## MATH2420 Multiple Integrals and Vector Calculus

### 10 creditsClass Size: 200

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2005/06

### Pre-requisites

MATH 1400. Exclusion: not with MATH2360.

This module is approved as an Elective

### 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

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. 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

Lectures (22 hours) and examples classes (11 hours).

### Methods of assessment

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

2 hour written examination at end of semester (85%), coursework (15%).