404 CHAPTER 22. CALCULUS OF VECTOR FIELDS

(a) 2xy

(b) x2− y2

(c) sinxcoshy

(d) ln(x2 + y2

)(e) 1/

√x2 + y2 + z2

10. Verify the formula given in (22.1) is a vector potential for g assuming that divg = 0.

11. Show that if ∇2uk = 0 for each k = 1,2, · · · ,m, and ck is a constant, then

∇2

(m

∑k=1

ckuk

)= 0

also.

12. In Theorem 22.1.5, why is ∇2(

ε |x|2)= 2nε?

13. Using Theorem 22.1.5, prove the following: Let f ∈ C (∂U) ( f is continuous on∂U .) where U is a bounded open set. Then there exists at most one solution u ∈C2 (U)∩C

(U)

and ∇2u = 0 in U with u = f on ∂U . Hint: Suppose there are two

solutions ui, i = 1,2 and let w = u1−u2. Then use the maximum principle.

14. Suppose B is a vector field and ∇×A=B. Thus A is a vector potential for B.Show that A+∇φ is also a vector potential for B. Here φ is just a C2 scalar field.Thus the vector potential is not unique.

22.3 The Divergence TheoremThe divergence theorem relates an integral over a set to one on the boundary of the set. Itis also called Gauss’s theorem.

Definition 22.3.1 A subset V of R3 is called cylindrical in the x direction if it is of the form

V = {(x,y,z) : φ (y,z)≤ x≤ ψ (y,z) for (y,z) ∈ D}

where D is a subset of the yz plane. V is cylindrical in the z direction if

V = {(x,y,z) : φ (x,y)≤ z≤ ψ (x,y) for (x,y) ∈ D}

where D is a subset of the xy plane, and V is cylindrical in the y direction if

V = {(x,y,z) : φ (x,z)≤ y≤ ψ (x,z) for (x,z) ∈ D}

where D is a subset of the xz plane. If V is cylindrical in the z direction, denote by ∂V theboundary of V defined to be the points of the form (x,y,φ (x,y)) ,(x,y,ψ (x,y)) for (x,y) ∈D, along with points of the form (x,y,z) where (x,y)∈ ∂D and φ (x,y)≤ z≤ψ (x,y). Pointson ∂D are defined to be those for which every open ball contains points which are in D aswell as points which are not in D. A similar definition holds for ∂V in the case that V iscylindrical in one of the other directions.