MAS1603 : Multivariable Calculus and Differential Equations

Semester 2 Credit Value: 15
ECTS Credits: 8.0


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 MAS1601, 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 ODEs, singular points and integral curves of first-order ODEs.

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

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
Scheduled Learning And Teaching ActivitiesLecture11:001:00Assignment laboratory
Scheduled Learning And Teaching ActivitiesLecture11:001:00Class test
Scheduled Learning And Teaching ActivitiesLecture31:003:00Problem classes
Scheduled Learning And Teaching ActivitiesLecture31:003:00Revision lectures
Scheduled Learning And Teaching ActivitiesLecture341:0034:00Formal lectures
Guided Independent StudyAssessment preparation and completion19:009:00Revision for class test
Guided Independent StudyAssessment preparation and completion119:0019:00Revision for unseen exam
Guided Independent StudyAssessment preparation and completion12:302:30Unseen exam
Scheduled Learning And Teaching ActivitiesDrop-in/surgery41:004:00Tutorials in the lecture room
Scheduled Learning And Teaching ActivitiesDrop-in/surgery120:102:00Office hours
Guided Independent StudyIndependent study132:3032:30Studying, practising and gaining understanding of course material
Guided Independent StudyIndependent study53:0015:00Review of coursework assignments and course test
Guided Independent StudyIndependent study46:0024:00Preparation for coursework assignments
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. Tutorials are used to identify and resolve specific queries raised by students and to allow students to receive individual feedback on marked work. In addition, office hours (two per week) will provide an opportunity for more direct contact between individual students and the lecturer: a typical student might spend a total of one or two hours over the course of the module, either individually or as part of a group.

Assessment Methods

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

Description Length Semester When Set Percentage Comment
Written Examination1502A80N/A
Written Examination402M10Class test
Exam Pairings
Module Code Module Title Semester Comment
PHY1029Multivariate Calculus and Differential Equations2N/A
Other Assessment
Description Semester When Set Percentage Comment
Prob solv exercises2M10Coursework assignments
Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The coursework assignments will consist of one written assignment (approximately 3%), one assignment laboratory (approximately 3%) and two computer based assessments (each approximately 2%). The coursework assignments and the (in class, therefore 40 minute) coursework test 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