1.4 CIRCULATION AND FLUX 11 (2 W-。2r+。 w=16(J. If the path is changed to the straight line y =2x,dy =2 dx,we have 2 2 w=。4dk+。4dx w÷16(J), which is the same result as before. It appears that the line integral of this particular F is indeed independent of the path.The preceding example offers evidence,but no proof,that the line integral is independent of the path.Proof will be given at an appropriate time. A particular surface integral that is frequently encountered in working with vector fields is that giving the scalar flux or flow of a vector through a specified open or closed surface s.Let the flux density vector be called D.As shown in Figure 1.10(for a closed surface),the normal component of the differential flux density is simply found as the component of D ons(=D)in the direction of ds. so if the flux is called平，then dΨ=Dls·ds,and the total flux out of the closed surface is flux of vector D. (1.20) ds dVg=Dl,ds Flgure 1.10.Differential flux at a point on a closed surface
12 1 VECTOR ANALYSIS Remember that the direction of ds is normal(outward)to the surface at the point and that the circle on the double integral symbol means that we are concerned with a closed surface,as opposed to an open surface.If we are calculating the flux through an open surface then the circle on the integral symbol in Equation(1.20) is omitted.Other integral forms that we may encounter in this text are listed on the inside back cover ■Example1.4 Find the flux of the vector field A =a,/r2 out of the spherer =a.0m, 0s中≤2π.We have ·a,a2sin8d0dΦ r =a Flux Jo o sin 0 de do=4m. Now find the flux out of a cube I m on a side,centered at the origin,with sides parallel to the coordinate axes.In this case the total flux is six times that out of one face because of the radial symmetry of the field.Take the face located in the z=0.5 plane.The total flux is given by /2 Fux=6k-2Jy-2a·adkd. Since we have a,.a:cos 0 =z/r =(x2 +y2+)-12,then (V2 (2 Flux =3 dx dy J-I2J-2(x2+y2+)32 1/2 Flux =3 -22+4)x2y2+4)”-2 dy x=6.02+402+) After using the change of variable u2=y2+,we obtain 1}21/v2 Flux -12.cos-16)
1.5 DIVERGENCE It is not an accident that this result is identical to the first,and this will be explained when Gauss's law is considered in Chapter 2. 1.5 DIVERGENCE Consider a small box2 with sides Ar,Ay,and Az.Let the vector field D at the geometric center be given by D=Diax D2ay +D3a:, (1.21) where D,D2,and D3 are the x,y,and z components of D.This arrangement is Rsp深we监 We now calculate the flux out of the box using Equation (1.20): realizing that we have the sum of six double integrals,one for each face.Consider the integral over the surface closest to the reader: △y△z (1.22) front Equation (1.22)is approximate because D is not,in general.constant over this face.In terms of the given flux density at the center of the box,we may write 2A rectangular parallelepiped. d -y D=D+ Flgure 1.11.Incremental Gaussian surface for deriving Gauss's law in point form (Maxwell's equa- ion
14 1 VECTOR ANALYSIS o叫+告2 (1.23) front In words,Equation (1.23)states that D,on the front face is approximately equal to D at the center of the box (i.e.,D1)plus the rate of change of Dx withx times the distance(Ax/2)over which this change occurs.It is worth mentioning that Equation(13)may be obtained more rigorously as the first wo terms in the Taylor's series expansion for D.about the center of the box.We now have (1.24) Proceeding in a similar manner for the back face,we have back where DD-告设 so that Thus dx For all six faces we have 知0+0+兴 Dividing by the differential volume Av,we get D+D+D=每aD。·dk ax dy+az If we now take the limit as Av approaches zero,we obtain exactly
1.5 DIVERGENCE 15 设+兴+识=一 (1.25) dy Av 2m (independent of coordinate system),and this quantity is called the divergence of D;that is, Divergence of D-div D=limds (1.26) Consider a situation such as that shown in Figure 1.12 where flux density lines appear to diverge from the small volume va,suggesting the presence within va of some kind of source.On the other hand,flux density lines appear to converge (negative divergence)into the small voumesuggesting the pre esence within v of some kind of sink(negative source).It is apparently true that the flux out of va is positive,and the flux out of vo is negative.It appears that the flux out of ve is zero.All of these conclusions could be verified by means of Equation(1.25) if we were given an explicit expression for the vector field D at every point in space.A nonzero value for the divergence of a vector field at a point is intimately associated with the presence fa scalar source a that poin.Gauss's law,obe presented in Chapter 2,will shed more light on this subject. It is apparent from Equations (1.25)and (1.26)that div D=D:+aDDs (1.27) z for Cartesian coordinates,but this result occurred merely because we started with a small rectangular box.The result given by Equation (126)is general,and is independent of coordinate system.In order to emphasize these points and to show explicitly that taking the divergence of a vector is a vector operation,we introduce the vector del operator: D Flgure 1.12.Flux density streamlines indicating flux in(or out of)volumes