VECTOR ANALYSIS This introductory chapter will begin with a brief review of the vector algebra to which the junior level engineering student has previously been exposed.More complicated operations,such as divergence of a vector,gradient of a scalar,curl of a vector,line integral,flux of a vector,and others are also presented and explained.More meaningful uses for these vector operations will occur in the following chapters when Maxwell's equations are presented and used in more practical applications such as lines,guidcs,and antennas. The three most common coordinate systems,Cartesian,circular cylindrical,and spherical,that are used amost exclusively in this text,are presented along with the information necessary to move from one coordinate system to another.Vector relations and coordinate relations are conveniently summarized on the inside front and back covers. 1.1 VECTOR ADDITION A scalar field may be thought of as an ensemble of point scalars,and temperature is an example that we can all appreciate.A vector field may,in a similar manner, be thought of asan ensemble of point vectors,and the arth's gravitational field is an example that we are all familiar with.On the other hand,a(point)ve is independent of any coordinate system that is being used to describe it,and is therefore unique Two vectors A and B are equal (A =B)if they have the same magnitude direction,and sense.Note that this means that they are not necessarily coincident in space,but one may be translated so that they can become coincident.On the other hand,if two vector fields are equal,then they are coincident.When adding two vectors A and B to produce the sum C.we mean that one of the two vectors may be moved parallel to itself(translated)so that its initial point(tail of the arrow)
2 -1 VECTOR ANALYSIS Flgure 1.1.Vector addition:C A B. coincides with the final point (head of the arrow)of the other.This is illustrated in Figure 1.1,where we see C=A B=B+A (commutative law), (1.1) demonstrating that vector addition is commutative.If we add another vector D to C=A B,obtaining a new vector E,then we get E=D C=D +(A B) or E=D+(A B)=(D+A)+B (associative law). (1.2) Thus.vector addition is also associative. The subtraction of vector r'from vector r to produce vector R means we have R=r-r'=r+(-r') (1.3) This is demonstrated in Figure 1.2. 1.2 VECTOR MULTIPLICATION The results of Section 1.1 indicate that we have A+A-2A and B+B+B=3B. More generally,if a is a scalar,then aA is a vector in the direction of A if a>0 or in the direction of-A if<.The magnitude (length)of the vector aA is times the magnitude of A.and this is symbolized by aA=aA=aA (1.4) R=r-r' Figure 1.2.Vector subtraction:R r-r
1.2 VECTOR MULTIPLICATION multiplication is to be carried out must be stated. The scalar or dot ()product of two vectors is defined to be a scalar such that A B A B cos [A,B](scalar product), (1.5) where [A,B]is the angle between A and B measured from A to B.Since cos [A,B]cos [B,A],it follows that BA BA cos [B.A]AB|cos [A,B], B·A=A·B(commutative law), (1.6) and the scalar product is commutative.It is also distributive: A·(B+C)=A·B+A·C(distributive law). (1.7 The scalar product is utilized very nicely in the familiar example of finding the differential work (dW)done in moving an object a distance dwith a force (F) applied at an angle [F,dl] dw =F.dl Flldl cos [F,di] (1.8) Note that Fcos [F,dl]is the magnitude of the component of F in the direction of dl. The vector or cross(x)product of two vectors is defined as the vector given A x B an AB sin [A,B](vector product), (1.9) where an is a unir vector (unit magnitude)normal to the plane containing A and B in the right-handed sense.That is,if the fingers of the right hand are rotated from A to B through the smaller of the two possible angles [A,B],then the thumb points in the direction of an and in the direction of A x B.Since B×A=a BA sin[B,A] and since a=-an from the preceding sentence,we have B×A=-A×B, (1.10) and vector multiplication is not commutative.It is distributive: A×(B+C)=A×B+A×C(distributive law). (1.11) As an example of the vector product,consider a cone that is spinning about its axis with angular velocity whose direction is indicated in Figure 1.3.For this
g 1 VECTOR ANALYSIS Flgure 1.3.Vector product:u =w x r. configuration,the velocity u of a point on the base of the cone at the periphery (tip of r)is given by U=Xr (1.12) and u is normal to both w and r. The scalar triple product is A·(B×C)=C·(A×B)=B·(C×A) (1.13) and is commutative.The reader can easily show how the scalar triple product can be used to give the volume of a parallelepiped.Note that the parentheses in the scalar triple product are not necessary,but in the vector triple product A×(B×C), (1.14) the parentheses are necessary since we have Ax(BxC)(Ax B)xC in general.There are many vector identities that involve vector addition and multiplication.Vector relations are summarized on the inside front and back cov er The division of one vector by another is permissible if the vectors are parallel. 1.3 COORDINATE SYSTEMS It is absolutely necessary that the reader be familiar with the coordinate systems we will be using in this textbook.They are the Cartesian(rectangular),circular cylindrical,and spherical coordinate systems.Figure 1.4shows apoint Pescribed by the three coordinates associated with these systems. Also shown are coordinate transformations between the systems.These are easy to derive with Figure 1.4 and simple trigonometry. The point P is also the locus of the intersection of the three e orthogonal surface on which each of the coordinates is constant.This is shown in Figure 1.5.Note that
1.3 COORDINATE SYSTEMS t=pcos=rsin8cas中 e=√2+y2=rsin=√2+2+z2√02+2 r=p sin =r sin 0 sin 。=Pcos8 =tan1(yx) 0=tan-1()/z=tan-I (plz) P(x.y.2) ·P(p,z) AP,8,) 0<x<∞ 0<0<a 0≤y<∞ <y< 0≤中≤2 0≤8≤T (a) (6) 6 Flgure 1.4.General point P desc coordinates;(b)P(p,中，z)in dinates plane) (plane) a) (b) fe) Flgure 1.5.Orthogonal surfaces for (a)Cartesian.(b)cylindrical,and (c)spherical coordinates. in Cartesian coordinates=constant is the same infinite plane as in cylindricall coordinates,and in cylindrical coordinates=constant is the same half-plane as in spherical coordinates. Figure 1.6 shows the three orthogonal unit vectors for each system.These systems are all right handed.That is,if the fingers of the right hand are rotated from one unit vector toward the next unit vector in the correct order (a to ay aly to a top toto to pi to to) then the correct direction for the third unit vector is obtained from the direction of the thumb.This is also demonstrated by the vector products listed in Figure 1.6.Note that each unit vector is normal to its respective coordinate surface,and the direction of the unit vector is that for which its coordinate is increasing.It When we refer to cylindrical coordinates in what There are other types of cylindrical ircular eyindnc coor