# MATHS 332 : Real Analysis

## Science

### Course Prescription

A standard course for every student intending to advance in pure mathematics. It develops the foundational mathematics underlying calculus, it introduces a rigorous approach to continuous mathematics and fosters an understanding of the special thinking and arguments involved in this area. The main focus is analysis in one real variable with the topics including real fields, limits and continuity, Riemann integration and power series.

### Course Overview

An introduction to the basic theory of analysis in one real variable. Students will have seen  the topics covered in the prerequisite courses, such as Maths250 and Maths130 (or equivalent), but in this course you understand how to actually develop and prove this theory.

Apart from enabling a thorough understanding of the results treated, this course lays the foundations of this type of mathematical thinking, reasoning, and arguing. In this sense it is important for development of a rounded mathematical student and essential for those wishing to advance in pure mathematics or any part of mathematics that uses mathematical analysis or its understanding. This includes many parts of applied mathematics,  mathematical physics, and statistics.    It is a prerequisite for Maths333, and then many of the 700 level mathematics courses.

### Course Requirements

Prerequisite: MATHS 250, and MATHS 254 or 255 or an A or higher in MATHS 253 and 260

### Capabilities Developed in this Course

 Capability 1: Disciplinary Knowledge and Practice Capability 2: Critical Thinking Capability 3: Solution Seeking Capability 4: Communication and Engagement Capability 5: Independence and Integrity

### Learning Outcomes

By the end of this course, students will be able to:
1. Demonstrate how analysts use the real numbers and their properties (Capability 1, 2, 3 and 4)
2. Develop and apply the theory of convergent and divergent sequences (Capability 1, 2, 3, 4 and 5)
3. Develop and apply the theory of (power) series and convergence tests. (Capability 1, 2, 3, 4 and 5)
4. Develop and apply the theory of continuous functions and the associated classical theorems. (Capability 1, 2, 3, 4 and 5)
5. Develop and apply the theory of differentiable functions and the associated classical theorems (Capability 1, 2, 3, 4 and 5)
6. Develop and apply the theory of Riemann integrals and the associated classical theorems (Capability 1, 2, 3 and 4)
7. Develop and apply the theory of uniform convergence and the associated classical theorems (Capability 1, 2, 3 and 5)

### Assessments

Assessment Type Percentage Classification
Assignments 20% Individual Coursework
Test 20% Individual Coursework
Final Exam 60% Individual Coursework
1 2 3 4 5 6 7
Assignments
Test
Final Exam
Plussage applies. The final mark will be  based on: 20% assignments, 20% test and 60% exam; or 100% exam, whichever is the maximum.

Must achieve a mark of 35% in the Final Exam in order to pass the course.

### Key Topics

Real numbers;  sequences and series; limits of function; continuous
functions; differentiation;  the Riemann integral; sequences of functions; and infinite series.

### Learning Resources

Recommended: Basic Analysis: Introduction to Real Analysis, by Jiřı́ Lebl, May
2018. Freely available at http://www.jirka.org/ra/

Introduction to Real Analysis, by R. G. Bartle and D. Sherbert, 4 th edition, 2011.

### Special Requirements

Must achieve a mark of 35% in the Final Exam in order to pass the course.

This course is a standard [15] point course and students are expected to spend 10 hours per week involved in each 15 point course that they are enrolled in.

For this course, you can expect [3] hours of lectures, a [1] hour tutorial, [2] hours of reading and thinking about the content and [4] hours of work on assignments and/or test preparation.

### Digital Resources

Course materials are made available in a learning and collaboration tool called Canvas which also includes reading lists and lecture recordings (where available).

Please remember that the recording of any class on a personal device requires the permission of the instructor.

The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.

You may copy the course content for the purposes of private study or research, but you may not upload onto any third party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.

The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. This requirement also applies to sources on the internet. A student's assessed work may be reviewed against online source material using computerised detection mechanisms.

### Inclusive Learning

All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.

Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website at http://disability.auckland.ac.nz

### Special Circumstances

If your ability to complete assessed coursework is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due.

If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page: https://www.auckland.ac.nz/en/students/academic-information/exams-and-final-results/during-exams/aegrotat-and-compassionate-consideration.html.

This should be done as soon as possible and no later than seven days after the affected test or exam date.

### Student Feedback

During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.

At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.

Your feedback helps to improve the course and its delivery for all students.

### Student Charter and Responsibilities

The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter (https://www.auckland.ac.nz/en/students/forms-policies-and-guidelines/student-policies-and-guidelines/student-charter.html).

### Disclaimer

Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.

In this course you may be asked to submit your coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. The final decision on the completion mode for a test or examination, and remote invigilation arrangements where applicable, will be advised to students at least 10 days prior to the scheduled date of the assessment, or in the case of an examination when the examination timetable is published.

Published on 09/07/2020 03:22 p.m.