Mathematical Methods - 2019 entry

During your mathematics degree, you will be solving problems and proving theories in several branches of mathematics such as in pure mathematics, in applications to science and engineering, and in statistics. Inevitably you need to be able to calculate. That is what gives the mathematics its great power. This module covers developed bodies of useful techniques as a toolkit of common knowledge. It brings emphasis on the techniques rather than the applications of the techniques. Such techniques will enable you to deepen your familiarity with, and generalise, methods that you have seen at school level mathematics. This module will study topics that include the geometry of conic sections, properties of functions such as continuity and differentiability, differential and integral calculus, limits and convergence of sequences and series including Power Series and Taylor Series. The module also develops the fundamentals of vector and matrix theory, multivariate calculus, and the classification of various types of differential equations as well as analytical methods for solving them. The material in this module provide intuition for, and examples of, many of the mathematical structures that will be discussed in the module MTH1001 Mathematical Structures, and supply a firm understanding of methods required in future modules in the mathematics degree.

This module aims to develop your skills and techniques in calculus, geometry and algebra. It is primarily focused on developing methods and skills for accurate manipulation of the mathematical objects that form the basis of much of an undergraduate course in mathematics. Whilst the main emphasis of the module will be on practical methods and problem solving, all results will be stated formally and each sub-topic will be reviewed from a mathematically rigorous standpoint. The techniques developed in this course will be essential for much of your undergraduate degree programme, particularly the second-year streams of Analysis, Differential Equations & Vector Calculus, and Mathematical Modelling.

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1 explain how techniques in differential and integral calculus are underpinned by formal rigour;

2 apply techniques in geometry and algebra to explore three dimensional analytic geometry;

3 perform accurate manipulations in algebra and calculus of several variables using a variety of standard techniques;

4 solve some specific classes of ordinary differential equations;

Discipline Specific Skills and Knowledge:

5 demonstrate a basic knowledge of functions, sequences, series, limits and differential and integral calculus necessary for progression to successful further studies in the mathematical sciences;

Personal and Key Transferable/ Employment Skills and Knowledge:

6 reason using abstract ideas, and formulate and solve problems and communicate reasoning and solutions effectively in writing;

7 use learning resources appropriately;

8 exhibit self-management and time-management skills.

- Geometry: lines; planes; conic sections;

- Functions: single- and multivariate; limits; continuity; intermediate value theorem;

- Sequences: algebra of limits; L'Hopital's rule

- Series: convergence/divergence tests; power series;

- Differential calculus: simple and partial derivatives; Leibniz' rule; chain rule; Taylor approximation; implicit differentiation; minima and maxima;

- Integral calculus: substitution; integration by parts; multiple integrals; applications;

- Differential equations: linear and separable ordinary DEs; basic partial DEs;

- Vectors, matrices: Gaussian elimination; transformations; eigenvalues/eigenvectors.

In the case of module referral, the higher of the original assessment and the reassessment will be recorded for each component mark. In the case of module referral, the final mark for the module reassessment will be capped at 40%..

Basic reading: Any A-Level on mathematics and further mathematics

ELE: http://vle.exeter.ac.uk

Reading list for this module: