### Introduction

**
HELM booklet
**
28 considered the differentiation of scalar and vector fields. Here we consider how to integrate such fields along a line. Firstly, integrals involving scalars along a line will be considered. Subsequently, line integrals involving vectors will be considered. These can be scalar or vector depending on the form of integral used. Of particular interest are the integrals of conservative vector fields.

#### Prerequisites

- have a thorough understanding of the basic techniques of integration
- be familiar with the operators div, grad and curl

#### Learning Outcomes

- integrate a scalar or vector quantity along a line

#### Contents

1 Line integrals1.1 Line integrals in two dimensions

2 Line integrals of scalar products

3 Conservative vector fields

4 Engineering Example 1

4.1 Work done moving a charge in an electric field

5 Vector line integrals

6 Engineering Example 2

6.1 Force on a loop from a magnetic field

7 Engineering Example 3

7.1 Magnetic field from a line current