082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

27 Lineintegralsand
multipleintegrals
Contents 27.1 Introduction 867
27.2 Lineintegrals 867
27.3 Evaluationoflineintegralsintwodimensions 871
27.4 Evaluationoflineintegralsinthreedimensions 873
27.5 Conservativefieldsandpotentialfunctions 875
27.6 Doubleandtripleintegrals 880
27.7 Somesimplevolumeandsurfaceintegrals 889
27.8 ThedivergencetheoremandStokes’theorem 895
27.9 Maxwell’sequationsinintegralform 899
Reviewexercises27 901
27.1 INTRODUCTION
In this chapter a number of new sorts of integral are introduced. These are intimately
connected with the developments of the previous chapter on differential vector calculus.
The chapter starts by explaining the physical significance of line integrals and how
theseareevaluated.Thisleadsnaturallyintothetopicsofconservativevectorfieldsand
potentialfunctions.Theseareimportantinthestudyofelectrostatics.Doubleandtriple
integrals are then introduced; these generalize the earlier work on integration to integrands
which contain two and threeindependent variables.
Finallysomesimplevolumeandsurfaceintegralsareintroduced,togetherwiththedivergence
theorem and Stokes’ theorem. These enable Maxwell’s equations to be
expressed inintegral form.
27.2 LINEINTEGRALS
Consider an object of massmplaced in a gravitational field. Because the force of gravity
is a vector the gravitational field is an example of a vector field. The gravitational
force on the mass is known as its weight and is given by mg where g is a constant