MATH201 Real Analysis

Overview

This paper is an introduction to the basic techniques of real analysis in the familiar context of single-variable calculus.

Analysis is, broadly, the part of mathematics which deals with limiting processes. The main examples students have met in school and first-year university are from calculus, where the derivative and integral are defined using quite different limiting processes. Real analysis specialises to real-valued functions of a real variable. The methods of analysis have been developed over the past two centuries to give mathematicians rigorous methods for deciding whether a formal calculation is correct or not. This paper discusses the basic ideas of analysis. At the end of the semester, students should have a grounding in the methods of analysis which will prove invaluable in later years.

About this paper

Prerequisite

MATH 140 or MATH 170

Restriction

MATH 353

Schedule C

Arts and Music, Science

Eligibility

MATH 201 is compulsory for the Mathematics major and is of particular relevance, also, for students majoring in Statistics, Physics or any discipline requiring a quantitative analysis of systems and how they change with space and time.

Contact

maths@otago.ac.nz

Teaching staff

Teaching staff to be advised.

Paper Structure

This paper will begin by developing an axiomatic description of the real line. We will then use this one-dimensional construction to develop the n dimensional Euclidean spaces and understand their properties. The bulk of this course will be taken up with understanding rigorous definitions of limits in a variety of settings.

Main topics:

• A review of the real number system
• The completeness axiom
• The Euclidean spaces
• The distance function and open and closed sets
• Limits of sequences and the algebra of limits
• Limits of series and the algebra of limits
• Continuous functions
• Limits of functions and the algebra of limits
• Applications of real analysis in one dimensional calculus

Textbooks

Textbooks are not required for this paper.

Graduate Attributes Emphasised

Critical thinking.
View more information about Otago's graduate attributes.

Learning Outcomes

Students who successfully complete this paper are expected to:

• Understand the formal definition of Euclidean spaces, particularly the real number line and plane
• Understand the notion of open and closed sets in the Euclidean setting
• Understand the rigorous definition of convergence for a sequence or series and apply appropriate tools determine whether example sequences/series are convergent
• Understand the definition of a continuous function and the key properties of such functions
• Understand convergence of functions both pointwise and uniform

Timetable

Overview

This paper is an introduction to the basic techniques of real analysis in the familiar context of single-variable calculus.

Analysis is, broadly, the part of mathematics which deals with limiting processes. The main examples students have met in school and first-year university are from calculus, where the derivative and integral are defined using quite different limiting processes. Real analysis specialises to real-valued functions of a real variable. The methods of analysis have been developed over the past two centuries to give mathematicians rigorous methods for deciding whether a formal calculation is correct or not. This paper discusses the basic ideas of analysis. At the end of the semester, students should have a grounding in the methods of analysis which will prove invaluable in later years.

About this paper

Prerequisite

MATH 140 or MATH 170

Restriction

MATH 353

Schedule C

Arts and Music, Science

Eligibility

MATH 201 is compulsory for the Mathematics major and is of particular relevance, also, for students majoring in Statistics, Physics or any discipline requiring a quantitative analysis of systems and how they change with space and time.

Contact

For more information, contact MATH and COMO200-300 Level Advisor Jõrg Hennig at joerg.hennig@otago.ac.nz

Teaching staff

Teaching staff to be advised.

Paper Structure

This paper will begin by developing an axiomatic description of the real line. We will then use this one-dimensional construction to develop the n dimensional Euclidean spaces and understand their properties. The bulk of this course will be taken up with understanding rigorous definitions of limits in a variety of settings.

Main topics:

• A review of the real number system
• The completeness axiom
• The Euclidean spaces
• The distance function and open and closed sets
• Limits of sequences and the algebra of limits
• Limits of series and the algebra of limits
• Continuous functions
• Limits of functions and the algebra of limits
• Applications of real analysis in one dimensional calculus

Textbooks

Textbooks are not required for this paper.

Graduate Attributes Emphasised

Critical thinking.
View more information about Otago's graduate attributes.

Learning Outcomes

Students who successfully complete this paper are expected to:

• Understand the formal definition of Euclidean spaces, particularly the real number line and plane
• Understand the notion of open and closed sets in the Euclidean setting
• Understand the rigorous definition of convergence for a sequence or series and apply appropriate tools determine whether example sequences/series are convergent
• Understand the definition of a continuous function and the key properties of such functions
• Understand convergence of functions both pointwise and uniform

Timetable