION Cartesian form, change of order, polar form. Applications: pressure and density distributions, moment of inertia. SYSTEMS OF LINEAR ODE'S USING DIAGONALISATION METHOD Determine eigenvalues and eigenvectors of matrices up to order 3. Solution of systems of ODEs. FOURIER SERIES Definition. Determination using analytical and numerical techniques.

### Learning Outcomes

Upon completion of this module the student should be able to:- apply appropriate tabular and graphical methods of data presentation and calculate and interpret appropriate summary statistics;- calculate probabilities using the rules of probability and the Normal distribution;- calculate and interpret point and interval estimates of population parameters based on empirical data;- use regression techniques to model relationships between physical quantities from experimental data;- solve 1st order separable and linear, and 2nd order linear, constant coefficient ordinary differential equations analytically;- solve 1st order ordinary differential equations numerically;- use table to determine Laplace Transforms and inverse Laplace Transforms;- manipulate unit step function- apply Laplace Transforms to solve 1st and 2nd order linear, constant coefficient ordinary differential equations, possibly containing step or Dirac delta functions;- evaluate double integrals expressed in Cartesian form;- interchange the order of integration of a double integral;- convert a double integral from Cartesian to polar form;- determine the (real, distinct) eigenvalues and eigenvectors of a matrix up to order 3;- determine the solution to systems of ordinary differential equations using matrix methods;- define a Fourier series of a periodic function;- determine Fourier coefficients using an appropriate computer package;- approximate the Fourier coefficients of a sampled function (or signal) using numerical integration techniques.Through the study of a variety of mathematical techniques, the student should continue to develop his or her analytical and numerical skills. The statistical element of the module should enhance the student's ability in organising and interpreting data. The continuous assessment will give the student an opportunity to apply and develop skills in written communications and information technology.

### Teaching / Learning Strategy

Combination of lectures and tutorials. Coursework will normally be numerically-based. Lectures, 3, Tutorials/Labs, 3 hours/week. Students are encouraged to use appropriate software.

Stroud K A (2007) "Engineering Mathematics" (6th Ed) Palgrave Macmillan Croft A and Davidson R (2008) "Mathematics for Engineers - A Modern Interactive Approach" (3rd Ed) Prentice Hall

### Transferrable Skills

- Critical thinking and problem solving; - Cognitive / intellectual skills; - Knowledge and understanding in the context of the subject; - Independent working; - IT Skills - Communication skills, written, oral and listening.

### Module Structure

Activity Total Hours
Lectures (PT) 36.00
Independent Learning (FT) 122.00
Assessment (PT) 18.00
Practicals (FT) 4.00
Assessment (FT) 18.00
Tutorials (FT) 20.00
Lectures (FT) 36.00
Practicals (PT) 4.00
Independent Learning (PT) 128.00
Tutorials (PT) 14.00

### Assessment Methods

Component Duration Weighting Threshold Description
Coursework 1 n/a 30.00 35% As detailed by module/coursework leader
Exam (Exams Office) 2.00 70.00 35% Examination