## MATHS 2100 - Real Analysis II

### North Terrace Campus - Semester 2 - 2014

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, least upper bounds, completeness and its consequences. Sequences: convergence, subsequences, Cauchy sequences. Open, closed, and compact sets of real numbers. Continuous functions, uniform continuity. Differentiation, the Mean Value Theorem. Sequences and series of functions, pointwise and uniform convergence. Power series and Taylor series. Metric spaces: basic notions generalised from the setting of the real numbers. The space of continuous functions on a compact interval. The Contraction Principle. Picard's Theorem on the existence and uniqueness of solutions of ordinary differential equations.

• General Course Information
##### Course Details
Course Code MATHS 2100 Real Analysis II Mathematical Sciences Semester 2 Undergraduate North Terrace Campus 3 Up to 3.5 hours per week MATHS 1012 ongoing assessment 30%, exam 70%
##### Course Staff

Course Coordinator: Associate Professor Nicholas Buchdahl

##### 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 fundamental properties of the real numbers that lead to the formal development of real analysis;

2. comprehend rigorous arguments developing the theory underpinning real analysis;

3. demonstrate an understanding of limits and how they are used in sequences, series, differentiation and integration;

4. present an overview of the basic properties of metric spaces;

5. construct rigorous mathematical proofs of basic results in real analysis;

6. appreciate how abstract ideas and rigorous methods in mathematical analysis can be applied to important practical problems.

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

University Graduate Attribute Course Learning Outcome(s)
Knowledge and understanding of the content and techniques of a chosen discipline at advanced levels that are internationally recognised. all
The ability to locate, analyse, evaluate and synthesise information from a wide variety of sources in a planned and timely manner. 1,3,4
An ability to apply effective, creative and innovative solutions, both independently and cooperatively, to current and future problems. 2,5,6
Skills of a high order in interpersonal understanding, teamwork and communication. 5
A commitment to continuous learning and the capacity to maintain intellectual curiosity throughout life. 6
• Learning Resources
None.
##### Recommended Resources
There are many books on real analysis available in the library. Amongst those, the following is a selection of some that are very compatible with the level and objectives of this course:
1. Belding & Mitchell: "Foundations of Analysis";
2. Fitzpatrick: "Real Analysis";
3. Gaughan: "Introduction to Analysis";
4. Lárusson: "Lectures on Real Analysis".
##### Online Learning
Assignments, tutorial exercises, handouts, and course announcements will be posted on MyUni.

• Learning & Teaching Activities
##### Learning & Teaching Modes
This course relies on lectures as the primary delivery mechanism for the material. Tutorials supplement the lectures by providing exercises and example problems to enhance the understanding obtained through lectures. A sequence of written assignments provides assessment opportunities for students to gauge their progress and understanding.

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

 Activity Quantity Workload hours Lecture 36 90 Tutorials 6 18 Assignments 6 42 Test 1 8 TOTALS 158
##### Learning Activities Summary
Lecture Outline

1. Basic set theory. The real numbers and their defining properties. (Lectures 1-3)
2. Sequences: convergence, properties of limits, subsequences, Cauchy sequences. (Lectures 4-6)
3. Limits superior and inferior. The topology of the real numbers. (Lectures 7-9)
4. Continuity. The key properties of continuous real-valued functions of a real variable. (Lectures 10-12)
5. The Riemann integral. (Lectures 13-14)
6. Differentiation. Mean Value theorems, l'Hôpital's rules. (Lectures 16-18)
7. The inverse function theorem. The fundamental theorem of calculus. (Lectures 19-21)
8. Series. Tests for convergence.  (Lectures 22-24)
9. Power series. Taylor's theorem. Sequences and series of functions. (Lectures 25-27)
10. Convergence of sequences and series of functions. (Lectures 28-30)
11. Metric spaces: basic definitions and properties. (Lectures 31-33)
12. The contraction mapping theorem and Picard's theorem. (Lectures 33-36)

Tutorials

Tutorials will be held in each even week, covering material from the preceding two weeks of lectures. Students will be encouraged to play an active role in tutorials, but will not be assessed on this. Attendance at tutorials will comprise 3% of the final mark.
• 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 Objective Assessed Tutorial attendance Formative Even weeks 3 All Assignments Formative and summative Odd weeks 21 All Test Summative Lecture 15 20 1,2,3,5 Exam Summative Exam period 56 All
##### Assessment Related Requirements
An aggregate score of at least 50% is required to pass the course.
##### Assessment Detail
With the exception of the first assignment, assignments will be distributed in the middle of each even week and due at the end of each odd week.  Each will mainly cover material from the previous weeks' lectures, but may include material from the next one or two lectures.

The first assignment will be distributed at the start of the first week and due at the end of that week. It will largely cover material from Mathematics I for the purpose of refreshing the key ideas from a first course in Calculus.

The first assignment will be worth 1% of the final mark.  The remaining five assignments will each be worth 4% of the final mark.

The midsemester test will be held during the 15th lecture period. It will cover the material covered in lectures up to the end of Lecture 12.  It will be worth 20% of the final mark.
##### Submission

All written assignments are to be submitted to the designated hand-in boxes within the School of Mathematical Sciences.

Late assignments will not be accepted.

Assignments will have a two week turn-around time for feedback to students.

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
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