Unit outline_

# MATH2061: Linear Mathematics and Vector Calculus

## Overview

This unit starts with an investigation of linearity: linear functions, general principles relating to the solution sets of homogeneous and inhomogeneous linear equations (including differential equations), linear independence and the dimension of a linear space. The study of eigenvalues and eigenvectors, begun in junior level linear algebra, is extended and developed. The unit then moves on to 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, though cylinders, spheres and parametrised surfaces), Gauss' Divergence Theorem and Stokes' Theorem.

### Unit details and rules

Academic unit Mathematics and Statistics Academic Operations 6 (MATH1X21 or MATH1011 or MATH1931 or MATH1X01 or MATH1906) and (MATH1014 or MATH1X02) and (MATH1X23 or MATH1933 or MATH1X03 or MATH1907) None MATH2961 or MATH2067 or MATH2021 or MATH2921 or MATH2022 or MATH2922 None No

### Teaching staff

Coordinator Nathan Brownlowe, nathan.brownlowe@sydney.edu.au

## Assessment

Type Description Weight Due Length
Tutorial quiz Quizzes
2x Quizzes in Weeks 3 and 4
30% Multiple weeks 40 mins each
Outcomes assessed:
Assignment Assignments
2x assignment due Weeks 2 and 3
10% Multiple weeks 1 week per assignment
Outcomes assessed:
Final exam (Open book) Final Exam
Online open book without invigilation
60% Week 05 2 hours
Outcomes assessed:
= Type C final exam

### Assessment summary

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.

### 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.

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 Learning outcomes
Week 01 Linear systems, Gaussian elimination, vector spaces and subspaces Block teaching (3 hr)
Subspaces, linear combinations, span,linear dependence and independence Block teaching (3 hr)
Linear dependence and independence, span, basis and dimension Block teaching (1 hr)
Week 02 Linear dependence and independence, span, basis and dimension Block teaching (2 hr)
Basis and dimension, Lagrange interpolation, column space, null space, rank, nullity and linear transformations Block teaching (3 hr)
Eigenvalues and eigenvectors, diagonalisation theorem and Leslie population model Block teaching (3 hr)
Recurrence relations and systems of linear differential equations Block teaching (2 hr)
Week 03 Vector equations of lines and curves (revision), arc length, two types of line integrals and work done by a force Block teaching (3 hr)
Vector fields, grad and curl, normals to surfaces, conservative fields and potential functions Block teaching (3 hr)
Double integrals, area, volume and mass. Div (divergence of a vector field), green’s theorem and flux across a curve Block teaching (3 hr)
Week 04 Green’s theorem continued., surface area, surface integrals, flux across a surface, polar, cylindrical and spherical coordinates Block teaching (3 hr)
Triple integrals., volume and mass revisited and Gauss’ divergence theorem Block teaching (3 hr)
Triple integrals in cylindrical/spherical coordinates, stokes’ theorem and connections between different types of integrals Block teaching (1 hr)
Triple integrals in cylindrical/spherical coordinates, stokes’ theorem and connections between different types of integrals Block teaching (2 hr)
Revision Block teaching (7 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.

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

• LO1. solve a system of linear equations
• LO2. apply the subspace test in several different vector spaces
• LO3. calculate the span of a given set of vectors in various vector spaces
• LO4. test sets of vectors for linear independence and dependence
• LO5. find bases of vector spaces and subspaces
• LO6. find a polynomial of minimum degree that fits a set of points exactly
• LO7. find bases of the fundamental subspaces of a matrix
• LO8. test whether an n × n matrix is diagonalisable, and if it is find its diagonal form
• LO9. apply diagonalisation to solve recurrence relations and systems of DEs
• LO10. extended (from first year) their knowledge of vectors in two and three dimensions, and of functions of several variables
• LO11. evaluate certain line integrals, double integrals, surface integrals and triple integrals
• LO12. evaluate certain line integrals, double integrals, surface integrals and triple integrals
• LO13. understand the physical and geometrical significance of these integrals
• LO14. know how to use the important theorems of Green, Gauss and Stokes.

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

GQ1 GQ2 GQ3 GQ4 GQ5 GQ6 GQ7 GQ8 GQ9

## Responding to student feedback

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

No changes have been made since this unit was last offered.

### 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.

General Laboratory Safety Rules

• No eating or drinking is allowed in any laboratory under any circumstances
• A laboratory coat and closed-toe shoes are mandatory
• In case of fire, follow instructions posted outside the laboratory door
• First aid kits, eye wash and fire extinguishers are located in or immediately outside each laboratory
• As a precautionary measure, it is recommended that you have a current tetanus immunisation. This can be obtained from University Health Service: unihealth.usyd.edu.au/

### 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.