Begin2.DVI

I7-34. (a)x,y-plane, coordinate curves~r(u 0 ,v)vertical lines,~r(u,v 0 )horizontal lines

(b) x,y-plane polar coordinates,~r(u 0 ,v) circles of radius u 0 ,~r(u,v 0 ) rays at

anglev 0.

I7-35. I= 4

∫ 1

0

∫ 1 −x

0

dydx= 2

I7-36.

I=

∫∫

S

F~·ˆendS=

[∫∫

OCDG

+

∫∫

GF A 0

+

∫∫

F ABE

+

∫∫

BEDC

+

∫∫

EDGF

+

∫∫

ABC 0

]

F~·ˆendS

On face 0CDG ˆen=−eˆ 1 , F~·ˆen=−x^2

x=0

= 0, dS=dydz

On face GFA0 ˆen=−eˆ 2 , F~·ˆen=−y^2

y=0

= 0, dS=dxdz

On face FABE eˆn=ˆe 1 , F~·eˆn=x^2

x=1

= 1, dS=dydz

On face BEDC, ˆen=ˆe 2 , F~·ˆen=y^2

y=1

= 1, dS=dxdz

On face EDGF, ˆen=ˆe 3 , F~·ˆen=z^2

z=1

= 1, dS=dxdy

On face ABC0, ˆen=−ˆe 3 , F~·ˆen=−z^2

z=0

= 0, dS=dxdy

Add the above surface integrals over each face and showI=

∫∫

S

F~·ˆendS= 1+1+1 = 3

I7-37.
φ=x^2 +y^2 −1 = 0, 0 ≤z≤ 3 , N~ = gradφ= 2xˆe 1 + 2yeˆ 2

eˆn=xˆe 1 +yeˆ 2 sincex^2 +y^2 = 1, dS= dxdz

|ˆen·ˆe 2 |

=dxdz

y

I=

∫∫

S

f(x,y,z)dS=

∫ 3

0

∫ 1

0

2(x+ 1)ydxdzy = 9

I7-38. Method I Use cartesian coordinates

ˆen=x

ˆe 1 +yˆe 2 +zˆe 3

3

dS

=

dxdy

|ˆen·eˆ 3 |

= 3dxdy

z

F~·ˆen=x(x+z) +y(y+z)−(x+y)z

3

=x

(^2) +y 2
3
I=
∫∫
S
F~·ˆendS=
∫ 3
− 3
∫+√ 9 −x 2
y=−√ 9 −x^2
√x^2 +y^2
9 −x^2 −y^2
dydx= 36π
Solutions Chapter 7