Phys 2920, Spring 2009
Problem Set #8
1. Go back to Problem 5 on the last set and find the divergence of that vector field, ∇ · a.
Then for the cylindrical volume of that problem, evaluate RV(∇ · a)dV .
Did you get what you expected?
2. (VA 6.55) If Sis any closed surface enclosing a volume Vand A=ax i+by j+cz k,
prove that
Z ZS
A·ndS = (a+b+c)V
3. (VA 6.53) Verify the divergence theorem for A= 2x2yi+−y2j+ 4xz2ktaken over the
region in the first octant bounded by y2+z2= 9 and x= 2.
4. (VA 6.63) Verify Stokes’ theorem for A= (y−z+ 2) i+ (yz + 4) j−xz k, where Sis the
surface of the cube x= 0, y= 0, z= 0, x= 2, y= 2, z= 2 above the xy plane.
5. Check Stokes’ theorem using the function v= 2yi+ 3xjwhere the path is the unit circle
in the xy plane. (This is the same thing as checking “Green’s theorem in a plane” for this
case.
6. Evaluate:
a) R1
0cos x δ(x−π
4)dx
b) R4
0(3x2−2x−1)(δ(x−2) + δ(x−5)) dx
c) RV(5r2−2r·c−7) δ3(r−2k)dV where c= 3i−5kand Vis the sphere of radius 3
centered at the origin.
7. (CV 1.53 g)) Evaluate, in simple x+iy form,
(2 + i)(3 −2i)(1 + 2i)
(1 −i)2
8. (CV 1.54 b, j)) If z1= 1 −i,z2=−2 + 4i,z3=√3−2i, find
(a)
z1+z2+ 1
z1−z2+i
(b) Im {z1z2/z3}
1