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.
|Paper title||Real Analysis|
|Teaching period||Semester 2 (On campus)|
|Domestic Tuition Fees (NZD)||$955.05|
|International Tuition Fees||Tuition Fees for international students are elsewhere on this website.|
- MATH 140 or MATH 170
- MATH 353
- Schedule C
- Arts and Music, Science
- 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.
- 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.
- 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 are not required for this paper.
- Graduate Attributes Emphasised
- Critical thinking.
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- 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