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 with sequences and series of real numbers, introducing the concept of Cauchy sequences and results for bounded sequences. Subsequently, sequences and series of functions are introduced and concepts of uniform convergence and power series are discussed. The concept of Lebesgue integral on real line is then developed. Finally, the rudiments of Fourier series are introduced.
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 of 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 online videos, at least one weekly lecture 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.
Sequences and series of real numbers: Cauchy sequences, Cauchy's criterion for series, absolute convergence implies convergence.
Uniform convergence of sequences and series of functions, power series.
The Lebesgue integral: construction of the Lebesgue integral on the real line, the class of Lebesgue integrable functions and their integrals. Basic properties. Integration of continuous functions and the fundamental theorem of calculus. Integration and convergence.
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 2021/22, 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)||MATH10068 Honours Analysis||2:00|
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||Miss Greta Mazelyte