Short Title:  Mathematical Methods 

Full Title:  Mathematical Methods 

Language of Instruction:  English 

Field of Study:  Mechanics and metal work 

Reviewed By:  DIARMUID RUSH 

Module Author:  NOEL GORMAN 

Module Description:  This module aims to equip students with the calculus and linear algebra based mathematical methods needed in the analysis of many engineering systems 

Learning Outcomes 
On successful completion of this module the learner will be able to: 
LO1 
Use Taylorâ€™s Theorem and Maclaurin series to construct approximation schemes. 
LO2 
Test a power series for convergence using the Ratio Test. 
LO3 
Calculate derivatives implicitly and partial derivatives with the nabla operator. 
LO4 
Calculate single integrals by reduction formulas and multiple integrals by changing variables. 
LO5 
Use integral transform methods to solve ordinary and partial differential equations occurring in engineering. 
LO6 
Apply the algebra of matrices to engineering applications. 
LO7 
Relate the concepts of matrices and vectors in both geometric and algebraic applications. 
LO8 
Solve dense and tridiagonal systems of equations occurring in engineering modelling directly. 
LO9 
Solve sparse systems of equations occurring in engineering modelling using efficient iterative methods. 
LO10 
Recognise the importance of conditioning and sensitivity in the use of Linear Algebra in the solution of real engineering models using computers. 
Prerequisite learning 

Corequisite Modules
 No Corequisite modules listed 
Module Content & Assessment
Content (The percentage workload breakdown is inidcative and subject to change) 
% 
Limits and Series. Properties of limits. Differentiability. Implicit differentiation. Derivative of an inverse function. Taylorâ€™s theorem. Maclaurin series. Application to approximation schemes for solving nonlinear equations and calculating integrals. Convergence of power series and the Ratio Test. Integration by parts. Reduction formulas. The Gamma function.

20.00% 
Calculus of Several Variables Chain rule for partial derivatives. Cylindrical and spherical polar coordinates. Directional derivative. The nabla operator. Calculation of gradient of a scalar field and divergence and curl of a vector field. Double and Triple integrals. Change of variables.

25.00% 
Laplace Transforms Review of Laplace transform solution of secondorder linear differential equations, and systems of firstorder differential equations, with constant coefficients. Applications to engineering systems. Solution of classical partial differential equations by separation of variables. Laplace transform solution of partial differential equations occurring in engineering.

20.00% 
Vectors Vectors in two and three dimensions. Geometric interpretation. Norm of a vector. Dot Product. Lines and planes. Extension to higher dimensions.

5.00% 
Vector spaces. Definition of a Vector Space. Basis. Dimension. Spanning set. Linear independence.

5.00% 
Inner Products Inner products. Norms. Orthogonality. Inner product spaces.

5.00% 
Linear Transformations Definition of a Linear Transformation. Rotations in 2D and 3D. Kernel and Range. Matrices as Linear Transformations. The four fundamental subspaces.

5.00% 
Matrices Matrices. Revision of matrix algebra. Classes of matrices. Inverse of a matrix. Determinants.

5.00% 
Systems of Equations Systems of Equations. Partial pivoting. Gaussian Elimination in the context of LU Decomposition, Thomas Algorithm. Uniqueness of solutions.

5.00% 
Numerical Linear Algebra Condition of a matrix. Iterative methods for sparse systems: Jacobi, GaussSeidel, SOR.

5.00% 
Assessment Breakdown  % 
Course Work  20.00% 
End of Module Formal Examination  80.00% 
Course Work 
Assessment Type 
Assessment Description 
Outcome addressed 
% of total 
Assessment Date 
Continuous Assessment 
In class tests in calculus methods and linear algebra. 
1,2,3,6,7,8 
20.00 
n/a 
End of Module Formal Examination 
Assessment Type 
Assessment Description 
Outcome addressed 
% of total 
Assessment Date 
Formal Exam 
EndofSemester Final Examination 
1,2,3,4,5,6,7,8,9,10 
80.00 
EndofSemester 
Reassessment Requirement 

Repeat examination Reassessment of this module will consist of a repeat examination. It is possible that there will also be a requirement to be reassessed in a coursework element. 
IT Tallaght reserves the right to alter the nature and timings of assessment Module Workload
Workload: Full Time 
Workload Type 
Workload Description 
Hours 
Frequency 
Average Weekly Learner Workload 
Lecture 
No Description 
6.00 
Every Week 
6.00 
Independent Learning Time 
No Description 
4.00 
Every Week 
4.00 
Total Weekly Learner Workload 
10.00 
Total Weekly Contact Hours 
6.00 
Workload: Part Time 
Workload Type 
Workload Description 
Hours 
Frequency 
Average Weekly Learner Workload 
Lecture 
4 
4.00 
Every Week 
4.00 
Independent Learning 
3 
3.00 
Every Week 
3.00 
Total Weekly Learner Workload 
7.00 
Total Weekly Contact Hours 
4.00 
Module ResourcesRequired Book Resources 

 Recommended Book Resources 

 Erwin Kreyszig 2011, Advanced Engineering Mathematics, 10th Ed. Ed., John Wiley & Sons [ISBN: 978047064613]
 K A Stroud 2011, Advanced Engineering Mathematics, 7th Ed. Ed., Palgrave Macmillan [ISBN: 978023027548]
 Glyn James, David Burley, Dick Clements, Phil Dyke, John Searl 2010, Advanced Modern Engineering Mathematics, Prentice Hall [ISBN: 9780273719236]
 Gilbert Strang, Introduction to Linear Algebra, 10th ed. Ed., Wellesley Cambridge Pr [ISBN: 9780980232714]
 Gilbert Strang 2006, Linear algebra and its applications, Thomson Brooks/Cole Belmont, Calif. [ISBN: 9780030105678]
 Howard Anton, Robert C. Busby 2003, Contemporary linear algebra, Wiley New York [ISBN: 0471163627]
 Howard Anton, Chris Rorres, Elementary Linear Algebra with Supplemental Applications, Wiley [ISBN: 9780470561577]
 This module does not have any article/paper resources 

This module does not have any other resources 

Module Delivered in
