# Engineering Institute of Technology

 Unit Name ENGINEERING MATHEMATICS 3 Unit Code BSC202C

 Unit Duration Term Award Bachelor of Science (Engineering) Duration 3 years Year Level Two Unit Creator/Reviewer Core/Elective Core Pre/Co-requisites BSC104C Credit Points 3 Total Course Credit Points 81 (27 x 3) Mode of Delivery Online or on-campus. Unit Workload (Total student workload including “contact hours” = 10 hours per week; 5 hours per week for 24 week delivery)Pre-recordings / Lecture – 1.5 hours (0.75 hours for 24 week delivery)Tutorial – 1.5 hours(0.75 hours for 24 week delivery)Guided labs / Group work / Assessments – 2 hours (1 hour for 24 week delivery)Personal Study recommended – 5 hours (2.5 hours for 24 week delivery)

• This unit may be delivered over 24 weeks (2 Terms) because the nature of the content is deemed suitable (from a pedagogical perspective) for a longer duration than the standard 12 week (1 Term). In addition, these 24-week duration Units require half the student workload hours, 5 hours per week, which allows the total load to be kept at 15 hours per week when combined with a typical 10 hours per week, 12-week Unit. EIT has extensive data to demonstrate that if the load is higher than 15 hours per week the attrition rate for part time students dramatically increases.

# Unit Description and General Aims

This unit builds on the fundamentals discussed in Mathematics units 1 and 2 by providing the student with a sound understanding of advanced engineering mathematical concepts involving vector calculus, Laplace and Fourier transforms, complex numeric functions and statistics. Students will be able to solve problems related to engineering applications by applying these techniques. The topics in the unit are so structured that the student is able to achieve proficiency in all three phases of problem solving viz. modelling, solving the model by applying a suitable mathematical model, and interpreting the results.

# Learning Outcomes

On successful completion of this Unit, students are expected to be able to:

1. Apply Laplace and Fourier transforms

2. Acquire knowledge of vector calculus concepts needed to solve problems across all engineering disciplines

3. Perform complex integration

4. Use Conformal mapping for solving engineering problems

5. Find solutions for linear systems using numerical methods

Professional Development

Completing this unit may add to students professional development/competencies by:

1. Fostering personal and professional skills and attributes in order to:

1. Conduct work in a professionally diligent, accountable and ethical manner.

2. Effectively use oral and written communication in personal and professional domains.

3. Foster applicable creative thinking, critical thinking and problem solving skills.

4. Develop initiative and engagement in lifelong learning and professional development.

5. Enhance collaboration outcomes and performance in dynamic team roles.

6. Effectively plan, organise, self-manage and manage others.

7. Professionally utilise and manage information.

8. Enhance technologist literacy and apply contextualised technologist skills.

2. Enhance investigatory and research capabilities in order to:

1. Develop an understanding of systematic, fundamental scientific, mathematic principles, numerical analysis techniques and statistics applicable to technologists.

2. Access, evaluate and analyse information on technologist processes, procedures, investigations and the discernment of technologist knowledge development.

3. Foster an in-depth understanding of specialist bodies of knowledge, computer science, engineering design practice and contextual factors applicable to technologists.

4. Solve basic and open-ended engineering technologist problems.

5. Understand the scope, principles, norms, accountabilities and bounds associated with sustainable engineering practice.

3. Develop engineering application abilities in order to:

1. Apply established engineering methods to broadly-defined technologist problem solving.

2. Apply engineering technologist techniques, tool and resources.

3. Apply systematic technologist synthesis and design processes.

4. Systematically conduct and manage technologist projects, work assignments, testing and experimentation.

Engineers Australia

The Australian Engineering Stage 1 Competency Standards for Engineering Technologists, approved as of 2013. This table is referenced in the mapping of graduate attributes to learning outcomes and via the learning outcomes to student assessment.

 Stage 1 Competencies and Elements of Competency 1. Knowledge and Skill Base 1.1 Systematic, theory based understanding of the underpinning natural and physical sciences and the engineering fundamentals applicable to the technology domain. 1.2 Conceptual understanding of the, mathematics, numerical analysis, statistics, and computer and information sciences which underpin the technology domain. 1.3 In-depth understanding of specialist bodies of knowledge within the technology domain. 1.4 Discernment of knowledge development within the technology domain. 1.5 Knowledge of engineering design practice and contextual factors impacting the technology domain. 1.6 Understanding of the scope, principles, norms, accountabilities and bounds of sustainable engineering practice in the technology domain. 2. Engineering Application Ability 2.1 Application of established engineering methods to broadly-defined problem solving within the technology domain. 2.2 Application of engineering techniques, tools and resources within the technology domain. 2.3 Application of systematic synthesis and design processes within the technology domain. 2.4 Application of systematic approaches to the conduct and management of projects within the technology domain. 3. Professional and Personal Attributes 3.1 Ethical conduct and professional accountability. 3.2 Effective oral and written communication in professional and lay domains. 3.3 Creative, innovative and pro-active demeanour. 3.4 Professional use and management of information. 3.5 Orderly management of self and professional conduct. 3.6 Effective team membership and team leadership.

Successfully completing this Unit will contribute to the recognition of attainment of the following graduate attributes aligned to the AQF Level 7 criteria, Engineers Australia Stage 1 Competency Standards for Engineering Technologists and the Sydney Accord:

 Graduate Attributes(Knowledge, Skills, Abilities, Professional and Personal Development) EA Stage 1 Competencies Learning Outcomes A. Knowledge of Science and Engineering Fundamentals A1. Breadth of knowledge of engineering and systematic, theory-based understanding of underlying principles, and depth of knowledge across one or more engineering sub- disciplines 1.1, 1.3 1, 2, 3, 4, 5 A2. Knowledge of mathematical, statistical and computer sciences appropriate for engineering technology 1.2 1, 2, 3, 4, 5 A3. Discernment of knowledge development within the technology domain 1.4 1, 2, 3, 4, 5 A4. Knowledge of engineering design practice and contextual factors impacting the technology domain 1.5 B. Problem Solving, Critical Analysis and Judgement B1. Ability to research, synthesise, evaluate and innovatively apply theoretical concepts, knowledge and approaches across diverse engineering technology contexts to effectively solve engineering problems 1.4, 2.1, 2.3 B2. Technical and project management skills to design complex systems and solutions in line with developments in engineering technology professional practice 2.1, 2.2, 2.3, 3.2 C. Effective Communication C1. Cognitive and technical skills to investigate, analyse and organise information and ideas and to communicate those ideas clearly and fluently, in both written and spoken forms appropriate to the audience 3.2 C2. Ability to engage effectively and appropriately across a diverse range of cultures 3.2 D. Design and Project Management D1. Apply systematic synthesis and design processes within the technology domain 2.1, 2.2, 2.3 D2. Apply systematic approaches to the conduct and management of projects within the technology domain 2.4 E. Accountability, Professional and Ethical Conduct E1. Innovation in applying engineering technology, having regard to ethics and impacts including economic; social; environmental and sustainability 1.6, 3.1, 3.4 E2. Professional conduct, understanding and accountability in professional practice across diverse circumstances including team work, leadership and independent work 3.3, 3.4, 3.5, 3.6 2, 4

Unit Competency and Learning Outcome Map

This table details the mapping of the unit graduate attributes to the unit learning outcomes and the Australian Engineering Stage 1 Competency Standards for the Engineering Technologist.

 Graduate Attributes A1 A2 A3 A4 B1 B2 C1 C2 D1 D2 E1 E2 Engineers Australia Stage 1 Competency Standards for Engineering Technologist 1.1  1.2  1.3  1.4   1.5  1.6  2.1    2.2   2.3    2.4  3.1  3.2    3.3  3.4   3.5  3.6  Unit Learning Outcomes LO1    LO2     LO3    LO4     LO5   

# Student assessment

 Assessment Type When assessed Weighting (% of total unit marks) Learning Outcomes Assessed Assessment 1Type: Multi-choice test / Group work / Short answer questionsExample Topic: Laplace transformsStudents may complete a quiz with MCQ type answers and solve some simple equations to demonstrate a good understanding of the fundamental concepts. Week 3(Week 6 for24 week delivery) 15% 1 Assessment 2Type: Multi-choice test / Group work / Short answer questions / PracticalExample Topic: Fourier transforms, Fourier integral, FFTStudents will provide solutions to problems on vector differential and integral calculus and complex integration to show evidence of their understanding of the concepts involved or complete a practical. Week 6(Week 12for 24 week delivery) 20% 1 Assessment 3Type: Multi-choice test / Group work / Short answer questions / PracticalExample Topic: Vector differential and integral calculusStudents will provide solutions to simple problems related to conformal mapping and use numeric methods to solve problems Week 9(Week 18for 24 week delivery) 20% 2, 3 Assessment 4Type: ExaminationExample Topic: All topics, mainly numerical methodsQuestions predominantly related to conformal mapping, interpolation, numeric integration and differentiation, tridiagonalization and QR− factorizationAn examination where the student will complete a quiz with MCQ type answers and perform simple calculations and provide solutions to mathematical problems to be completed in 3 hours Final Week 40% 1 to 5 Attendance / Tutorial ParticipationExample: Presentation, discussion, group work, exercises, self-assessment/reflection, case study analysis, application. Continuous 5% 1 to 5

## Textbook

Bird, J 2014, Higher Engineering Mathematics, 7th Edn, Routledge, ISBN-13: 978- 0415662826

## Reference

Kreyszig, A 2012, Advanced Engineering Mathematics Student Solutions Manual, 10th edn, John Wiley \$ Sons, ISBN-13: 978-1118007402

## Journal, website

https://www.elsevier.com/physical-sciences/mathematics/mathematics-journals

## Notes and Reference texts

Open Textbook Library: https://open.umn.edu/opentextbooks/ Knovel library: https://app.knovel.com

IDC Technologies

Other material advised during the lectures

# Unit Content

One topic is delivered per contact week, with the exception of part-time 24-week units, where one topic is delivered every two weeks.

## Topic 1

Mathematical Induction Proofs

1. Notation

2. Axioms

3. Sums, Series and Sequences

4. Binomial theorem

5. Calculus value theorems

6. Taylor series

7. Green’s theorem

8. Change of variables and the Jacobian

9. Convergence and continuity

10. Measures of lengths and sets

11. Inequalities

## Topic 2

Laplace Transforms 1

1. Laplace Transform and inverse

2. Transforms of derivatives ad integrals

3. Transfer function

## Topic 3

Laplace Transforms 2

1. Unit step function

2. Short impulses, Dirac's delta function

3. Convolution

4. Frequency response

5. Poles

6. Differentiation and integration of transforms

## Topic 4

Fourier Series, Integrals and Transforms 1

1. Fourier series

2. Functions having points of discontinuity

3. Change of interval

4. Even and Odd functions

5. Half−Range expansions

## Topic 5

Fourier Series, Integrals and Transforms 2

1. Approximation by trigonometric polynomials

2. Fourier integral

3. Fourier Cosine and Sine transforms

4. Discrete and Fast Fourier transforms

5. Frequency response of a system

6. Signal processing

## Topic 6

Vector Differential Calculus

1. Vectors in 2−space and 3−space

2. lnner Product (Dot product)

3. Vector Product (Cross product)

4. Curves, Arc length and Curvature

5. Gradient of a scalar field

6. Divergence and curl of a vector field

## Topic 7

Vector Integral Calculus

1. Path independence of line integrals

2. Green's Theorem in the plane

3. Surface integrals

4. Triple integrals, Divergence theorem of Gauss

5. Stokes' theorem

## Topic 8

Complex Integration

1. Line integral in the complex plane

2. Cauchy's integral theorem

3. Cauchy's integral formula

4. Derivatives of analytic functions

## Topic 9

Conformal Mapping

1. Geometry of analytic functions

2. Linear fractional transformation

3. Special linear fractional transformations

4. Conformal mapping by other functions

5. Modelling and use of conformal mapping

## Topic 10

Numerical Methods 1

1. Solution of equation by iteration

2. Interpolation

3. Spline interpolation

4. Numeric integration and differentiation

5. Linear Systems: Gauss elimination, LU factorization, Matrix inversion

6. Least Squares method

## Topic 11

Numerical Methods 2

1. Eigen values and eigenvectors

2. Matrix Eigenvalues

3. Tridiagonalization and QR− Factorization

## Topic 12

Numerical Methods 3

1. Numeric methods for First−Order ODEs

2. Methods for Systems and Higher Order ODEs

3. Introduction to methods for Elliptic, Parabolic PDEs, Hyperbolic PDEs

4. Exam revision

The Engineering Institute of Technology (EIT) is dedicated to ensuring our students receive a world-class education and gain skills they can immediately implement in the workplace upon graduation. Our staff members uphold our ethos of honesty and integrity, and we stand by our word because it is our bond. Our students are also expected to carry this attitude throughout their time at our institute, and into their careers.