- Find div f and curlf where f is
- Prove formula (2) of Theorem 17.1.3.
- Show that if u and v are C2 functions, then curl =
- Simplify the expression f× +
- Simplify ∇× where
T = xi + yj + zk and v is a constant vector.
- Discover a formula which simplifies ∇⋅.
- Verify that ∇⋅
u∇2v − v∇2u.
- Verify that ∇2 =
v∇2u + 2 +
- Functions u, which satisfy ∇2u = 0 are called harmonic functions. Show that the following functions
are harmonic where ever they are defined.
- x2 − y2
- Verify the formula given in (17.1) is a vector potential for g assuming that div g = 0.
- Show that if ∇2uk = 0 for each k = 1,2,
,m, and ck is a constant, then
- In Theorem 17.1.5, why is ∇2 = 2
- Using Theorem 17.1.5, prove the following: Let f ∈ C (
f is continuous on ∂U.) where U is a
bounded open set. Then there exists at most one solution u ∈ C2
∇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
- 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