Outline

Overview

This unit opens with topics from vector calculus, including vector-valued functions (parametrised curves and surfaces; vector fields; div, grad and curl; gradient fields and potential functions), line integrals (arc length; work; path-independent integrals and conservative fields; flux across a curve), iterated integrals (double and triple integrals, polar, cylindrical and spherical coordinates; areas, volumes and mass; Green's Theorem), flux integrals (flow through a surface; flux integrals through a surface defined by a function of two variables, through cylinders, spheres and other parametrised surfaces), Gauss' and Stokes' theorems. The unit then moves to topics in solution techniques for ordinary and partial differential equations (ODEs and PDEs) with applications. It provides a basic grounding in these techniques to enable students to build on the concepts in their subsequent courses. The main topics are: second order ODEs (including inhomogeneous equations), higher order ODEs and systems of first order equations, solution methods (variation of parameters, undetermined coefficients) the Laplace and Fourier Transform, an introduction to PDEs, and first methods of solutions (including separation of variables, and Fourier Series).

Unit details and rules

Academic unit	Mathematics and Statistics Academic Operations
Credit points	6
Prerequisites ?	(MATH1X21 or MATH1931 or MATH1X01 or MATH1906) and (MATH1XX2) and (MATH1X23 or MATH1933 or MATH1X03 or MATH1907)
Corequisites ?	None
Prohibitions ?	MATH2921 or MATH2065 or MATH2965 or (MATH2061 and MATH2022) or (MATH2061 and MATH2922) or (MATH2961 and MATH2022) or (MATH2961 and MATH2922) or MATH2067
Assumed knowledge ?	None
Available to study abroad and exchange students	Yes

Teaching staff

Coordinator	Zhou Zhang, zhou.zhang@sydney.edu.au

Type	Description	Weight	Due	Length
Final exam (Take-home short release)	Final Exam Type D	60%	Formal exam period	2 hours
Outcomes assessed:
Tutorial quiz	Quiz 1 Take home quiz	15%	Week 06	16 pages
Outcomes assessed:
Assignment	Assignment 1 Written assignment	5%	Week 06	2 weeks
Outcomes assessed:
Tutorial quiz	Quiz 2 Take home quiz	15%	Week 11	16 pages
Outcomes assessed:
Assignment	Assignment 2 Written assignment	5%	Week 11	2 weeks
Outcomes assessed:
= Type D final exam ?

Assessment summary

Assignment: assignments will require you to integrate information from lectures and practicals to create a concise written solution.
Quiz: quizzes will be take home and test your understanding of material covered in the past few weeks. Online assigning and submission. A shorter time is allowed than Assignments 1 and 2. The aim is to have a quiz-like environment at home
Exam: exam will cover all material in the unit from both lectures and practical classes. The exam will mostly have short answer questions.

Detailed information for each assessment can be found on Canvas.

Assessment criteria

The University awards common result grades, set out in the Coursework Policy 2014 (Schedule 1).

As a general guide, a high distinction indicates work of an exceptional standard, a distinction a very high standard, a credit a good standard, and a pass an acceptable standard.

Result name	Mark range	Description
High distinction	85 - 100	Representing complete or close to complete mastery of the material;
Distinction	75 - 84	Representing excellence, but substantially less than complete mastery;
Credit	65 - 74	Representing a creditable performance that goes beyond routine knowledge and understanding, but less than excellence;
Pass	50 - 64	Representing at least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course.
Fail	0 - 49	When you don’t meet the learning outcomes of the unit to a satisfactory standard.

For more information see guide to grades.

Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

Deduction of 5% of the maximum mark for each calendar day after the due date.
After ten calendar days late, a mark of zero will be awarded.

Academic integrity

The Current Student website provides information on academic integrity and the resources available to all students. The University expects students and staff to act ethically and honestly and will treat all allegations of academic integrity breaches seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic integrity breach. If such matches indicate evidence of plagiarism or other forms of academic integrity breaches, your teacher is required to report your work for further investigation.

Use of generative artificial intelligence (AI) and automated writing tools

You may only use generative AI and automated writing tools in assessment tasks if you are permitted to by your unit coordinator. If you do use these tools, you must acknowledge this in your work, either in a footnote or an acknowledgement section. The assessment instructions or unit outline will give guidance of the types of tools that are permitted and how the tools should be used.

Your final submitted work must be your own, original work. You must acknowledge any use of generative AI tools that have been used in the assessment, and any material that forms part of your submission must be appropriately referenced. For guidance on how to acknowledge the use of AI, please refer to the AI in Education Canvas site.

The unapproved use of these tools or unacknowledged use will be considered a breach of the Academic Integrity Policy and penalties may apply.

Studiosity is permitted unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission as detailed on the Learning Hub’s Canvas page.

Outside assessment tasks, generative AI tools may be used to support your learning. The AI in Education Canvas site contains a number of productive ways that students are using AI to improve their learning.

Learning support

Simple extensions

If you encounter a problem submitting your work on time, you may be able to apply for an extension of five calendar days through a simple extension.  The application process will be different depending on the type of assessment and extensions cannot be granted for some assessment types like exams.

Special consideration

If exceptional circumstances mean you can’t complete an assessment, you need consideration for a longer period of time, or if you have essential commitments which impact your performance in an assessment, you may be eligible for special consideration or special arrangements.

Special consideration applications will not be affected by a simple extension application.

Using AI responsibly

Co-created with students, AI in Education includes lots of helpful examples of how students use generative AI tools to support their learning. It explains how generative AI works, the different tools available and how to use them responsibly and productively.

Weekly schedule

WK	Topic	Learning activity
Week 01	Vectors-valued functions, curves, and the line integral.	Lecture (3 hr)
Week 01	Introduction into Mathematica	Computer laboratory (1 hr)
Week 02	Work done by a force, Vector fields, gradient and curl-vector, normals to surfaces, conservative fields, and potential functions.	Lecture (3 hr)
	Vectors-valued functions, curves, and the line integral.	Computer laboratory (1 hr)
	Vectors-valued functions, curves, and the line integral.	Tutorial (1 hr)
Week 03	Double integrals. Area, volume and mass, divergence of a vector field. Green’s theorem. Flux across a curve.	Lecture (3 hr)
	Work done by a force, Vector fields, gradient and curl-vector, normals to surfaces, conservative fields, and potential functions.	Computer laboratory (1 hr)
	Work done by a force, Vector fields, gradient and curl-vector, normals to surfaces, conservative fields, and potential functions.	Tutorial (1 hr)
Week 04	Surface area. Surface integrals. Flux across a surface.	Lecture (3 hr)
	Double integrals. Area, volume and mass, divergence of a vector field. Green’s theorem. Flux across a curve.	Computer laboratory (1 hr)
	Double integrals. Area, volume and mass, divergence of a vector field. Green’s theorem. Flux across a curve.	Tutorial (1 hr)
Week 05	Double and Triple integrals, Change of coordinates systems: Polar, cylindrical, or spherical coordinates.	Lecture (3 hr)
	Surface area. Surface integrals. Flux across a surface.	Computer laboratory (1 hr)
	Surface area. Surface integrals. Flux across a surface.	Tutorial (1 hr)
Week 06	Gauss’ divergence theorem and Stokes’ theorem.	Lecture (3 hr)
	Double and Triple integrals, Change of coordinates systems: Polar, cylindrical, or spherical coordinates.	Computer laboratory (1 hr)
	Double and Triple integrals, Change of coordinates systems: Polar, cylindrical, or spherical coordinates.	Tutorial (1 hr)
Week 07	Revision on 1st- and 2nd-order linear differential equations.	Lecture (3 hr)
	Gauss’ divergence theorem and Stokes’ theorem.	Computer laboratory (1 hr)
	Gauss’ divergence theorem and Stokes’ theorem.	Tutorial (1 hr)
Week 08	Method of variation of constants, method of undetermined coefficients, method of reduction of order.	Lecture (3 hr)
	Solving 1st- and 2nd-order linear differential equations with Mathematica.	Computer laboratory (1 hr)
	Revision on 1st- and 2nd-order linear differential equations.	Tutorial (1 hr)
Week 09	The Frobenius method (or power series solutions) and Laplace transform to solve 2-nd order linear ordinary differential equations.	Lecture (3 hr)
	Numerical methods to solve ODEs: Runge Kutta, Euler iteration scheme.	Computer laboratory (1 hr)
	Method of variation of constants, method of undetermined coefficients, method of reduction of order.	Tutorial (1 hr)
Week 10	The Fourier transform and solving 2nd-order linear ordinary differential equations with discontinuous forcing terms.	Lecture (3 hr)
	The Frobenius method (or power series solutions) and Laplace transform to solve 2-nd order linear ordinary differential equations.	Computer laboratory (1 hr)
	The Frobenius method (or power series solutions) and Laplace transform to solve 2-nd order linear ordinary differential equations.	Tutorial (1 hr)
Week 11	Boundary-value problems, eigenvalue equations, and the Sturm-Liouville operator on an interval.	Lecture (3 hr)
	Computing the Fourier-transform of functions with Mathematica.	Computer laboratory (1 hr)
	The Fourier transform and solving 2nd-order linear ordinary differential equations with discontinuous forcing terms.	Tutorial (1 hr)
Week 12	Separation of variables and solving the Laplace equation on a rectangle, the heat and wave equation on a rod.	Lecture (3 hr)
	Solving boundary-value problems and eigenvalues in Mathematica.	Computer laboratory (1 hr)
	Boundary-value problems, eigenvalue equations, and the Sturm-Liouville operator on an interval.	Tutorial (1 hr)
Week 13	Revision and exam preparation!	Lecture (3 hr)
	Solving the Laplace equation on a rectangle and the heat and wave equation on a rod by using Mathematica.	Computer laboratory (1 hr)
	Solving the Laplace equation on a rectangle, the heat and wave equation on a rod.	Tutorial (1 hr)

Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

Learning outcomes

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University's graduate qualities and are assessed as part of the curriculum.

Outcomes
Graduate qualities

At the completion of this unit, you should be able to:

LO1. demonstrate a conceptual understanding of vector-valued functions, partial derivatives, curves, and integration over a region, volume, and surface as well as solving basic differential equations thorough background in a variety of techniques and applications of mathematical analysis.
LO2. understanding the definitions, main theorems, and corollaries for path integrals, conservative fields, multiple integrals Green's theorem, Gauss' theorem, and Stokes' theorem, but also to understand the structure of the solutions of linear differential equations, the method of series solutions, the Laplace transform, solving boundary-value problems involving Sturm-Liouville operators on either a bounded interval or a rectangle, and to understand what is an eigenvalue.
LO3. be fluent with substitutions in integrals and changing coordinate systems from cartesian into polar, cylindrical, or spherical ones.
LO4. develop appreciation and strong working knowledge of the theory and applications of elementary Vector Analysis and Differential Equations.
LO5. be fluent with important examples, theorems, and applications of the theory and be able to implement these into Mathematica software.

Graduate qualities

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

GQ1	Depth of disciplinary expertise Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline.
GQ2	Critical thinking and problem solving Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem.
GQ3	Oral and written communication Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context.
GQ4	Information and digital literacy Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies.
GQ5	Inventiveness Generating novel ideas and solutions.
GQ6	Cultural competence Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues.
GQ7	Interdisciplinary effectiveness Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries.
GQ8	Integrated professional, ethical, and personal identity An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context.
GQ9	Influence Engaging others in a process, idea or vision.

Outcome map

Learning outcomes	Graduate qualities
Learning outcomes

Responding to student feedback

This section outlines changes made to this unit following staff and student reviews.

In this year, we swapped the Vector Calculus part and the Differential Equations part.

Additional information

During the semester, students will receive emails via Canvas about news, changes and updates about the lecture and tutorials.

Work, health and safety

We are governed by the Work Health and Safety Act 2011, Work Health and Safety Regulation 2011 and Codes of Practice. Penalties for non-compliance have increased. Everyone has a responsibility for health and safety at work. The University’s Work Health and Safety policy explains the responsibilities and expectations of workers and others, and the procedures for managing WHS risks associated with University activities.

Disclaimer

The University reserves the right to amend units of study or no longer offer certain units, including where there are low enrolment numbers.

To help you understand common terms that we use at the University, we offer an online glossary.

Current students

Overview

Overview

Unit details and rules

Teaching staff

Assessment

Assessment

Assessment summary

Assessment criteria

Late submission

Academic integrity

Learning support

Learning support

Simple extensions

Special consideration

Using AI responsibly

Weekly schedule

Weekly schedule

Study commitment

Learning outcomes

Learning outcomes

Graduate qualities

Outcome map

Responding to student feedback

Responding to student feedback

Additional information

Additional information

Work, health and safety

Disclaimer

On this page

Useful links

Media

Student links

About us

Connect

Current students

MATH2021: Vector Calculus and Differential Equations

Semester 1, 2021 [Normal day] - Remote

Overview

Overview

Unit details and rules

Teaching staff

Assessment

Assessment

Assessment summary

Assessment criteria

Late submission

Academic integrity

Learning support

Learning support

Simple extensions

Special consideration

Using AI responsibly

Weekly schedule

Weekly schedule

Study commitment

Learning outcomes

Learning outcomes

Graduate qualities

Outcome map

Responding to student feedback

Responding to student feedback

Additional information

Additional information

Work, health and safety

Disclaimer

On this page

Useful links

Media

Student links

About us

Connect