### Module Details

 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.
 Title METRIC SPACES AND CALCULUS Code MATH241 Coordinator Dr NT Pagani Mathematical Sciences Nicola.Pagani@liverpool.ac.uk Year CATS Level Semester CATS Value Session 2023-24 Level 5 FHEQ First Semester 15

### Aims

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

### 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

### 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.

### Pre-requisites before taking this module (other modules and/or general educational/academic requirements):

MATH101 Calculus I 2022-23; MATH102 CALCULUS II 2022-23; MATH103 Introduction to Linear Algebra 2022-23

### Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
written exam  120    60
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