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MAS1607 : Multivariable Calculus & Differential Equations

  • Offered for Year: 2023/24
  • Module Leader(s): Professor Paul Bushby
  • Owning School: Mathematics, Statistics and Physics
  • Teaching Location: Newcastle City Campus

Your programme is made up of credits, the total differs on programme to programme.

Semester 2 Credit Value: 20
ECTS Credits: 10.0
European Credit Transfer System


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. 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 Methods

Teaching Activities
Category Activity Number Length Student Hours Comment
Guided Independent StudyAssessment preparation and completion301:0030:00Completion of in course assessments
Scheduled Learning And Teaching ActivitiesLecture111:0011:00Problem Classes
Scheduled Learning And Teaching ActivitiesLecture311:0031:00Formal Lectures
Scheduled Learning And Teaching ActivitiesLecture21:002:00Revision Lectures
Scheduled Learning And Teaching ActivitiesSmall group teaching51:005:00Group Tutorials
Guided Independent StudyIndependent study1211:00121:00N/A
Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking. 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 Methods

The format of resits will be determined by the Board of Examiners

Description Length Semester When Set Percentage Comment
Written Examination1502A80N/A
Other Assessment
Description Semester When Set Percentage Comment
Prob solv exercises2M20Problem-solving exercises
Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of the programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

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.

Reading Lists