MATHS 7100 - Real Analysis

North Terrace Campus - Semester 2 - 2020

Much of mathematics relies on our ability to be able to solve equations, if not in explicit exact forms, then at least in being able to establish the existence of solutions. To do this requires a knowledge of so-called "analysis", which in many respects is just Calculus in very general settings. The foundations for this work are commenced in Real Analysis, a course that develops this basic material in a systematic and rigorous manner in the context of real-valued functions of a real variable. Topics covered are: Basic set theory. The real numbers and their basic properties. Sequences: convergence, subsequences, Cauchy sequences. Open, closed, and compact sets of real numbers. Continuous functions and uniform continuity. The Riemann integral. Differentiation and Mean Value theorems. The Fundamental Theorem of Calculus. Series. Power series and Taylor series. Convergence of sequences and series of functions.

• General Course Information
Course Details
Course Code MATHS 7100 Real Analysis School of Mathematical Sciences Semester 2 Postgraduate Coursework North Terrace Campus 3 Up to 4 hours per week Y MATHS 1012 Much of mathematics relies on our ability to be able to solve equations, if not in explicit exact forms, then at least in being able to establish the existence of solutions. To do this requires a knowledge of so-called "analysis", which in many respects is just Calculus in very general settings. The foundations for this work are commenced in Real Analysis, a course that develops this basic material in a systematic and rigorous manner in the context of real-valued functions of a real variable. Topics covered are: Basic set theory. The real numbers and their basic properties. Sequences: convergence, subsequences, Cauchy sequences. Open, closed, and compact sets of real numbers. Continuous functions and uniform continuity. The Riemann integral. Differentiation and Mean Value theorems. The Fundamental Theorem of Calculus. Series. Power series and Taylor series. Convergence of sequences and series of functions.
Course Staff

Course Coordinator: Professor Finnur Larusson

Course Timetable

The full timetable of all activities for this course can be accessed from Course Planner.

• Learning Outcomes
Course Learning Outcomes

On successful completion of this course, students will be able to:

1. describe the fundamental properties of the real numbers that underpin the formal development of real analysis;

2. demonstrate an understanding of the theory of sequences and series, continuity, differentiation and integration;

3. demonstrate skills in constructing rigorous mathematical arguments;

4. apply the theory in the course to solve a variety of problems at an appropriate level of difficulty;

5. demonstrate skills in communicating mathematics.

This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:

University Graduate Attribute Course Learning Outcome(s)
Deep discipline knowledge
• informed and infused by cutting edge research, scaffolded throughout their program of studies
• acquired from personal interaction with research active educators, from year 1
• accredited or validated against national or international standards (for relevant programs)
1, 2, 3, 4
Critical thinking and problem solving
• steeped in research methods and rigor
• based on empirical evidence and the scientific approach to knowledge development
• demonstrated through appropriate and relevant assessment
3, 4
Teamwork and communication skills
• developed from, with, and via the SGDE
• honed through assessment and practice throughout the program of studies
• encouraged and valued in all aspects of learning
3, 5
• technology savvy
• professional and, where relevant, fully accredited
• forward thinking and well informed
• tested and validated by work based experiences
1, 2, 3, 4, 5
Self-awareness and emotional intelligence
• a capacity for self-reflection and a willingness to engage in self-appraisal
• open to objective and constructive feedback from supervisors and peers
• able to negotiate difficult social situations, defuse conflict and engage positively in purposeful debate
3, 4, 5
• Learning Resources
Required Resources
The textbook for the course is Lectures on Real Analysis by Finnur Lárusson, freely available to students via the library catalogue.
Recommended Resources
There are many good real analysis books out there for students who want to look at other sources, for example Understanding Analysis by Stephen Abbott, which is freely available via the library catalogue.
Online Learning
All course materials (except the textbook) will be made available on MyUni.

• Learning & Teaching Activities
Learning & Teaching Modes
Each week's material is presented in two sources that complement each other: the textbook and lecture videos that are posted on MyUni at the beginning of the week. Having studied the material from both sources, students test their initial understanding with an online quiz. The following week students deepen their understanding of the material and their skills in applying it by working on tutorial exercises and attending a tutorial (face to face or online). Biweekly assignments provide students with further opportunities to practise and get feedback on their work. Students interact with the lecturer and with each other on a MyUni discussion platform. In addition, the lecturer offers weekly consulting.

The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

 Activity Quantity Workload hours Study of textbook and videos 62 Tutorial 12 24 Quiz 11 22 Assignment 6 36 Test 2 12 TOTAL 156
Learning Activities Summary
Topics

Numbers, sets, and functions.
The real numbers.
Sequences.
Open, closed, and compact sets.
Continuity.
Differentiation.
Integration.

Tutorials

Tutorials will be held every week, covering material from the previous week. The first tutorial, in Week 1, will be a review of first-year calculus.
• Assessment

The University's policy on Assessment for Coursework Programs is based on the following four principles:

1. Assessment must encourage and reinforce learning.
2. Assessment must enable robust and fair judgements about student performance.
3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
4. Assessment must maintain academic standards.

Assessment Summary
 Component Task type Due Weighting Outcomes Assessed Assignments Formative and summative Even weeks 20% All Quizzes Formative and summative Weekly 10% All Test 1 Summative Week 4-6 15% All Test 2 Summative Week 8-10 15% All Exam Summative Exam period 40% All

More details will be announced later.

Assessment Detail

No information currently available.

Submission

No information currently available.

Grades for your performance in this course will be awarded in accordance with the following scheme:

M10 (Coursework Mark Scheme)
FNS   Fail No Submission
F 1-49 Fail
P 50-64 Pass
C 65-74 Credit
D 75-84 Distinction
HD 85-100 High Distinction
CN   Continuing
NFE   No Formal Examination
RP   Result Pending

Further details of the grades/results can be obtained from Examinations.

Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

Final results for this course will be made available through Access Adelaide.

• Student Feedback

The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

• Student Support
• Policies & Guidelines
• Fraud Awareness

Students are reminded that in order to maintain the academic integrity of all programs and courses, the university has a zero-tolerance approach to students offering money or significant value goods or services to any staff member who is involved in their teaching or assessment. Students offering lecturers or tutors or professional staff anything more than a small token of appreciation is totally unacceptable, in any circumstances. Staff members are obliged to report all such incidents to their supervisor/manager, who will refer them for action under the university's studentâ€™s disciplinary procedures.

The University of Adelaide is committed to regular reviews of the courses and programs it offers to students. The University of Adelaide therefore reserves the right to discontinue or vary programs and courses without notice. Please read the important information contained in the disclaimer.

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