Undergraduate Course: Honours Analysis (MATH10068)
|School||School of Mathematics
||College||College of Science and Engineering
|Credit level (Normal year taken)||SCQF Level 10 (Year 3 Undergraduate)
||Availability||Available to all students
|Summary||Core course for Honours Degrees involving Mathematics.
This is a second course in real analysis and builds on ideas in the analysis portion of Fundamentals of Pure Mathematics. The course begins by introducing uniform convergence and the Riemann integral and leads on to the abstract concept of a metric space. Finally we look at some applications to solving equations via the fixed-point method, and at the rudiments of Fourier series.
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 practise 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.
The course has a main section and a skills section. The main section consists of 3 online weekly lectures and a weekly workshop designed to augment and extend understanding the material covered in the lectures in a smaller-group setting. There will be frequent reading assignments and exercise assignments which students will be expected to have completed before the lecture.
The syllabus for the main part of the course is:
Review of material from Fundamentals of Pure mathematics and covering parts that were omitted (Cauchy sequences + Bolzano Weierstrass theorem)
Uniform convergence of sequences and series of functions, power series.
The Riemann integral: step functions, the class of Riemann-integrable functions and their integrals. Basic properties. Integration of continuous functions via uniform continuity and the fundamental theorem of calculus. Integration and uniform convergence.
Metric spaces: definition and examples, especially R^n and C([a,b]) with various metrics. Convergence, Cauchy sequences; completeness, examples and non-examples. The sup metric on C([a,b]) and uniform convergence. Equivalent metrics. Point set topology: open balls, open sets, closed sets, interior, closure, boundary, limit points. Continuity. Connectedness. Compactness: examples and non-examples; relationships between compactness, closedness and boundedness, Heine-Borel theorem in R^n.
Banach contraction mapping theorem and applications to algebraic equations, differential equations and integral equations.
Fourier series (time permitting).
Skills: The content will be chosen appropriate to the learning outcomes. (10h)
Information for Visiting Students
|Pre-requisites||Visiting students are advised to check that they have studied the material covered in the syllabus of each pre-requisite course before enrolling.
|High Demand Course?
Course Delivery Information
|Academic year 2020/21, Available to all students (SV1)
|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
|Assessment (Further Info)
|Additional Information (Assessment)
||Coursework 30%, Examination 70%
||Hours & Minutes
|Main Exam Diet S2 (April/May)|| Honours Analysis (MATH10068)||3:00|
On completion of this course, the student will be able to:
- State definitions and principal results of the course accurately, to present and explain standard proofs, and to use these together with the principal concepts, arguments and methods presented in the course in standard situations, and also to tackle problems analogous to or extending seen examples.
- Find, explain and discuss examples that illustrate the theory in the course or which do or do not satisfy a definition or the conditions or conclusions of a theorem.
- Use the theory and methods of the course to address and solve unseen problems in both concrete and abstract situations.
- Combine purposeful private study with help from fellow students, tutors and lecturers and to use lecture notes and alternative sources to build a sound basis of understanding of the areas of mathematical analysis presented in the course.
- Demonstrate the ability to read and write mathematics using advanced notation accurately and appropriately, demonstrating the role of axioms, definitions, conjectures, theorems etc. in mathematical practice.
|Students are expected to have a personal copy of :|
Wade, W R, 'An Introduction to Analysis', 4th Edition, ISBN 9780136153702
|Graduate Attributes and Skills
|Course organiser||Dr Martin Dindos
|Course secretary||Mr Christopher Palmer
Tel: (0131 6)50 5060