Undergraduate Course: Honours Analysis (MATH10068)
|School||School of Mathematics
||College||College of Science and Engineering
||Availability||Available to all students
|Credit level (Normal year taken)||SCQF Level 10 (Year 3 Undergraduate)
|Home subject area||Mathematics
||Other subject area||None
||Taught in Gaelic?||No
|Course description||Core course for Honours Degrees involving Mathematics.
This is a second course in real analysis and builds on ideas in the analysis portion of Foundations of Pure Mathematics. The course begins by introducing the abstract concept of a metric space to study analytic ideas such as continuity, but the majority of the course is spent on using these ideas to explore the concrete situation in n real dimensions.
In the 'skills' section of this course we develop and start to use some of the fundamental tools of a professional mathematician that are often only glimpsed in lecture courses. Mathematicians formulate definitions (rather than just reading other people's), they make conjectures and then try and prove or disprove them.
Mathematicians find their own examples to illustrate their own and other people's ideas, and they find new ways of developing the theory and new connections. We will explore and practice these activities in the context of material drawn from some of the lectures in the course and related subjects. We will practise explaining mathematics and also consider 'metacognitive skills': the ability that an experienced mathematician has to step back from a calculation or problem, to 'zoom out' and consider whether it is developing well or whether perhaps there is a flaw in the approach. A typical example is the habit stopping and asking whether a proof one is working on is actually using all the assumptions of the theorem.
Entry Requirements (not applicable to Visiting Students)
|| Students MUST have passed:
Fundamentals of Pure Mathematics (MATH08064)
||Other requirements|| Students must not have taken :
MATH10008 Pure & Applied Analysis or MATH10049 Metric Spaces
|Additional Costs|| None
Information for Visiting Students
|Displayed in Visiting Students Prospectus?||No
Course Delivery Information
|Delivery period: 2013/14 Semester 2, Available to all students (SV1)
||Learn enabled: Yes
|Course Start Date
|Breakdown of Learning and Teaching activities (Further Info)
Lecture Hours 35,
Seminar/Tutorial Hours 10,
Supervised Practical/Workshop/Studio Hours 10,
Summative Assessment Hours 3,
Programme Level Learning and Teaching Hours 4,
Directed Learning and Independent Learning Hours
Students must pass exam and course overall.
|Breakdown of Assessment Methods (Further Info)
|Main Exam Diet S2 (April/May)||Honours Analysis||3:00|
Summary of Intended Learning Outcomes
|1. An ability to perform simple abstract arguments involving metric spaces.
2. An ability to demonstrate an understanding of notions such as openness, closedness, continuity, completeness, equivalence of metrics, compactness and path-connectedness as applied in the context of general and specific metric spaces.
3. Facility in working with concrete metric spaces based upon real ndimensional space and the discrete metric. In particular, able to work with open, closed, connected, and compact sets in real n-dimensional space and to work with continuous functions in real n-dimensional space using topological notions.
4. Understanding the definitions and concepts of convergence, Cauchy
sequences, and completeness in R
5. Ability to test sequences and series of functions for uniform convergence, including the Weierstrass M test.
6. Familiarity with the idea that problems of analysis such as convergence of Fourier series can be couched in and resolved using the language of metric spaces.
7. Familiar with the supremum metric and with other metrics on the set of continuous functions and able to demonstrate an understanding of the inequivalence of these metrics.
8. Experience of mathematical activities such as formulating and exploring definitions and conjectures, identifying interesting questions and constructing illustrative examples.
9. Improved abilities in talking about and explaining mathematics
10. An awareness of and some development in a range of metacognitive
skills in a mathematical context, for example considering the relation of a proof to the conditions of a theorem, identifying the key points in a proof or choosing to work on a simplified example of a problem first.
11. Deeper understanding of some representative material from lectures.
12. Experience of mathematical activities such as formulating and exploring definitions and conjectures, identifying interesting questions and constructing illustrative examples.
|See 'Breakdown of Assessment Methods' and 'Additional Notes' above.|
||A suggested syllabus for this course is:
Weeks 1-4: The metric spaces structure of the real line. This is to include a review of Fundamentals of Pure Maths 2 (including sets and sequences), the definition of a metric space; limits, interior, compactness, and connectedness in a metric space, with emphasis on the situation in the real line; and previously omitted material on Cauchy sequences and uniform continuity in the real line.
Weeks 5-6: Riemann integration.
Weeks 7-8: Series of functions (including the completeness of C[a,b]).
Week 9: Other metrics on C[a,b].
Weeks 10-11: Fourier series.
Skills: The content will be chosen appropriate to the learning outcomes. (10h)
||Students are expected to have a personal copy of :
Wade, W R, 'An Introduction to Analysis'
(continuing students should already have a copy from Year 2)
||See 'Breakdown of Learning and Teaching activities' above.
|Course organiser||Prof A Carbery
Tel: (0131 6)50 5993
|Course secretary||Mrs Kathryn Mcphail
Tel: (0131 6)50 4885
© Copyright 2013 The University of Edinburgh - 10 October 2013 4:52 am