- Find div f and curlf where f is
- Prove formula 2 of Theorem 27.1.3.
- 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
- 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 27.1 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,
- In Theorem 27.1.5, why is ∇2 = 2
- Using Theorem 27.1.5, 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 ∈
∩ C 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
- 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.