Begin2.DVI

I8-32. If origin outside ofS, then use the divergence theorem and show

∫∫

S

ˆen·~r

r^3 dS=

∫∫∫

V

∇·

(

~r

r^3

)

dV

and then show∇·

(

~r

r^3

)

= 0

If origin is inside ofS, then place a sphere of radiusabout the origin and show

∫∫

S

ˆen·~r

r^3 dS+

∫∫

S

ˆen· ~r

r^3 dS=

∫∫∫

V

∇·

~r

r^3 dV= 0

On sphere of radiusshow that

∫∫

S

ˆen·

~r

r^3

dS=− 4 π

I8-33. 5 x^2 yz^3 + 3xy^2 z^2

I8-35. Area=^12 (base) (height)

I8-36. − 162 π

I8-37. I= 216

I8-38. 4

Problems 8-40 to 8-49See equations (8.74) to (8.82)

I8-50. (c) If~r=~r(u,v,w), thend~r=∂u∂~rdu+∂~r∂vdv+∂w∂~rdwis the diagonal of a volume

element in the shape of a parallelepiped having sides A~ = ∂~r∂udu, B~ = ∂~r∂vdv and

C~=∂w∂~rdw. The volume of this elemental parallelepiped is

dV =|A~·(B~×C~)|=|

∂~r

∂u·(

∂~r

∂v×

∂~r

∂w)|dudvdw

Use vector identity and orthogonality property∂~r∂v·∂w∂~r = 0to show

|

∂~r

∂u·(

∂~r

∂v×

∂~r

∂w)|=huhvhw

I8-51. Calculategradv×gradwand then make use of the fact that mixed partial

derivatives are equal to showdiv (gradv×gradw) = 0

Solutions Chapter 8