2011/12 Undergraduate Module Catalogue

MATH2420 Multiple Integrals and Vector Calculus

10 creditsClass Size: 180

Module manager: Dr N Morrison
Email: morrison@maths.leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2011/12

Pre-requisite qualifications

MATH1050 and MATH1055 and (MATH1410 or MATH1400)

This module is mutually exclusive with

MATH2365

Vector Calculus

This module is approved as an Elective

Module summary

Vector calculus, an extension of ordinary differential and integral calculus, is the normal language used in applied mathematics for solving problems in two and three dimensions.

Objectives

- To develop methods for evaluating multiple integrals
- To discuss the basic tools of vector calculus and the theorems of Gauss and Stokes.

On completion of this module, students should be able to:
(a) evaluate line, surface and volume integrals using Cartesian and polar co-ordinates;
(b) change variables in double integrals using Jacobians;
(c) calculate the gradient of a scalar field and the divergence and curl of a vector field, together with associated quantities such as the Laplacian;
(d) use the divergence theorem and Stokes' theorem in the manipulation of multiple integrals.

Syllabus

This module starts by extending the familiar idea of integration in one dimension along the (straight) x-axis to integration along a curve and then considers integration over surfaces (2 dimensions), and through volumes (3 dimensions).

Then follows a discussion of the differential properties of scalar and vector functions in 3 dimensions (gradient, divergence and curl) and of the relations between them contained in the famous integral theorems of Gauss and Stokes.

These theorems show the close connections which exist between the line, surface and volume integrals studied at the start of this module.

A knowledge of vector calculus is essential for further study in many areas of applied mathematics.

Topics covered include:
1. Line integrals
2. Surface integrals: change of limits in repeated integrals, transformation of co-ordinates and Jacobians, normal to a general surface and evaluation of surface integrals by projection.
3. Volume integrals: Use of cylindrical and spherical polar co-ordinates.
4. Scalar and vector fields.
5. Gradient and directional directive: Divergence and curl.
6. Expansion formulae: Second order differential functions, the Laplacian.
7. Flux and the divergence theorem.
8. Circulation and Stokes' Theorem.

Teaching methods

Delivery type	Number	Length hours	Student hours
Workshop	10	1.00	10.00
Lecture	22	1.00	22.00
Private study hours			68.00
Total Contact hours			32.00
Total hours (100hr per 10 credits)			100.00

Opportunities for Formative Feedback

Regular problem solving assignments

Methods of assessment

Coursework

Assessment type	Notes	% of formal assessment
In-course Assessment	.	15.00
Total percentage (Assessment Coursework)		15.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Exams

Exam type	Exam duration	% of formal assessment
Standard exam (closed essays, MCQs etc)	2 hr	85.00
Total percentage (Assessment Exams)		85.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Reading list

The reading list is available from the Library website

Last updated: 27/02/2012

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