36 1 VECTOR ANALYSIS Vx M +Vx (Va)=Vx M by Equation (1.39).The magnetic flux density field and the velocity field for an incompressible fluid are examples of solenoidal fields. 1.9 THE DIVERGENCE THEOREM Consider a volume in space bounded by a closed surface s with a well-behaved vector field F present as shown in Figure 1.19.Let dv be a volume element and let ds be a vector surface element of the external surface s.Now imagine that the entire volume is partitioned into N small volumes,Av,Av2,.Av. △vN each having closed surfaces△si,△s2,.,△sr,.,△sN.For each small zero.That is, (7·F)n=lim △yn or (pn≈$a,E,·ds, △yn or (V·F)n△yn≈Fn·dsn Writing similar expressions for all the small volumes and summing,we get element Closed surfaces Cross section of closed surface Flgure 1.19.for establishing the divergence theorem(Gauss's integral theorem)
.9 THE DIVERGENCE THEOREM 之 >(7·F)n△vn≈>，Fn·dsn n=1 n=15, where the right-hand side of the equation is the flux out of the external surface s, that is, ·ds. because the contributions from all the surfaces (or partial surfaces)internal to s cancel.They cancel,because at interface surfaces common to adjacent small volumes,ds will be the same in magnitude,but opposite in direction.Thus,we have >(F)n△vneF·ds n=1 Now if we let N approach infinity so that Avn approaches zero,the sum on the left becomes a volume integral and the approximation becomes exact V·Fdw=Fs·ds divergence theorem. (1.43) Equation(1.43)is known as the divergence theorem.The long arrow means that the surface on the right defines the volume on the left. ■Example1.8 and Jaaa'sin dedd= Jr=0J8=0JΦ=0 sin e dr dedφ Or 4ma =4ma. Suppose the surface is that of a cone of radius a and cone angle=00.In this case,ds=ara2sinθd0d中+aer sin0drd中，butA·ds=asin0d0dd,so that
28 1 VECTOR ANALYSIS fa f8of2 Jo Jo Jo sin 0dr d6dΦ 2ma(1-cos 0o)=2ma(1 cos 0o). 1.10 STOKES'THEOREM Consider an open surface s whose periphery is c with a well-behaved vector field A present.Figure 1.20 demonstrates the geometry and the positive sense for the vector quantities involved.The surface s(not necessarily planar)is subdivided into△s1,△s2, .AsN (incremental vector areas).If each Asn is sufficiently small,then A may be assumed to be constant over it.We may write. for the circulation, Adl=∑And n=1Jc This result occurs because the line integrals over segments of the c interior to c are zero since the line integrals along a common boundary for adjacent areas are taken in directions.Only those parts of the's that coincide with c contribute to the sum.Letting N approach infinity so that Asn approaches zero gives (1.44) Open surface Closed contour (line)c Flgure 1.20.Geometry for establishing Stokes'theorem
1.10 STOKES'THEOREM 29 The component in the direction of the curl of A.(A).at a point is defined by Equation(1.37)as the limit of the circulation per unit area as the area (as)approaches zero.Thus,we have 1 a(×A)=ima中A xAd or,applied to Asn 1 An'dl. a(W×)≈asn Therefore,we have aa·(xA)asn=$Aa·l (1.45) Substituting Equation(1.45)into Equation(1.44),we get A·dl=im∑a·(×A)Asn As-0n=1 =lim∑(V×A)·Asa 5n0n-1 but the limit is an (open surface)integral in the usual way,so Stokes'theorem, (1.46) where Equation(1.46)is known as the Stokes theorem
30 1 VECTOR ANALYSIS ■Example1.9 Show that the Stokes theorem holds for the vector field F=pa,x F=2a:, when the closed path is a circle of radius p=a with its center at the origin.The open surface is the disk0≤p≤a,z=0.We have F·dl=aao'asa do=a2db, so that the Stokes theorem is oJ。2pdpd, 2Tu2 =2Ta? Suppose that the open surface is the upper hemispherer=a,0/2. In this case F.dl is unchanged.and the Stokes theorem gives 「T2f2r 2ma2= 2a:·ara2sin0d6db 00 Now we have a:a,cos 0,so that we get 7/2 2ma2=2ma2 2 sin 0 cos 0 d0 =2ma2 Jo sin 20 d0, and,once again,we have 2ra2=2ra2. Any other open surface whose opening is defined by a closed path that is the circle p=a (center at the origin)will give the same result. 1.11 CONCLUDING REMARKS Vecrdion and multiplication were briefly reviewed in the first part of this chapter.This material should be familiar to the engineering student.Next,the three coordinate systems that are used exclusively in this text were introduced and related. The concepts of circulation (line integrals)and flux were defined and given physical meaning by means of familiar examples.The vector operations of diver- gence,gradient,and curl that we will have much use for in the following chapters, were introduced and explained. There are many vector identities that appear in the study of vector analysis.and the two most important of these that we will use were introduced.The divergence theorem(the Gauss integral theorem)and the Stokes theorem are fundamental to field theory.The derivation of these theorems was presented.A summary of vector relations is listed on the inside front and back covers