The Chemistry Maths Book, Second Edition

(Grace) #1

640 Solutions to exercises







  1. (i)


Section 8.7



  1. 125 50. 6265


Chapter 9


Section 9.1



  1. (i) 0 (ii) 16 (iii) 36

  2. (i) 1 (ii)− 322 (iii) 2


Section 9.3



  1. 4 x,− 2 y 4. 2 x 1 − 13 , 4 y 1 + 12 5. 2 z, 3 z

  2. 2 x 1 cos(x


2

1 − 1 y


2

),− 2 y 1 cos(x


2

1 − 1 y


2

)










9.u


x

1 = 16 x 1 + 12 y


3

,u


y

1 = 12 y 1 + 16 xy


2

,


u


xx

1 = 16 ,u


yx

1 = 1 u


xy

1 = 16 y


2

,u


yy

1 = 121 + 112 xy,


u


yyx

1 = 1 u


yxy

1 = 1 u


xyy

1 = 112 y,u


yyy

1 = 112 x,


u


xyyy

1 = 1 u


yxyy

1 = 1 u


yyxy

1 = 1 u


yyyx

1 = 112












=+−+









z


y


xy xye


xy

cos( ) sin( ) ,




=+++









z


x


xy xye


xy

sin( ) cos( ) ,



∂∂


=− +=



∂∂


22

4


z


xy


xxy


z


yx


cos( )




=− +


2

2

z


y


cos(xy),




=− +


2

2

4


z


x


yxycos( ),




=− +


z


y


2 xxy


2

sin( ),




=− +


z


x


4sin( ),xy x y



∂∂


=



∂∂∂


=



∂∂


=


3

2

33

2

8


z


xy


z


yxy


z


yx



∂∂


=



∂∂∂


=



∂∂


=−


3

2

33

2

6


z


xy


z


xyx


z


yx



∂∂


=− + =



∂∂


22

68


z


xy


xy


z


yx




=−




=


2

2

2

2

26 8


z


x


y


z


y


, x




=− +




=− +


z


x


xxyy


z


y


26 4 3 8xxy


22

,


(cos sin), sin2


22

xxyyxye xe xy


xx

−−


12 π


1


1


2


1


1


2


,(),,( ),±±ii −±− 11 i











  1. (i)


(ii)


(iii)nR 2 (V 1 − 1 nb)


(iv) 2 n


2

a 2 V


3

1 − 1 (p 1 + 12 n


2

a 2 V


2

) 2 (V 1 − 1 nb)


Section 9.4


21.(− 223 , 42 3) 22.(1,2),(− 1 ,2)






24.(− 223 , 42 3): maximum


25.(1,2): minimum; (− 1 ,2) : saddle point



  1. : saddle points; (1,2) : minimum;


(− 1 ,−2): maximum


27.f 1 = 1 1 at (x,y,z) 1 = 1 (1 22 , 123 , 12 6)


28.f 1 = 1 c


6

2 27 atx


2

1 = 1 y


2

1 = 1 z


2

1 = 1 c


2

23



  1. (i)f 1 = 1 4 at (x,y,z) 1 = 1 (1 23 , 223 , 22 3);


f 1 = 1 16 at (x,y,z) 1 = 1 (− 123 ,− 223 ,− 22 3)



  1. (ii)


Section 9.5



  1. 2 x 1 dx 1 + 12 y 1 dy


32.[6x 1 + 1 cos(x 1 − 1 y)]dx 1 − 1 cos(x 1 − 1 y)dy



  1. 3 x


2

y


2

dx 1 + 1 (2x


3

y 1 + 112 y)dy






35.sin 1 θ 1 sin 1 φdr 1 + 1 r 1 cos 1 θ 1 sin 1 φdθ 1



  • 1 r 1 sin 1 θ 1 cos 1 φdφ


































Tpn Tp


V


n


dn


n


V


n


,, ,,


B

A

A

A

B

ddn


B

dV


n


V


T


dT


n


V


p


pn Tn


=

























,,,,


AB AB




dp



++


++


1


22232

()


()


xyz


xdx ydy zdz


λ=± : ,± ,


()


ab 2121212


λ=: ,,−


()


a 12012 ,


(, )03±


( ,±−−), ( , ), ( , )031212


−− − +










()Vnb p


na


V


nab


V


2

2

3

3

2


nR p


na


V


nab


V


−+










2

2

3

3

2











2


2


xyz


yxz




=




=




=


r


x


x


r


r


y


y


r


r


z


z


r


,,



∂∂


=



∂∂


=− +



22

2


z


xy


z


yx


xye


xy

sin( )




=− +



2

2

2


z


y


xye


xy

cos( ) ,




=+



2

2

2


z


x


xye


xy

cos( ) ,

Free download pdf