CHAPTER 1 Poblems Pob. 1.1 Pob. 1.2 () Given P(1,-4,-3), convet to cylindicl nd spheicl vlues; 4
x y = + = + = = 1 ( 4) 17 4.123. 1 y 1 4 = tn = tn = 284.04. x 1 P(,, ) = (4.123, 284.04, 3). Spheicl : x y 1 16 9 5.099. 2 = + + = + + = 1 1 4.123 = tn = tn = 126.04. 3 P (,, ) = P(5.099, 126.04, 284.04 ). (b) (c ) 1 y 1 0 = 3, = tn = tn = 0 x 3 o Q(,, ) = Q(3,0,5) o 1 1 3 = 9 + 0 + 25 = 5.831, = tn = tn = 30.96 5 o o Q (,, ) = Q(5.831,30.96,0 ) = 4 + 36 = 6.325, = tn = 108.4 2 o R(,, ) = R(6.325,108.4,0) 1 6 1 1 6.325 = = 6.325, = tn = tn = 90 0 o o R (,, ) = R(6.325,90,108.4 ) o o o Pob. 1.3 5
6
Pob. 1.4 () x = cos, y = sin, 2 V = cos sincos + sin (b) U = x + y + + y + 2 2 = + sin sin + 2 cos = [1+ sin sin + 2 cos ] Pob. 1.5 7
8
9
10
11
Pob. 1.6 () x + F cos sin 0 y F sin cos 0 = + F 0 0 1 4 + F F F 1 = + = + + [ cos sin ] ; 1 = [ cossin+ cossin ] = 0; + = 4 + 1 F = ( + 4 ). + ; In Spheicl: x sin cos sin sin cos F y F coscos cossin sin = F sin cos 0 4 4 2 4 F = sin cos + sin sin + cos = sin + cos ; 4 4 F = sin cos cos + sincos sin sin = sincos sin ; F = sincos sin + sinsincos = 0; 2 4 4 F = (sin + cos ) + sin (cos ). 12
(b) 2 x + G cos sin 0 2 y G = sin cos 0 + G 0 0 1 2 + G G G 2 3 = [ cos + sin ] = ; + + = 0; = 2 + 2 G = ( + ). + ; Spheicl : Pob. 1.7 2 G = ( xx + yy + ) = = sin sin 13
14
15
Pob. 1.8 () D cos sin 0 0 D = sin cos 0 x + D 0 0 1 0 D = ( x + ) sin = ( cos + ) sin D = ( x+ ) cos = ( cos + ) cos D = ( cos + )[sin + cos ] Spheicl : Since D = D = 0, we my leve out the fist nd x thid column of the tnsfomtion mtix. Thus, D... sinsin... 0 D... cossin... x = + D... cos... 0 D = ( x+ )sinsin = (sincos+ cos )sinsin. D = ( x+ )cossin = (sincos + cos )cossin. D = ( x+ )cos = (sincos+ cos ) cos. D= (sincos+ cos )[sinsin + cossin + cos ]. (b) Cylindicl: E cos sin 0 y x E sin cos 0 = xy E 0 0 1 x E = y x + xy ( )cos sin = (sin cos ) cos + cos sin 2 = cos 2 cos + sin cos. 2 E = y x + xy ( )sin cos 2 = cos 2 sin + sin cos. 2 E = x = cos. 2 E = cos ( sin cos 2 ) + sin ( cos + cos 2 ) + ( cos ). 16
In spheicl: E E = y x + xy + x E sincos sinsin cos y x E coscos cossin sin = xy E sin cos 0 x ( )sincos sinsin ( )cos ; but x= sincos, y = sinsin, = cos ; 2 3 = sin (sin cos + (sin cos cos ) cos ; 3 3 2 )cos + sin cos sin cos E = ( y x )coscos + xy cossin ( x )sin ; 3 2 = sin cos2coscos + sin cos sin cos (sin cos 2 cos 2 ) sin ; E = ( x y )sin + xy cos 3 = sin cos2sin + sin cos sin cos ; 2 3 3 3 2 [ sin cos 2 cos sin cos sin cos (sin cos cos ) cos ] E = + + + 3 2 [ sin cos 2coscos + sin cos sin cos sin (sin cos cos )] 3 [ sin cos 2 sin sin cos sin cos ] + + + Pob. 1.9 () H cos sin 0 3 H sin cos 0 2 = H 0 0 1 4 H = 3cos + 2sin, H = 3sin + 2cos, = 4 H H = (3cos + 2sin ) + ( 3sin + 2cos ) 4 o (b) At P, = 2, = 60, = 1 H = (3cos60 + 2sin60 ) + ( 3sin60 + 2cos60 ) 4 o o o o 3.232 1.598 4 = 17
Pob. 1.10 () 18
19
Pob. 1.11 () = 2 x + y + = +. 1 = tn ; =. o = x + y = sin cos + sin sin. = sin ; = cos ; =. (b) Fom the figues below, - cos sin sin ( ) = sin + cos ; = cos sin ; = Hence, sin 0 cos = cos 0 sin 0 1 0 Fom the figues below, = cos + sin ; = cos sin ; =.. 20
sin ( ) cos cos sin sin cos cos sin = 0 0 0 1 0 Pob. 1.12 21
Pob. 1.13 Using eq. (1.32), d = 2 + 1 212cos( 2 1) + ( 2 1) o o 2 = 10 + 5 2(5)(10)cos(60 30 ) + ( 4 2) = 74.4 d = 74.4 = 8.625 Pob. 1.14 2 () d = ( 6 2) + ( 1 1) + ( 2 5) = 29 = 5. 385 (b) d = 3 + 5 2()()cos 3 5 π + ( 1 5) = 100 d = 100 = 10 22
(c) d π π π π π π = 10 + 5 2(10)(5)cos cos 2(10)(5)sin sin cos(7 ) 4 6 4 6 4 4 π π π π o = 125 100(cos cos sin sin ) = 125 100 cos75 = 99.12 4 6 4 6 2 3 d = 99.12 = 9.956. Pob. 1.15 A x Ay A cos sin 0 A = sin cos 0 A 0 0 1 A x y 0 x + y x + y y x = 0 x + y x + y 0 0 1 A A A Ax sincos coscos sin A = Ay sin sin cos sin cos A A cos sin 0 A x x y x + y + x + y x + y + x + y y y x = x + y + x + y x + y + x + y x + y 0 x + y + x + y + A A A 23
Pob. 1.16 24
Pob. 1.17 () A B = (5,2, 1) (1, 3,4) = 5 (b) (c ) 5 2 1 AxB= = 5 21 17 1 3 4 A B 5 cosab = = = 0.179 AB = 100.31 AB 25 + 4 + 1 1+ 9 + 16 o 25
(d) n AB x (5, 21, 17) (5, 21, 17) = = = AB x 2 5 + 21 + 17 755 = 0.182 0.7643 0.6187 ( A BB ) 5B (e) AB = ( A B) B = = = 0.1923 + 0.5769 0.7692 ` 2 B 26 Pob. 1.18 26
Pob. 1.19 () At Q, = 10, = π / 2, = π / 3 x = sin cos = 10sin π / 2cos π /3 = 10(1)(1/ 2) = 5 y = sinsin = 10sin π / 2 sin π / 3 = 10(1)( 3 / 2) = 8.66 = cos = 10 cos π / 2 = 0 Q(x,y,) = Q(5,8.66,0) (b) At P, = 9+ 16+ 4 = 5.385 2 o cos = = = 0.3714 = 68.2 5.385 cos = x 3 o = = 0.6 = 306.87 x + y 5 o o P (,, ) = P (5.385,68.2,306.87 ) (c ) d = ( x x ) + ( y y ) + ( y y ) = 2 + 12.66 + 2 = 168.2756 2 p Q p Q p Q d = 12.97 Pob. 1.20 () An infinite line pllel to the -xis. (b) Point (2,-1,10). (c) A cicle of dius sin = 5, i.e. the intesection of cone nd sphee. (d) An infinite line pllel to the -xis. (e) (f) A semi-infinite line pllel to the x-y plne. A semi-cicle of dius 5 in the y- plne Pob. 1.21 27
Pob. 1.22 3 ( ) At T, x = 3, y = 4, = 1, = 5,cos = 5 3 A= 0 5(1)( ) + 25(1) 5 = 3 + 25 5 1 = 26, sin = cos = 26, 26 3 5 B = 26( ) + 2( 26) 5 26 = 15.6 + 10 28
( b) In cylindicl coodintes, B B B sin cos 0 15.6 B = 0 0 1 0 B cos sin 0 10 5 = 15.6 sin = 15.6( ) = 15.3 26 = 10, B = 15.6 cos = 3.059 B(,, ) = ( 15.3,10, 3.059) AB = ( A B) B = ( A B) B 1 (30 76.485)( 15.3,10, 3.059) = 2 343.45 B = 2.071 1.354 + 0.4141. () c In spheicl coodintes, A sin 0 cos 0 A = cos 0 sin 3 A 0 1 0 25 25 A = 25cos = = 4.903 26 5 A = 25 sin = 25( ) = 24.51 26 A = 3 A B = 4.903 24.51 3 = 245.1 95.83 382.43 15.6 0 10 ± A B AxB = = ± + (0.528 0.2064 0.8238. 464.23 29
Pob. 1.23 At P( 02,, 5), = 90 ; B B B x y cos sin 0 = sin cos 0 0 0 1 0 1 0 5 = 1 0 0 1 0 0 1 3 B = 5 3 x y B B B ( ) A+ B = ( 2410,, ) + ( 1, 5, 3) = + 7 x y A B 52 ( b) cosab = = AB 4200 1 52 AB = cos ( ) = 143.36. 4200 A B 52 () c AB = A B = = = 8. 789. B 35. 30
Pob. 1.24 cossin 2 G= cos x + y+ (1 cos ) sin = cos + 2 cot sin + sin x y 2 G sin cos sin sin cos cos = G cos cos cos sin sin 2 cot sin 2 G sin cos 0 sin 3 G = sincos + 2cossin + cossin G G 3 2 = sincos + 3cos sin 3 = coscos + 2cotcossin sinsin 2 = sincos + 2cotsincos 3 2 G = [sin cos + 3 cos sin ] 3 + [ cos cos + 2 cot cos sin sin sin ] + sincos (2cot cos ) 31