### Introduction

Various theorems exist equating integrals involving vectors. You have already met the fundamental theorem of line integrals. Those involving line, surface and volume integrals are introduced here. They are the multivariable calculus equivalent of the fundamental theorem of calculus for single variables (“integration and differentiation are the reverse of each other”).

Often, use of these theorems can make certain vector integrals easier. This Section introduces the theorems known as Gauss’ theorem, Stokes’ theorem and Green’s theorem.

#### Prerequisites

- be able to find the gradient of a scalar field and the divergence and curl of a vector field
- be familiar with the integration of vector functions

#### Learning Outcomes

- use vector integral theorems to facilitate vector integration

#### Contents

1 Stokes’ theorem1.1 Justification of Stokes’ theorem

2 Gauss’ theorem

3 Engineering Example 6

3.1 Gauss’ law

4 Engineering Example 7

4.1 Field strength around a charged line

5 Engineering Example 8

5.1 Field strength on a cylinder

6 Green’s Identities (3D)

6.1 Proof of Green’s identities

6.2 Green’s theorem in the plane

6.3 Justification of Green’s theorem in the plane