I8-19. IfV~=∇φ, thendiv~V =∇^2 φ= 0andcurlV~= curl (∇φ) =~ 0
I8-21. Only flux is from top surface and
∫∫SF~·dS~ = 16π and from divergencetheorem∫∫∫VdivF dV~ = 16πI8-22. (i) Evaluating the left and right-hand sides of the Green’s theorem one finds
the value -16/3. (ii) See page 192I8-23. Both sides of the Stokes theorem yield the valueπ/ 4
I8-24. (i)φ=x^2 y+xy^2 =Cis family of solution curves.
I8-25. Area=π
I8-26.
∫∫SF~·ˆendS=∫∫∫VdivF dV~ = 4πa^3I8-27. On sphere with radiusr= 1,
∫∫SF~·ˆendS=− 4 π/ 3 and on sphere with radiusr= 2one finds∫∫SF~·ˆendS= 32π/ 3 Total flux= 32π/ 3 − 4 π/3 = 28π/ 3I8-28. On inner surface flux is− 2 πand on outer surface flux is 8 π. Zero flux across
top and bottom surfaces. Total flux= 8π− 2 π= 6πI8-29. The divergence of F~ is zero and so the sum of the fluxes associated with
the±ˆenfaces must sum to zero. For example,∫z 1
z 0∫y 1
y 0 y dydz−∫z 1
z 0∫y 1
y 0 y dydz= 0I8-31. 3 V
Solutions Chapter 8