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MATH2020 – Vector Calculus and Complex Analysis

2022 – Session 2, In person-scheduled-weekday, North Ryde

General Information

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Unit convenor and teaching staff Unit convenor and teaching staff
Paul Bryan
Ross Moore
Credit points Credit points
10
Prerequisites Prerequisites
MATH2010 or MATH2055 or MATH235
Corequisites Corequisites
Co-badged status Co-badged status
Unit description Unit description

The topics covered in this unit lay the foundations for further study in modern areas of mathematics (such as partial differential equations, fluid mechanics, and mathematical biology). This unit builds on the first year single variable calculus units by extending calculus to several variables, and focuses primarily on integration techniques for complex functions and vector fields. Complex analysis is the study of complex-valued functions of complex variables. The main properties of complex functions of a single complex variable will be presented, including the important concepts of analyticity and singularity structure. This will be followed by a treatment of Cauchy's theorem and the residue theorem to evaluate contour integrals of complex functions around various curves in the complex plane. Vector calculus is the study of vector fields in two and three dimensions, and facilitates the modelling of a variety of physical phenomena, for example in fluid mechanics and electromagnetism. By introducing the gradient, divergence and curl operators, the main properties of vector fields can be analysed. A variety of integrals of vector fields over paths, surfaces and volumes will be performed, and the application of three important integral theorems of vector calculus due to Green, Stokes and Gauss to evaluate these integrals will be demonstrated.

Important Academic Dates

Information about important academic dates including deadlines for withdrawing from units are available at https://www.mq.edu.au/study/calendar-of-dates

Learning Outcomes

On successful completion of this unit, you will be able to:

  • ULO1: Analyse the main properties of functions of a single complex variable, such as analyticity and singularity structure.
  • ULO2: Evaluate contour integrals of complex functions by applying Cauchy's theorem and the residue theorem.
  • ULO3: Analyse the main properties of vector fields using the gradient, divergence and curl operators.
  • ULO4: Evaluate path, surface and volume integrals of vector fields.
  • ULO5: Apply the important theorems due to Green, Stokes and Gauss to physical applications.

General Assessment Information

Late Assessment Submission Penalty   

From 1 July 2022, Students enrolled in Session based units with written assessments will have the following late penalty applied. Please see https://students.mq.edu.au/study/assessment-exams/assessments for more information.   

Unless a Special Consideration request has been submitted and approved, a 5% penalty (of the total possible mark) will be applied each day a written assessment is not submitted, up until the 7th day (including weekends). After the 7th day, a grade of '0' will be awarded even if the assessment is submitted. Submission time for all written assessments is set at 11:55 pm. A 1-hour grace period is provided to students who experience a technical concern.   

For any late submission of time-sensitive tasks, such as scheduled tests/exams, performance assessments/presentations, and/or scheduled practical assessments/labs, students need to submit an application for Special Consideration.    

Assessments where Late Submissions will be accepted 

In this unit, late submissions will be accepted as follows: 

Assignment 1, Assignment 2 – YES, Standard Late Penalty applies  Test 1, Test 2, Final Exam - NO, unless Special Consideration is granted

Assessment Tasks

Name Weighting Hurdle Due
Test 2 (Online) 15% No Week 10
Final exam 50% No Examination Period
Assignment 2 10% No Week 12
Assignment 1 10% No Week 7
Test 1 (Online) 15% No Week 4

Test 2 (Online)

Assessment Type 1: Quiz/Test
Indicative Time on Task 2: 10 hours
Due: Week 10
Weighting: 15%

 

Online test

 


On successful completion you will be able to:
  • Analyse the main properties of functions of a single complex variable, such as analyticity and singularity structure.
  • Evaluate contour integrals of complex functions by applying Cauchy's theorem and the residue theorem.
  • Analyse the main properties of vector fields using the gradient, divergence and curl operators.
  • Evaluate path, surface and volume integrals of vector fields.
  • Apply the important theorems due to Green, Stokes and Gauss to physical applications.

Final exam

Assessment Type 1: Examination
Indicative Time on Task 2: 15 hours
Due: Examination Period
Weighting: 50%

 

Summative examination, held during the university examination period.

 


On successful completion you will be able to:
  • Analyse the main properties of functions of a single complex variable, such as analyticity and singularity structure.
  • Evaluate contour integrals of complex functions by applying Cauchy's theorem and the residue theorem.
  • Analyse the main properties of vector fields using the gradient, divergence and curl operators.
  • Evaluate path, surface and volume integrals of vector fields.
  • Apply the important theorems due to Green, Stokes and Gauss to physical applications.

Assignment 2

Assessment Type 1: Problem set
Indicative Time on Task 2: 10 hours
Due: Week 12
Weighting: 10%

 

The assignments reinforce and build on material from lectures, and involve calculations and explanations.

 


On successful completion you will be able to:
  • Analyse the main properties of functions of a single complex variable, such as analyticity and singularity structure.
  • Evaluate contour integrals of complex functions by applying Cauchy's theorem and the residue theorem.
  • Analyse the main properties of vector fields using the gradient, divergence and curl operators.
  • Evaluate path, surface and volume integrals of vector fields.
  • Apply the important theorems due to Green, Stokes and Gauss to physical applications.

Assignment 1

Assessment Type 1: Problem set
Indicative Time on Task 2: 10 hours
Due: Week 7
Weighting: 10%

 

The assignments reinforce and build on material from lectures, and involve calculations and explanations.

 


On successful completion you will be able to:
  • Analyse the main properties of functions of a single complex variable, such as analyticity and singularity structure.
  • Evaluate contour integrals of complex functions by applying Cauchy's theorem and the residue theorem.
  • Analyse the main properties of vector fields using the gradient, divergence and curl operators.
  • Evaluate path, surface and volume integrals of vector fields.
  • Apply the important theorems due to Green, Stokes and Gauss to physical applications.

Test 1 (Online)

Assessment Type 1: Quiz/Test
Indicative Time on Task 2: 10 hours
Due: Week 4
Weighting: 15%

 

Online test

 


On successful completion you will be able to:
  • Analyse the main properties of functions of a single complex variable, such as analyticity and singularity structure.
  • Evaluate contour integrals of complex functions by applying Cauchy's theorem and the residue theorem.
  • Analyse the main properties of vector fields using the gradient, divergence and curl operators.
  • Evaluate path, surface and volume integrals of vector fields.
  • Apply the important theorems due to Green, Stokes and Gauss to physical applications.

1 If you need help with your assignment, please contact:

  • the academic teaching staff in your unit for guidance in understanding or completing this type of assessment
  • the Writing Centre for academic skills support.

2 Indicative time-on-task is an estimate of the time required for completion of the assessment task and is subject to individual variation

Delivery and Resources

Classes

Lectures: there 2 x 1hr lectures and 1 x 2hr SGTA each week.

Unit Schedule

Vector Calculus

Week Topic Assessment
1 Differentiation, curves, vector fields  
2 Line integrals  
3 Green's theorem, Surfaces  
4 Surface integrals,flux Test 1
5 Divergence Theorem, Kelvin-Stokes' Theorem  
6 Review and catch up  

 

Complex Analysis

Week Topic Assessment
7 Intro. to Complex Analysis,  Complex Functions Assignment 1
8 Analytic Functions,  Complex Logarithm  
9  (Monday Holiday)    Complex Integration  
10 Cauchy's Integral Theorem,   Cauchy's Integral Formula Test 2
11 Taylor's Theorem,   Laurent Series  
12 Isolated Zeros,  Isolated Singularities Assignment 2
13  Cauchy's Residue Theorem                      Review  

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At Macquarie, we believe academic integrity – honesty, respect, trust, responsibility, fairness and courage – is at the core of learning, teaching and research. We recognise that meeting the expectations required to complete your assessments can be challenging. So, we offer you a range of resources and services to help you reach your potential, including free online writing and maths support, academic skills development and wellbeing consultations.

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