Last Updated S022020

### BSC202C

 Unit Name Engineering Mathematics 3
 Unit Code BSC202C
 Unit Duration 1 Semester
 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 hoursTutorial – 1.5 hoursGuided labs / Group work / Assessments – 2 hoursPersonal Study recommended – 5 hours

## 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 transform, 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

Bloom’s Level 3

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

Bloom’s Level 3

3. Evaluate complex integration

Bloom’s Level 3

4. Use Conformal mapping for solving engineering problems

Bloom’s Level 3

5. Find solutions for linear systems using numerical methods

Bloom’s Level 4

## Student Assessment

 Assessment Type When assessed Weighting (% of total unit marks) Learning Outcomes Assessed Assessment 1 Type: Multi-choice test / Group work / Short answer questions Example Topic: Laplace transforms, Fourier series Students may complete a quiz with MCQ type answers and solve some simple equations to demonstrate a good understanding of the fundamental concepts. Due after Topic 3 10% 1 Assessment 2 - mid-semester test Type: Multi-choice test / Group work / Short answer questions / Practical Example Topic: Fourier transforms, Fourier integral, FFT Students 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. Due after Topic 6 30% 1, 2 Assessment 3 Type: Multi-choice test / Group work / Short answer questions / Practical Example Topic: Vector differential and integral calculus Students will provide solutions to simple problems related to conformal mapping and use numeric methods to solve problems Due after Topic 9 10% 2, 3, 4 Assessment 4 Type: Examination Example Topic: All topics, numerical methods Questions predominantly related to conformal mapping, interpolation, numeric integration and differentiation, tridiagonalization and QR− factorization An 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 45% 1 to 5 Attendance / Tutorial Participation Example: Presentation, discussion, group work, exercises, self-assessment/reflection, case study analysis, application. Continuous 5% 1 to 5

#### Textbook

P. O'Neil, Advanced Engineering Mathematics, SI Edition, 8th Edition. Cengage, 2018. ISBN 9781337274524

Second Textbook

J. Bird, Higher Engineering Mathematics, 9th Edition. Routledge, 2021. ISBN: 9780367643737

Reference

A. Kreyszig, Advanced Engineering Mathematics Student Solutions Manual, 10th Edition. John Wiley & Sons, 2012. ISBN-13: 978-1118007402

Journal, website

http://www.elsevier.com/physical-sciences/mathematics/mathematics-journals
Notes and Reference texts
Open Textbook Library: http://open.umn.edu/opentextbooks/
Knovel library: http://app.knovel.com
IDC Technologies
Other material advised during the lectures

## Unit Content

#### Topic 1

Laplace Transforms 1

1. Laplace Transform and inverse
2. Elementary functions
3. Transforms of derivatives ad integrals
4. Initial and final value theorems
5. Laplace transform in a solution of initial value problems

#### Topic 2

Laplace Transforms 2

1. Unit step function
2. Short impulses, Dirac's delta function
3. Shifting theorems
4. Convolution
5. Differential equations with polynomial coefficients
6. Transfer function

#### Topic 3

Fourier Series, Integrals and Transforms 1

1. Fourier series
2. Functions having points of discontinuity
3. Convergence of Fourier series
4. Even and Odd functions

#### Topic 4

Fourier Series, Integrals and Transforms 2

1. Fourier cosine and sine series
2. Integration and differentiation of Fourier series
3. Phase angle form of Fourier series
4. Complex Fourier series

#### Topic 5

Fourier Series, Integrals and Transforms 3

1. Fourier integral
2. Fourier cosine and sine integral
3. Complex Fourier integral
4. Fourier transform
5. Fourier Cosine and Sine Transforms
6. Finite Fourier cosine and sine transform
7. Discrete and Fast Fourier transforms
8. Frequency response of a system

#### Topic 6

Vector Differential Calculus

1. Vectors in 2−space and 3−space
2. Velocity, acceleration and Curvature
3. Curves, Arc length
4. Streamlines
5. Gradient of a scalar field and directional derivatives
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. Independence of path
4. Surface integrals
5. Triple integrals, Divergence theorem of Gauss
6. Stokes' theorem

#### Topic 8

Complex Integration

1. Line integral in the complex plane
2. Properties of complex integrals
3. Cauchy's integral theorem
4. Consequences of Cauchy’s theorem
5. Deformation theorem
6. Cauchy's integral formula

#### Topic 9

Conformal Mapping

1. Functions as mapping
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 the equation by iteration
2. Regression
3. Numeric integration review

#### Topic 11

Numerical Methods 2

1. Eigenvalues and eigenvectors
2. Matrix Eigenvalues
3. Tridiagonalization
4. Orthogonal & symmetric matrices

#### Topic 12

Numerical Methods 3

1. Numeric methods for First−Order ODEs
2. Power series methods of solving ODEs
3. Exam revision

## Software/Hardware Used

#### Software

• Software: Python Jupyter Notebook or Google Colab

• Version: N/A

• Instructions:  N/A

• Additional resources or files: N/A

• N/A