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 Semester 1, 2020 Normal day 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 Yes

### Teaching staff and contact details

Coordinator Laurentiu Paunescu, laurentiu.paunescu@sydney.edu.au

## Assessment

Type Description Weight Due Length
Final exam Examination
See Canvas
60% Formal exam period 2 hours
Outcomes assessed:
Assignment Linear Algebra Assignment
Assignment
5% Week 05 Students given at least 1.5 weeks
Outcomes assessed:
Tutorial quiz Linear Algebra Quiz
Multiple choice quiz
15% Week 06 40 minutes
Outcomes assessed:
Assignment Vector Calculus Assignment
Assignment
5% Week 10 Students given at least 1.5 weeks
Outcomes assessed:
Tutorial quiz Vector Calculus Quiz
Multiple choice quiz
15% Week 13 40 minutes
Outcomes assessed:

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.

### 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 Lecture and tutorial (4 hr)
Linear systems, Gaussian elimination, vector spaces and subspaces Seminar (1 hr)
Week 02 Subspaces, linear combinations, span,linear dependence and independence Lecture and tutorial (4 hr)
Subspaces, linear combinations, span,linear dependence and independence Seminar (1 hr)
Week 03 Linear dependence and independence, span, basis and dimension Lecture and tutorial (4 hr)
Linear dependence and independence, span, basis and dimension Seminar (1 hr)
Week 04 Basis and dimension, Lagrange interpolation, column space, null space, rank, nullity and linear transformations Lecture and tutorial (4 hr)
Basis and dimension, Lagrange interpolation, column space, null space, rank, nullity and linear transformations Seminar (1 hr)
Week 05 Eigenvalues and eigenvectors, diagonalisation theorem and Leslie population model Lecture and tutorial (4 hr)
Eigenvalues and eigenvectors, diagonalisation theorem and Leslie population model Seminar (1 hr)
Week 06 Recurrence relations and systems of linear differential equations Lecture and tutorial (4 hr)
Recurrence relations and systems of linear differential equations Seminar (1 hr)
Week 07 Vector equations of lines and curves (revision), arc length, two types of line integrals and work done by a force Lecture and tutorial (4 hr)
Vector equations of lines and curves (revision), arc length and two types of line integrals and work done by a force Seminar (1 hr)
Week 08 Vector fields, grad and curl, normals to surfaces, conservative fields and potential functions Lecture and tutorial (4 hr)
Vector fields, grad and curl, normals to surfaces, conservative fields and potential functions Seminar (1 hr)
Week 09 Double integrals, area, volume and mass. Div (divergence of a vector field), green’s theorem and flux across a curve Lecture and tutorial (4 hr)
Double integrals, area, volume and mass. Div (divergence of a vector field), green’s theorem and flux across a curve Seminar (1 hr)
Week 10 Green’s theorem continued., surface area, surface integrals, flux across a surface, polar, cylindrical and spherical coordinates Lecture and tutorial (4 hr)
Green’s theorem continued., surface area, surface integrals, flux across a surface, polar, cylindrical and spherical coordinates Seminar (1 hr)
Week 11 Triple integrals., volume and mass revisited and Gauss’ divergence theorem Lecture and tutorial (4 hr)
Triple integrals., volume and mass revisited and Gauss’ divergence theorem Seminar (1 hr)
Week 12 Triple integrals in cylindrical/spherical coordinates, stokes’ theorem and connections between different types of integrals Lecture and tutorial (4 hr)
Triple integrals in cylindrical/spherical coordinates, stokes’ theorem and connections between different types of integrals Seminar (1 hr)
Week 13 Revision Lecture and tutorial (4 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