Aims

To develop an understanding of ordinary differential equations and a familiarity with relevant solution methods. To introduce the calculus of functions of several variables.

Module summary

This module, which continues and extends the work of MAS1605, develops many of the ideas that are needed when constructing mathematical models of phenomena in the real world. Many such models are formulated in terms of ordinary differential equations, and this module introduces the methods that are needed to solve problems of this type. The world where we live is multi-dimensional - three-dimensional if we consider spatial dimensions alone, or four-dimensional if we treat time as another variable. It is therefore essential to develop tools to describe and model objects and processes that occur in multi-dimensional spaces. In order to do this we require multidimensional calculus. This module introduces the partial derivative, and the multiple integral, as well as power series in two or more variables.

Outline Of Syllabus

Introduction to ordinary differential equations (ODEs): terminology and examples.

First-order ODEs: separable equations, homogeneous equations, integrating factor. Existence and uniqueness of the initial value problem for first-order linear ODEs; a brief introduction to isoclines.

Second-order ODEs: homogeneous equations with constant coefficients, particular integrals for inhomogeneous equations, method of reduction of order.

Introduction to functions of several variables: partial differentiation, gradient, chain rule and Jacobian matrices.

Taylor series in two (or more) variables, classification of stationary points.

Multiple Integrals: double and triple integrals, change of variables (including polar coordinates).

Teaching Rationale And Relationship

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.

In the event of on-campus examinations not being possible, an on-line alternative assessment will be used for written examination 1.
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