1.7 CURL Flgure 1.15.Line integral contourc and one open surfaces. (eaiA.=码fAn (1.37) where As=aAs,and the closed path c defines the area As.The unit vector a is normal to the surface As,and in a direction determined by the right-hand rule as explained in Figure 1.15. For example,following a procedure similar to that used in arriving at a formula for divergence,we may describe a rectangle parallel to theplane as shown in Figure 1.16.At the center of the rectangle (whose sides are Ax,Ay),A =axAx+ aA.+a.A:.It is easy to show by means of the Taylor series expansion that A =A+A+, A =A:+y Ix+Ax/2 x2 ly+Ay/2 av 2 and =Ax- y-4y12 Flgure 1.16.Geometry for evaluating the component of the curl of A
22 1 VECTOR ANALYSIS and we get 手aa=股-r4. Higher-order terms that would disappear in the limiting process have simply been dropped.In the limit then,we get m=验- Repeating this procedure for rectangles parallel to the y =0 plane and x =0 plane gives 《m许-沿 and (cuiA.-盼-脸 Then the curl of A,given by x A,may be written for Cartcsian coordinates as ×A=-}+A+映-的)测 or,in a form more easily remembered, ax ay az v×A=ax ay oz Ax Ay A: For cylindrical and spherical coordinates,we have 路-+叫院- V×A=apab-az} +a1a(42-1Ag pppa (cylindrical) (1.38b) and
1.7 CURL 23 ×A=,0.2480D-8 +e1 aA.a(rA r sin 0 dd dr +22A)-4 ar -a0 (spherical). (1.38c) There are at most three components of a vector in space,and each of these thrcc components can depend at most on three space variables.Thus,there are at most nine(first)derivatives involving a spatial vector.Six of these are involved in specifying the curl of a vector,as Equations(1.38a),(1.38b),and(1.38c)show, while the remaining three are involved in specifying the divergence of a vector,as Equations(1.31a),(1.31b),and(1.31c)show.If both the curl and the divergence of a vector are specified,then the vector can be uniquely determined. In order to attach so me physical significance to the curl of a vector,we will employ the small"paddlewheel"as suggested by Skilling.4 Let the vector field be a fluid velocity field,regardless of what it is physically.Place the small paddlewheel in this velocity field and move it about.For every point in the field,where the curl of the field is to be found,the paddlewheel axis should be oriented in all possible directions.The maximum angular velocity of the paddlewheel at a point is proportional to the curl,whil the axis of the paddlewheel points in the oftheurdntothe right-hand rue.That is,if th fingersfh point in the direction of the rotation of the paddlewheel blades,then the thumb of the right hand points in the direction of the axis of rotation or This is demonstrated in Figure 1.17.If the paddlewheel does not rotate,the vector field is irrotational,or has zero curl!A simple example will help clarify this paddlewheel concept. See references at end of chapter. .17.velcity field and the right-handu
24 1 VECTOR ANALYSIS gure 1.Demonstraion for finding increasing vector mall paddlewheel. ■Example1.7 Suppose A=Ka,where K is a constant.This field is sketched in Figure 1.18 for z>0.It might be analogous to water flow in a river.It is obvious that the paddlewheel rotates in a clockwise direction (regardless of )and the angular velocity of the paddlewheel is proportional toK(the slope)regardless of z.Furthermore,the maximum angular velocity occurs when the paddlewheel axis is parallel to the y axis and is zero when the paddlewheel axis is parallel to the x oraxis.According to the right-hand rule the direction of the curl is that of+ay Analytically,using Equation(1.38a),we have V×A=+a,A=a,K. dz which certainly agrees with our observations with the paddlewheel! ■ 1.8 SOME FUNDAMENTAL VECTOR IDENTITIES It is easy to show in Cartesian coordinates that V×(7Φ)=0 (1.39) for any scalar field Now,for example,any vector field F that can be written as (±)the gradient ofΦ(F=+7Φ)must have zero curl(×F=0).This is true even if a constant Cis added toΦsince V(Φ+C)=7Φ+Vc=vΦ(vC=0). a field that has no curl is said to be irrotational as mentioned at the end of section 1.7 ×F=0 irrotational field F (1.40)
1.8 SOME FUNDAMENTAL VECTOR IDENTITES 25 The work done by an external source in moving an object from P:to P in the vector force field F is given by the negative of Equation (1.18): w -pF.d. Now using Cartesian coordinates,with F=-V,we get 0=小驶+9+}a血++划 =4k++ 0qΦ 、aΦ dx where the right-hand side of the equation is the differential dd: dΦ=vΦ·dl=-Fdl, so that we get W=+ n地=-， Therefore,W depends on the endpoints of the path.but not on the path itself Furthermore,F must be a conservative field [see Equation (1.19)and the material preceding it).We conclude that any vector field that is the gradient of a scalar field is a conservative field.We can test a field to determine whether it is conservative or not by simply determining whether or not its curl is identically zero: 7×F=0 conservative field F. (1.41) The electrostatic field intensity,the gravitational force field,and the heat flux vector are all examples of conservative fields. Another useful vector identity is V·(7×M0=0 (1.42) for any vector field M.As a consequence,any vector field N.for example that can be written as the curl of M (N M)must have no divergence (V.N =0).Such fields are called divergenceless or solenoidal.This is true even if the gradient of any scalar ficld a is added to M,since Vx(M +Vo)-