## MATH2420 Multiple Integrals and Vector Calculus

### 10 creditsClass Size: 200

Module manager: Dr A. Courvoisier
Email: alice@maths.leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2008/09

### Pre-requisites

 MATH1400 Modelling with Differential Equations

### 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

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

 Delivery type Number Length hours Student hours Example Class 11 1.00 11.00 Lecture 22 1.00 22.00 Private study hours 67.00 Total Contact hours 33.00 Total hours (100hr per 10 credits) 100.00

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

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