Begin2.DVI

I9-13. (b)~V = [−(x−y)z−yz 0 +x 0 z 0 ]ˆe 2

I9-14. φ=−mgz, W=

∫h 2

h 1

F~·d~r=

∫h 2

h 1

−mgdz=−mg(h 2 −h 1 ) =φ(h 2 )−φ(h 1 )

I9-16. Ifφ=x^2 −y^2 , thengradφ=F~= 2xˆe 1 − 2 yˆe 2 with field linesxy=c

I9-18. Show
φ=y^2 sinx+xz^3 +f 1 (y,z)
φ=y^2 sinx− 4 y+f 2 (x,z)
φ=zx^3 +f 3 (x,y)
Selectf 1 ,f 2 ,f 3 such that all three integrations are the same to obtainφ=y^2 sinx+
xz^3 − 4 y

I9-19. φ=x^2 yz+xy

I9-20. π(2−

√

3)

I9-21. 3 V

I9-22. Use the results∇·(φC~) =∇φ·C~=C~·∇φ and φC~·ˆen=C~·(φˆen)to show

C~·





∫∫∫

V

∇φdV−

∫∫

S

φˆendS



= 0

and sinceC~ is arbitrary the term in the brackets must equal zero.

I9-23. (a)F~=∇φso thatdivF~==∇·(∇φ) =∇^2 φ= 0

(b) Use the Gauss divergence theorem to show

∫∫∫

VdivF dV~ = 0 =

∫∫

S∇φ·ˆendS=

∫∫

S

∂φ

∂ndS

Solutions Chapter 9