Begin2.DVI

I6-50. If dtd(B~×C~) =B~×d
C~
dt +

dB~

dt ×

C~, then

d

dt

[

A~×(B~×C~)

]

=A~×

d

dt(

B~×C~) +dA~

dt×(

B~×C~)

=A~×

[

B~×dC~

dt +

dB~

dt ×

C~

]

+

dA~

dt×(

B~×C~)

=A~×(B~×

dC~

dt

) +A~×(

dB~

dt

×C~) +

dA~

dt

×(B~×C~)

I6-51. The curves~r=~r(r 0 ,θ) =r 0 cosθˆe 1 +r 0 sinθˆe 2 are coordinate curves which are

circles of radiusr 0. The curves~r=~r(r,θ 0 ) =rcosθ 0 eˆ 1 +rsinθ 0 ˆe 2 are coordinate curves

which are the raysθ=θ 0 =a constant

∂~r

∂r

= cosθ+ sinθˆe 2 =ˆer

∂~r

∂θ

=−rsinθˆe 1 +rcosθˆe 2 =rˆeθ

I6-52.

∫

C

F~×d~r=

∫(2,6)

(1,3)

ˆe 1 (y−x)dz−ˆe 2 xydz+ˆe 3 (xy dy−(y−x)dx)

On the liney= 3x,z= 0, dz= 0, dy= 3dx

so that

∫

C

F~×d~r=

∫ 2

1

[x(3x) 3dx−(3x−x)dx]ˆe 3 = 18ˆe 3

I6-53.

∫

C

F~·d~r=

∫

C

[(xy+ 1)dx+ (x+z+ 1)dy+ (z+ 1)dz]

∫

C

F~·d~r=

∫

0 A

F~·d~r+

∫

AB

F~·d~r+

∫

BC

F~·d~r=I 1 +I 2 +I 3

On 0A,y= 0,dy= 0,z= 0,dz= 0andI 1 =

∫ 1

0 dx= 1

On AB,x= 1,dx= 0,z= 0,dz= 0andI 2 =

∫ 1

0

2 dy= 2

On BC,x= 1,dx= 0,y= 1,dy= 0andI 3 =

∫ 1

0

(z+ 1)dz= 3/ 2

Therefore

∫

C

F~·d~r= 9/ 2

I6-55.

∫

C

F~·d~r=

∫

0 A

F~·d~r+

∫

AB

F~·d~r=I 1 +I 2

On 0Ay=x,dy=dx,z= 0,dz= 0, 0 ≤x≤ 1 ,I 1 =

∫ 1

0

(x+ 2x^2 )dx= 7/ 6

On ABx= 1,y= 1,dx=dy= 0, 0 ≤z≤ 2 I 2 =

∫ 2

0 dz= 2

Therefore,

∫

C

F~·d~r= 19/ 6

Solutions Chapter 6