Unit outline_

# MATH2921: Vector Calculus and Differential Eqs (Adv)

## Overview

This is the advanced version of MATH2021, with more emphasis on the underlying concepts and mathematical rigour. The vector calculus component of the course will include: parametrised curves and surfaces, vector fields, div, grad and curl, gradient fields and potential functions, lagrange multipliers line integrals, arc length, work, path-independent integrals, and conservative fields, flux across a curve, double and triple integrals, change of variable formulas, polar, cylindrical and spherical coordinates, areas, volumes and mass, flux integrals, and Green's Gauss' and Stokes' theorems. The Differential Equations half of the course will focus on ordinary and partial differential equations (ODEs and PDEs) with applications with more complexity and depth. The main topics are: second order ODEs (including inhomogeneous equations), series solutions near a regular point, higher order ODEs and systems of first order equations, matrix equations and solutions, solution methods (variation of parameters, undetermined coefficients) the Laplace and Fourier Transform, elementary Sturm-Liouville theory, an introduction to PDEs, and first methods of solutions (including separation of variables, and Fourier Series). The unit then moves to topics in solution techniques for ordinary and partial differential equations (ODEs and PDEs) with applications. It provides a more thorough 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 6 [(MATH1921 or MATH1931 or MATH1901 or MATH1906) or (a mark of 65 or above in MATH1021 or MATH1001)] and [MATH1902 or (a mark of 65 or above in MATH1002)] and [(MATH1923 or MATH1933 or MATH1903 or MATH1907) or (a mark of 65 or above in MATH1023 or MATH1003)] None MATH2021 or MATH2065 or MATH2965 or (MATH2061 and MATH2022) or (MATH2061 and MATH2922) or (MATH2961 and MATH2022) or (MATH2961 and MATH2922) or MATH2067 None Yes

### Teaching staff

Coordinator Alexander Fish, alexander.fish@sydney.edu.au Robert Marangell

## Assessment

Type Description Weight Due Length
Final exam (Record+) Final take home exam
2 hour exam
70% Formal exam period 2 hours
Outcomes assessed:
Assignment Assignment 1
10% Week 06 3 weeks
Outcomes assessed:
online quiz
10% Week 08
Due date: 27 Apr 2021 at 23:59
1 hour
Outcomes assessed:
online quiz
10% Week 12
Due date: 25 May 2021 at 23:59
1 hour
Outcomes assessed:
= Type B final exam

### Assessment summary

Assignments: Will require you to integrate information from lectures and practicals to create a concise written argument. Test your understanding of material covered in the vector calculus and differential equations components of the course.

Final exam: The exam will cover all material in the unit from lectures, tutorials and practice classes.

### 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.

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.

## 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
Multiple weeks Review of partial differentiation techniques. Gradients, maxima, and minima, curves in space and the plane and the chain rule Lecture and tutorial (8 hr)
Multiple integration and their computation both in Euclidean and transcribed coordinates. Lecture and tutorial (8 hr)
Line integration and the main theorems of vector calculus Lecture and tutorial (8 hr)
Review of linear second order ordinary differential equations, methods for solving them and their applications Lecture and tutorial (8 hr)
Generalisation of the methods from second order linear equations to higher order equations. Lecture and tutorial (8 hr)
Advanced methods for solving linear ODEs. Lecture and tutorial (8 hr)
Fourier Series and applications to solving classical boundary value problems and PDEs Lecture and tutorial (8 hr)
Weekly Demonstration of interesting problems relating to the course material using the computer software package Mathematica Lecture and tutorial (11 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.

All readings for this unit can be accessed through the Library eReserve, available on Canvas.

• Textbook: Calculus III, by Jerrold Marsden and Alan Weinstein. Undergraduate Texts in Mathematics. Springer-Verlag, New York. 1985.
• Textbook: Elementary Differential Equations with Boundary Value Problems, by William F. Trench. Free Online edition 1.01 December 2013.
• Texbook: Introduction to Partial Differential Equations by Peter. J. Olver. Undergraduate Texts in Mathe- matics. Springer, New York. 2016.

## 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. demonstrate a conceptual understanding of vector-valued functions, partial derivatives, curves and integration over a regioun, volume and surface as well as solving basic differential equations with a background in a variet of texhniques and applications of mathematical analysis
• LO2. understand the definitions, main theorems, propositions, lemmata and corollaries for multivariate calculus as well as their applications to science. Also understand and the main theorems and methods of solving elementary linear differential equations as well as the applications.
• LO3. be fluent in the computation of integrals via substitution and coordinate transform methods.
• LO4. develop an appreiciation and strong working knowledge of the theory and applications of elementry vector analysis and differential equations.
• LO5. be fluent with important examples, theorems and applications and be able to implement these in a computational software package.
• LO6. present complete and mathematically rigorous solutions for problems in vector calculus and differential equations that include appropriate justification for their reasoning.
• LO7. recognise problems in mathematics and other areas of science and engineering that are amenable to mathematical analysis, and to apply th etechniques of mathematical analysis in solving them.

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