171 PROBLEMS 39 PROBLEMS origin Laws ofVector 3-1: Basic Section Find an expression for the unit vector directed the line described from an arbitrary point on toward the by x = z=-3. Algebra and directed toward the Find an expression for the unit vector axis at a heighth above the x-y plane point P located on the z z (x, y, -5) in the plane from an arbitrary point 3.10 Vector A starts at point (1, -1,-3) and ends at point Find a unit vector in the direction *3.1 1.0). Given = of A. vectors =*2 -y3 +2, B =$2-ý+23, A 4+ý2-22, show that C is perpendicular = to both A Find a unit vector 3.11 and = the area a4 Given A B, and (a) Find B,if A Find a relation b) to .5 C =$B,+2 +2B: and B is parallel to expressions: if A is Line A 2-24, =*+y2-23, B =$2 -Ý4, and find the following: The component of (c) BAc (d) AxC (Bx Given C) vectors =$2 -ý+23 and B A C Given vector parallel to unit 3.8 By (a) expansion The A starts that A is orthogonal is 9 $3 -22, and whose (a) Show The vector relation A at point P in Cartesian for the = (1, C find relation for the vector Eq.(3.33). Answer(s) between the lines by +2y 4. triple available in Appendix D. triple the and ends at point P on the line such Find an expression for A direction is line. two vectors is given C= â(B where â is given by (b) The product given D defined to A is in the a)=A(B A) A2 the unit vector of vector component of B by -1,2). product A and B, defined as the vector A direction of coordinates, prove: scalar to given that, perpendicular The described at the origin Eq. (3.29). (b) the smaller angle find is A =$(x+2y)-(y+3z)+2(3x-y), determine 3.7 a given line Vector 3.14 whose magnitude to both A and B. perpendicular a vector to C) (Ax ý)- 3.6 A 3.13 B along C $xB (h) vector algebra x Ax (Bx (g) 2 at their intersection point. (a) A andâ (b) Use = -6. = 8. x +2y 3x +4y 1 perpendicular Line vectors the following plane are described by x-y lines in the B. Given (e) A Two 3.12 B. between B, and Bz line +z =4. 2x = =å2-y3 +l either direction of the to parallel described by and B. In Cartesian coordinates, the three corners of a triangle (4, -4,4),and P3 (0,.4, 4), P2 (2,2, -4). Find are Pi of the triangle. #a 3 =-5. A as the vector givenby by D =B A(B A) A2 component of B A certain plane is Coordinate Sections 3-2 and 3-3: described by +42= 16. 2x +3y Find the unit vector normal to the surface from the origin. in the direction away B =(z-3y) +ý(2x -3z) -(x to B at point P parallel =(1, 0, -1). unit vector *3.17 Find direction is A 3.18 G a vector whose magnitude perpendicular =*+ý2-2 and F to both A at vectors =ý3-6. is E described is Cartesian y), find a (b) P= d) P= (-2,2,-2) (1, 1,3) (c) 4 and whose F, where the equation: =x-1. is orthogonal the to at point P2 on the Find an expression line. for A. Vector E E field is given by Determine the component of E surface R =2 P point at 3.20 When sketching a vector we of to (a) Pi (2, 7/4, -3) (b) = P2 = (3,0, c) P = (4,7, 5) 3.24 (a) =R 5R cos 0sin +$3sin d. cos the coordinates of the following points Cartesian coordinates: Convert -2) the coordinates of the following points from coordinates: cylindrical Convert spherical 3.19 to P=(0,0,2) 3.23 P = (0, 2) and ends at point starts which A by the coordinates of the following noi lowing points from and spherical coordinates: cylindrical (a) P=1,2,0) + and Systems Convert cylindrical given line y line, 3.22 Given 3.16 Vector VECTO ANALYSI 3.15 E 3 CHAPTER 172 to = = P= P (5, 0, 0) 7) (b) Pa (5,0, (c) (3, 7/2, 0) tangential to the spherical =(2,30°,60°). or demonstrating often use arrows, as the spatial variation Fig. P3.20, wherein the length of the arrow is made to be proportional to the strength of the field and the direction of the arrow is the same as that of the field's. the vector away from field, The sketch field E shown =fr, the origin in Fig. P3.20, which represents in consists of arrows and pointing radially their lengths increasing linearly in proportion to their distance away from the origin. Using this arrow representation, sketch each of the following vector fields: (a) Ej=-iy (b) E2 (c) (d) (e) E3= åx +9y =ix +y2y Es =dr E4 )E 3.21 =r =fsin o Use arrowsto Ej =îx ýy E2 = -¢ Es =(1/x) (d) E4 =fcos o sketch each of the following vector fields: (a) (b) (c) Figure P3.20 Arrow (Problem 3.20). representation for vector field E =tr 173 PROBLEMS 25 the TIse appropriate area ds a)r=3; the 3.30 differential ,9 S T/2 ST; =T/4; -2 c)2<r<, =f(cos d +3z) -¢(2r +4sin o) + (r-2) B=-fsino + cos o A z ==0 find <zs2 0 d) R 2:0s0sT/3; <dsT = 0 5; T/3; 0 <o < 27 e) 0 R Alsosketch 3.27 A (6) A volumes described o 05090, of a sphere 30 described is < 90°. P (c) by The enclosed volume. Also 3.28 A field vector is 0<R<2, points: = (2,7, 3) (b) described (a) (b) 3.29 point The vector component Transform by The vector component of At spherical a given point coordinates in into P the surface of the cylinder P, find: of E E = to (b) given spherical coordi- = Po T) (4, 7/2, the vector =R e cos sin? cylindrical (2, T/2, = d +8cos d and coordinates sin then A å(x +y) at Pi =(1,2,3) Ps=(1, -1,2) in =R4 +02-. D =R (e) E at P2 sin 6 +0cos 6 + coso at (a) (C) in the direction B the direction of A. The vector component of B perpendicular to A. of in A B (c) C (b) component =Rcos +0sin o + sin 0 at Ps = (3, 7/2, 7) Transform the following vectors into spherical nates and then evaluate them at the indicated points: Find: (D) The vector at = (1,0,2) 3.35 B=-2 +$3. of B it 7/2). B =(y -x) +ýx -y) (d) The scalar component, or projection, of A. o evaluate P=(2,7/2,7/4) a) pairs of c)C=y/(x +y*)- ýx-/«* +y)+24 tangential to the cylinder. A and B are in Transform the following vectors into cylindrical coordinates and then evaluate them at the indicated points: the cylinder. by A cylindrical 3.34 (a) perpendicular space, vectors in and P2 (0, 2,3) (2,7/3, 1) and P4= (4,7/2,3) (1, 1,2) P= 3.33 given in cylindrical coordinates on 4) (3, () Ps = (3, 7, 7/2) and section. is located byr =2. At 0) and = (a) Pi E =fr coso +¢rsin o +z' P Cartesian Determine the distance between the following A Point (3, P4 (4, nates. 3.32 Find thefollowing: (b) of the in 3) (1, /3,1) (2, T coordinates. The surface area of the spherical section. the outline B at and the following pairs of points: Find the distance between Ps (a) sketch A both to vector perpendicular unit coordinates. by the following: the outline of each volume. section and at (2, 7/2,0) = 2, and P2 = (-2, -3,-2) = 7/4, 6) =(1,7/4,3) and P = 7,0) Ps = 7/2, ST; 0<z<2 (a)2srs5; 5; 0 s6 ST/3; 0<o < 27 b) 0s R sketch eAB (a) T/2 Also (a) 3.31 the outline of each surface. Find the 1 26 Given vectors 0O S 7/3; -2szs 2 2r S b) for deter urtace Surtaces: expression termine the area of each of the following to y +ýxz +24 + y+z)- = fcos o-¢ sin Ps= (2,T/4,2) (d) at Pi =(1,-1,2) 2(+ y) o+ cos at osin d P =(-1,0,2) at D =y/(r2+ y*)-a/a+y)+24 P4=(1, -1,2) coordi- at CHAPTER3 VECTORANALY NSS 174 Sections 3-4 to 3-7: Gradient, Divergence, and Curl Operators 3.36 Pi=(0,3) Find the gradient of the following scalar functions: (a) T (b) V (c) U 3/(x +z) =xy24 zcoso/(1 +r2) = P2=3,0) W =eRsin e S =4xe +y (d) *(e) ) N g) Figure P3.41 For each of the analytical solution for scalar following VT and generate a obtain an fields, corresponding arrow 3.42 For = 10+x, -10 < x (b)T =x,for -10 <x 10 T for its 10 evaluate T =100+xy,for -10<x< 10 d) T xy, for -10<x, y < 10 (e) T 20+x +y, for -10 <x,y < 10 (c) 1+sin(7x/3),for -10<x< T , (g) T 1+cos(Tx/3), T (h) 3.43 = rcos 15 )T15+r cos x -10 < for for evaluate 10 3.44 =0, 10 atz 3.39 Follow find its A For the directional =( -ýz) 3.41 0Srs 0< 27. segment P = the range =(5, 7/4,7/2). sin s 6, determine it direction V A in the R and then vector fields is displayed in form of analytically and a vector representation. then compare the result expectations on the basis of the displayed scalar is given 7T -îcos x A = sin y +ý sinx cos Determine with your pattern. y, for by =ie z t similar to that leading given by function derivative along and then evaluate Evaluate U T(z). a procedure to derive the expression coordinates. 3.40 P along dete f and then function scalar , direction =(2, 7/4,3). Each of the following Fig. P3.44 10 The gradient of a scalar function = T at =zer/ cos radial d,for0 VT If P derivative it T 10 (a) 3.38 at For the directional function derivative along the directional it scalar the representation. (f) 3.41. M =R cosesino 3.37 (a) Problem =r cos it Eq. (3.83) for to V =xy -z, V the direction Eq. (3.82) in spherical determine of = -1,4). of E =*x -ýy at the line integral to Pa of the circular path P vector (1, along the shown in Fig. P3.41. Figure P3.44(a) -7 <X, y <T 175 PROBLEMS y b)A=Ns +ycos 2x,for -7x,y<T Figure P3.44(b) (c) -$xy +ýy5, for -10 < x,y < 10 (e) A =r,for -10 <r< ) A 10 10 Figure P3.44(e) =åxy>, for -10< x, y 10 -10 -10 Figure P3.44(c) d)A=-$cOs x+ý sin y, for -7 X, y Figure P3.44() ST ()A =Rxy2 +ýr*y, for -10<x, y < T0 Figure P3,44(d) Figure P3.44(g) 10 CHAPTER 3 176 VECTOR ANALYSIS h) A i sin (0)+$sin().for -10 x,y 10 field Vector 3.45 E is characterized by the ne ;(b) the magnitu followin points along properties: de of the distance fromthe function of only origin; (c) =12, everywhere. Find Find an vanishesa the origin; and (d) E that satisfies these properties. expression (a) E for For the vector 3.46 divergence (a) The each and (b) The A =fr+or cos d,for rs2 10 r V E vector field units over the cube's volume. = rl0e E field -23z, veri the cylindrical region enclosed for =2, z =0, and z =4 3.48 A the by D =ir° exists the region between in = two r surfaces defined by andr=2.wi and z both cylinders extending between z 5. Verifv th he the following: divergence theorem by evaluating concentric cylindrical =0 1 Figure P3.44(h) of theorem divergence verify ofa 2 the vector For 3.47 -ýyz-2rv flowing through the sure the origin and with sides equal to the Cartesian axes. at parallel integral = Xrz flux outward total cube centered E field by computing theorem Ei E VE = (a)D-ds (b)VD dv For D =R3R-, the vector field theorem the divergence 3.50 For the vector field (a) 10 A =fr?+or2 sind,for0Srs 27. o< E dl around = R sphericalshells defined by Figure P3.44(i) E the and R triangular sides of between the 2. =*ry-ý(x*+2y), contour calculate shown Fig. P3.50(a). (a) Figure P3.50 Problem 3.51. Figure P3.44) evaluate both for the region enclosed I 3.49 (b) Contours for (a) Problem 3.50 and (b) in 177 PROBLEMs TxE) ds over the area ofthe triangle. 3.50 for the Problem Prob contour Repeat shown in (0,3) .51 sob) eorem for the Stokes's vector Verify B (Tr cos field Li o+d sin o) -3,0) L3 the following: by evaluating the over B-dl semicircular contour shown Figure P3.55 in Problem 3.55. P3.52(a). Fig. 3.56 xB)-ds overthe surface ofthe semicircle. Determine if each of the following vector fields solenoidal, conservative, or both: (a) A =ir2-ý2xy i2-ýy2 +ΕΎ2 C= (Sin)/r2 +d (cos o)/r2 B (b) (c) D (d) R/R (e)E-(3-)+z ()F=Ry +ýx)/(r2 +y) G (g) (a) r2+z)-9y2 +x)-(y H=R(Re-R) (h) (b) Find the Laplacian of the following scalar functions: 3.57 P3.52 Fieure Problem 53 Fig. Contour paths for 3.52 for Problem V 4xy2z3 (b) V xy yz (c) V 3/(x2 (a) the contour shown (d) = + +zX +y) V = cos p 5e (e) V in =10e-Rsin P3.52(b). Stokes's Verify A=RCOS+¢ sin6 of unit for theorem by it evaluating the on vector field the hemisphere theorem Stokes's Verify + Bde circle, 3.58 Find the Laplacian of the following scalar functions: (a) Vi (b) Va radius B= (f cos a) Problem 3.52 and (b) 3.53 Repeat 3.54 3.55 (a) as (x sin o)by for Fig. P3.55, B).ds over vector field evaluating: over the path comprising shown in the +) a quarter section of a and the surface of the quarter section. = 2 =(2/R>)cos 10r3 sin sin o is