Module Specification

The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
1. Module Title METRIC SPACES AND CALCULUS
2. Module Code MATH241
3. Year Session 2023-24
4. Originating Department Mathematical Sciences
5. Faculty Fac of Science & Engineering
6. Semester First Semester
7. CATS Level Level 5 FHEQ
8. CATS Value 15
9. Member of staff with responsibility for the module
Dr NT Pagani Mathematical Sciences Nicola.Pagani@liverpool.ac.uk
10. Module Moderator
11. Other Contributing Departments  
12. Other Staff Teaching on this Module
Professor L Rempe Mathematical Sciences L.Rempe@liverpool.ac.uk
Dr SA Fairfax Mathematical Sciences Simon.Fairfax@liverpool.ac.uk
13. Board of Studies
14. Mode of Delivery
15. Location Main Liverpool City Campus
    Lectures Seminars Tutorials Lab Practicals Fieldwork Placement Other TOTAL
16. Study Hours 24

  24

      48
17.

Private Study

102
18.

TOTAL HOURS

150
 
    Lectures Seminars Tutorials Lab Practicals Fieldwork Placement Other
19. Timetable (if known)            
 
20. Pre-requisites before taking this module (other modules and/or general educational/academic requirements):

MATH102 CALCULUS II; MATH101 Calculus I; MATH103 Introduction to Linear Algebra
21. Modules for which this module is a pre-requisite:

 
22. Co-requisite modules:

 
23. Linked Modules:

 
24. Programme(s) (including Year of Study) to which this module is available on a mandatory basis:

25. Programme(s) (including Year of Study) to which this module is available on a required basis:

26. Programme(s) (including Year of Study) to which this module is available on an optional basis:

27. Aims
 

To introduce the basic elements of the theory of metric spaces and calculus of several variables.

 
28. Learning Outcomes
 

(LO1) Be familiar with a range of examples of metric spaces.

 

(LO2) Understand the notions of convergence and continuity.

 

(LO3) Understand the contraction mapping theorem and be familiar with some of its applications.

 

(LO4) Be familiar with the concept of the derivative of a vector valued function of several variables as a linear map.

 

(LO5) Understand the inverse function and implicit function theorems and appreciate their importance.

 

(LO6) Have developed their appreciation of the role of proof and rigour in mathematics

 

(S1) Problem solving skills

 
29. Teaching and Learning Strategies
 

Material is provided in advance of classes for students to study asynchronously. The contact hours consist of one 2-hour active learning session and one 2-hour supported study/drop-in session.

 
30. Syllabus
   

Metric spaces: Examples of metric spaces: R^n, the discrete metric, metric of uniform convergence on C[a, b].

Convergence and continuity.  Open and closed subsets.

Completeness of metic spaces. Infimum and supremum, lim inf and lim sup. The Bolzano-Weierstrass theorem and completeness of R^n.

The Contraction mapping theorem.

Point-wise and uniform convergence, and the completeness of C[a, b]. Term-by-term differentiation and integration of power series.  Local existence and uniqueness of solutions of first order ODEs. The Hausdorff metric. Iterated function systems and fractals.

Calculus: Revision of linear algebra: matrix product, determinant, and inverse. Continuity and differentiability of functions R^n -> R and R^n -> R^m.  The chain rule, inverse function theorem and implicit function theorem.

 
31. Recommended Texts
  Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.
 

Assessment

32. EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
  written exam 120 60
33. CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
  Class Test 1 60 20
  Class Test 2 60 20