The Chemistry Maths Book, Second Edition

(Grace) #1

Solutions to exercises 641


Section 9.6


37.− 21 − 12 t 2 (1 1 − 1 t


2

)


122

38.−(2 1 sin 1 t 1 + 1 3)e


2cost− 3 t

39.(−a 1 sin 1 t 1 + 1 b 1 cos 1 t 1 + 1 c) 2 (a 1 cos 1 t 1 + 1 b 1 sin 1 t 1 + 1 ct)


40.(3 1 cos 1 θ 1 − 121 sin 1 θ) 2 (2 1 cos 1 θ 1 + 131 sin 1 θ)



  1. (i)(1 1 − 1 sin 1 u) 1 cos(u 1 + 1 v),


(ii)−u(1 1 − 1 sin 1 u) 1 cos(u 1 + 1 v)


42.− 5 x


4

y 2 (x


5

1 − 1 cos 1 y)






44.2(y


2

1 − 1 x


2

) 2 y







  1. (i)


(ii)



  1. (ii)


(iii)


Section 9.7


53.exact 54.not exact 55.exact


Section 9.8



  1. 6423 58. 2 59. 20

  2. 723 62. 31

  3. (i)− 123 , (ii) 0


Section 9.9



  1. 24 65.2(1 1 − 1 e


−R

) 23


Section 9.10



  1. 4023 67.a


5

215



  1. (ii)− 5122


Section 9.11





    1. 315 π 28




71.π 28


1


44


1


π














yx


f


x


nx y


f


y


ny


nn n











=












=


−−

44


1

,


11

x


n

xu


f


y


ny


f


u


f


n











=




















− 1

v


v














yu


f


x


nx


f


u


f


n











=























− 1

v


v
















=+




=+


f


u


abu


f


22 ab


22 22

() (),


v


v




=












=







f


u


a


f


x


b


f


y


f


b


f


x


a


f


y


,


v


xy


xy


xy


cos − siny


















3


Vn


p


T


nR


Vnb


,












=



Chapter 10


Section 10.2


1.(0, 0 ,1) 2.(0, 2 ,0)



  1. 4.(1,π 22 ,0)


5.(1,π 22 ,π 2 2) 6.(3,cos


− 1

(2 2 3),tan


− 1

(2))


7.(9,cos


− 1

(− 82 9),tan


− 1

(−4) 1 + 1 π)


8.(7,cos


− 1

(6 2 7),tan


− 1

(3 2 2) 1 + 1 π)


9.(13,cos


− 1

(− 122 13),tan


− 1

(− 42 3) 1 + 1 π)


Section 10.3


10.r


2

1 sin


2

1 θ 1 cos 12 φ 11.tan


2

1 θ


12.r


2

(3 1 cos


2

1 θ 1 − 1 1) 13. (i) 12 r (ii)−x 2 r


3

Section 10.4



  1. 1 15.a


2

b


3

c


4

224 16. 122



  1. 1 18. 4 πa


5

25 19. 2 πa


2

23



  1. 480 π 22. 23. 6 a


0


  1. 180


Section 10.5



  1. 2 y


3

z


4

1 + 16 x


2

yz


4

1 + 112 x


2

y


3

z


2





27.e


−r

2 r










Section 10.6



  1. (i)x 1 = 1 a 1 cos 1 t,y 1 = 1 −a 1 sin 1 t,z 1 = 1 bt


(ii)ρ 1 = 1 a,φ 1 = 1 −t,z 1 = 1 bt


39.πa


4

216 40.π(1 1 + 1 R)e


−R





Chapter 11


Section 11.2









  1. 8.x


4

1 + 1 ax


2

1 + 1 bx 1 + 1 c
−++

1


9


cos 3 xaxb


−+



e


c


3 x

3


x


c


3

3






π


2


() 1 +



Re


R

xze


r


−r










2

1


4


3


1


9


2


3


2


2

3

−−











r
r

e


r

sin sinθ φ


nn


r


an


r


are


nar

() ()+






















12 1


2

2

1


81


2


π


(,,)−−32 32 1

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