Unit of study_

# 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.

### Details

Academic unit Mathematics and Statistics Academic Operations MATH2061 Linear Mathematics and Vector Calculus Intensive January, 2022 Block mode Camperdown/Darlington, Sydney 6

### Enrolment rules

 Prohibitions ? MATH2961 or MATH2067 or MATH2021 or MATH2921 or MATH2022 or MATH2922 (MATH1X21 or MATH1011 or MATH1931 or MATH1X01 or MATH1906) and (MATH1014 or MATH1X02) and (MATH1X23 or MATH1933 or MATH1X03 or MATH1907) None No

### Teaching staff and contact details

Coordinator Daniel Hauer, daniel.hauer@sydney.edu.au Brad Roberts

## Assessment

Type Description Weight Due Length
Assignment Assignment 1
written calculus and algebra
5% Week 02
Due date: 23 Jan 2022 at 23:59

Closing date: 30 Jan 2022
7 days
Outcomes assessed:
written calculus and algebra
15% Week 03
Due date: 24 Jan 2022 at 16:00
40 minutes
Outcomes assessed:
Assignment Assignment 2
written calculus and algebra
5% Week 04
Due date: 06 Feb 2022 at 23:59

Closing date: 13 Feb 2022
7 days
Outcomes assessed:
written calculus and algebra
15% Week 05
Due date: 09 Feb 2022 at 16:00
40 minutes
Outcomes assessed:
Final exam (Take-home short release) Final Exam
Written calculation.
60% Week 06
Due date: 14 Feb 2022 at 10:00
2 hours
Outcomes assessed:
= Type D final exam
• Quizzes: Two quizzes will be held online through Canvas. The quizzes are 40 Minutes and held during the given tutorial hour in a Zoom session. The better mark principle will be used for the quizzes so do not submit an application for Special Consideration or Special Arrangements if you miss a quiz. The better mark principle means that for each quiz, the quiz counts if and only if it is better than or equal to your exam mark. If your quiz mark is less than your exam mark, the exam mark will be used for that portion of your assessment instead.
• Assignments: There are two assignments. Each assignment must be submitted electronically, as one single typeset or scanned PDF file only via the Canvas by the deadline. Note that your assignment will not be marked if it is illegible or if it is submitted sideways or upside down. It is your responsibility to check that your assignment has been submitted correctly and that it is complete (check that you can view each page). Late submissions will receive a penalty. A mark of zero will be awarded for all submissions more than 7 days past the original due date. Further extensions past this time will not be permitted. Detailed information for each assessment can be found on Canvas. The better mark principle does not apply on assignments!
• Final Exam:  There is one final exam to this unit of study held in week 6. Further information about the exam will be made available at a later date on Canvas.

Detailed information for each assessment will be available 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.

### Special consideration

If you experience short-term circumstances beyond your control, such as illness, injury or misadventure or if you have essential commitments which impact your preparation or performance in an assessment, you may be eligible for special consideration or special arrangements.

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.

You may only use artificial intelligence and writing assistance tools in assessment tasks if you are permitted to by your unit coordinator, and if you do use them, you must also acknowledge this in your work, either in a footnote or an acknowledgement section.

Studiosity is permitted for postgraduate units unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission.

## 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

## Closing the loop

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