16 1 VECTOR ANALYSIS V=a, :0z (1.28) The del operator must operate on some quantity,and it is easy to show (treating V like any vector)that 7D= ++ ·{axDx+ayDy+aDz V、D三0Dx+aD之+9=lim aDis ds (1.29) Av for Cartesian coordinates. The left and right sides of Equation(1.29)give us a completely general math- ematical definition of del as an integral operator: .=lim dso △v (1.30) If the small circle ()becomes a dot,we obtain the divergence of a vector.If the small circle becomes a cross,we obtain the curl of a vector.Finally,if the small cirle disappears,s i ordinary The curl and gradient operations will be discussed shortly. It is useful to list V.D for Cartesian,circular cylindrical,and spherical coor- dinates(also listed on the inside front and back covers): V.D=ODx +D+a0 (1.31a) dy Fp品a0+0+0 7D=1a p ad (1.31b) v.n-10 1 a 1 dD ,京rr2D,)+,in90 (sinD）+,sin900 (1.31c) As can be seen,the divergence of a vector quantity is a scalar involving partial derivatives of a particular component with respect to the variable associated with that component. Some further remarks about the divergence are in order.An incompressible fluid.such as water,must have a velocity field u which has no divergence (V.u=0).Fluid is neither created nor destroyed at any point,and therefore has no sources or sinks.The magnetic flux density field,which we will encounter later,also has no sources or sinks because no isolated magneric charges have been found in nature.It too has no divergence.Such fields are said to be solenoidal or sourceless.The electrostatic flux density field,on the other hand,does have sources and sinks(the electric charges),and must have nonzero divergence at some point or points in space
1.6 GRADIENT 41 1.6 GRADIENT The gradient of a scalar field is a vector field that lies in the direction for which the scalar field is changing most rapidly.The magnitude of the gradient is the e ere时e shown)of constant In the general three-dimensional case these contours are called isothermal surfaces when,for example,(x,y,z)represents temperature and they are called equipotential surfaces if (x.y,)represents electric potential (to be introduced in Chapter 2).The direction n of the greatest rate of change of will be rormal to the contours of constant (from a smaller to a larger value) since the rate of change of is zero along or tangent to these contours.Thus,we have 8rad中-a, dn 132 Consider (in a completely general case)(x,y,z)at a point P(x,y,z)and x+△x,y+△y,z+△z)at a nearby point (x+△x,y+△y,z+△z).f is the distance betweenPand then we get △Φ=(x+Ax,y+△y,z+Az)-Φ(x,y,z) 、0 4y+4 dz 、,40V50 Flgure 1.13.Two-dimensional potential fieldin terms of equipotentials
18 1 VECTOR ANALYSIS Higher-order infinitesimals that will disappear in the limiting process to follow have been dropped.Then we have dz△ and 曾一0+留+骋盘 This quantity is the rate of change ofwith respect to distance at point P in a direction toward and is thus called the directional derivative.The directional derivative may also be written -+9+ aΦ.dl (directional derivative).(1.33) Note that the discussion in the preceding paragraph it is apparent that the directional derivative is maximum when ar=an,that is, when it is taken in a direction normal to the equipotential surface at P2 (as in Figure 1.13).Thus,we have (=ax dl max + aΦ 0Φ1 ·an (1.34) It is now obvious that the maximum directional derivative takes place in the direction of,and has the magnitude of the bracketed term in Equation(1.34). From the definition in the preceding paragraph,the bracketed term is the gradient of the scalar fieldΦ，so we get aΦ grad==a3x+a,0y ,a(Cartesian). +0 (1.35a) For cylindrical and spherical coordinates.we have Φ=apap 1∂Φ aΦ ap中 : (cylindrical) (1.35b) and 师=a +88+,8 1d①Φ (spherical). (1.35c) ar
.6 GRADIENT 19 20m 30m 40m h=50m Flgure 1.14.Contours of constant elevation h and-Vh. It is important to recognize that,in general,varies from point to point.A glance at a map having elevation contours (equipotentials)shows this very well Figure 1.14 shows a hill with elevation contours around it.We now ask ourselves a question."If we place a small ball at P,in what direction will it roll and what is the magnitude of the force acting to accelerate it in that direction?"This is equivalent to asking what is -grad h(x,y)? Example 1.5 Find(approximately)the gradient of at P and P2 in Figure 1.13.We have =+a.地 aΦ △中 +a:+a △y so,for P1,we get 10 m+a002=+313a and for P2,we get 0l,+a003+4004=+76a:+7146 10 ■ Temperature is a term that is familiar to us.The temperature in a region may vary from point to point,and can,therefore,be described as a three-dimensional scalar field.Surfaces of(the"quipote) surfaces.The heat flux vector s(W/m2)at a point in a conducting body is given by
20 1 VECTOR ANALYSIS the Fourier heat conduction law:s-KVT,where T is the temperature and K is the thermal conductivity.Thus,heat flows in a direction opposite to the positive temperature gradient,or,put more simply,heat flows from a region of higher temperature to a region of lower temperature. As in the case of the divergence,the gradient of a scalar field at a point P may be defined in integral form by =吗，$中s (1.36) where Av is a small volume surrounding P,and s is the surface area of Av. ■Example1.6 The vectorrthat appear rs in Figure 1.2 occurs very frequently in the super- position type integrals of field theory.This will be demonstrated in Chapter 2.If we define R =r-r',then R=R =r-r',and with r=xax y ay za and r'=x'ax +y'ay +z'a:,it follows that R=[c-x)2+0-y2+(2-z212. Then,we have (R)(R) 哈=-0”-4&-边 R3 R3 R3 哈-是 1 The same operation on the prime coordinates is called V'(1/R),and it should be obvious to the reader that we get 7(1/R)=-V'(1/R). These results will be used in Chapter 2. 1.7 CURL The component in thea direction(general)of the cur of A at a point Pmay be defined as the limit of the circulation of A per unit area as the area(aAs) approaches zero.Thus,we have Refer to Equation(1.30)