Undergraduate Programme and Module Handbook 2010-2011 (archived)

Module MATH4201: ANALYSIS IV

Department: Mathematical Sciences

MATH4201: ANALYSIS IV

Type	Open	Level	4	Credits	20	Availability	Available in 2011/12 and alternate years thereafter	Module Cap	None.	Location	Durham

Prerequisites

Complex Analysis II (MATH2011) and Analysis in Many Variables II (MATH2031)

Corequisites

None.

Excluded Combination of Modules

Analysis III (MATH3011).

Aims

To provide the student with basic ideas of differentiation and integration in n-dimensional real space, to explain conditions which guarantee existence of solutions of ordinary differential equations, to introduce the notion of a smooth manifold and its applications.

Content

Metric Spaces.
Differentiation on Manifolds.
Integration in n-dimensional real space.
Differential forms.
Integration on Manifolds.
Reading material on flows and vector fields on manifolds.

Learning Outcomes

Subject-specific Knowledge:

By the end of the module students will:
be able to solve novel and/or complex problems in Analysis.
have a systematic and coherent understanding of theoretical mathematics in the field of Analysis.
have acquired a coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas:
Completeness.
Contraction mappings.
Picard theorem.
Implicit Function Theorem.
Inverse Function Theorem
Integration: Riemann integral.
Sets of measure zero.
Fubini theorem.
Linear Algebra of alternating forms.
Differential forms.
Integration of Differential forms.
Stokes' Theorem.
Green's Theorem.
Divergence Theorem.
Knowledge and understanding of flows and vector fields on manifolds.

Subject-specific Skills:

In addition students will have highly specialised and advanced mathematical skills in the following areas which will be used with minimal guidance: Spatial awareness.
Ability to read independently to acquire knowledge and understanding in the area of analytic continuation.

Key Skills:

Students will have enhanced problem solving skills.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
Subject material assigned for independent study develops the ability to acquire knowledge and understanding without dependence on lectures.
Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and the lecturer on student's progress.
The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Teaching Methods and Learning Hours

Activity	Number	Frequency	Duration	Total/Hours
Lectures	40	2 per week for 19 weeks and 2 in term 3	1 hour	40
Preparation and Reading				160
Total				200

Summative Assessment

Component: Examination		Component Weighting: 100%
Element	Length / duration	Element Weighting	Resit Opportunity
Three hour written examination	3 hours	100%

Formative Assessment:

Four written assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University