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

(LO1) After completing the module students should: Be familiar with a range of examples of metric spaces. Have developed their understanding of the notions of convergence and continuity.

(LO2) Understand the contraction mapping theorem and appreciate some of its applications.

(LO3) Be familiar with the concept of the derivative of a vector valued function of several variables as a linear map.

(LO4) Understand the inverse function and implicit function theorems and appreciate their importance.

(LO5) Have developed their appreciation of the role of proof and rigour in mathematics.

(S1) problem solving skills

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. Pointwise 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.