MATH 174A: PROBLEM SET 6
DUE THURSDAY, MARCH 1, 2007
Problem 1. For f, g C(S1 ), let
1
f g =
2
Z
f ( )g() d.
S1
(1) Show that f g C(S1 ) and kf gk kf k kgk , where kf k =
supcfw_|f ()| : S1 .
(2) Show that f g = g f .
R
1
(3) Show that in fact kf gk kf k
MATH 174A: PROBLEM SET 7
DUE THURSDAY, MARCH 8, 2007
Problem 1. Use separation of variables to solve the Dirichlet problem for the
Laplacian: u = 0 in , u| = f given, where = cfw_z R2 : a < |z| < b,
a, b > 0, is an annulus. (Hint: remember that there are
MATH 174A: PROBLEM SET 8
DUE THURSDAY, MARCH 15, 2007
Problem 1. Do Taylor 3.3.15. (You can assume 3.3.14. We have discussed the
relevant part it in class.)
Problem 2. Do Taylor 3.3.16.
Problem 3. Do Taylor 3.4.1.
Problem 4. Do Taylor 3.4.2.
P
Problem 5.
MATH 174A: LECTURE NOTES ON THE INVERSE FUNCTION
THEOREM
Theorem 1. Suppose Rn is open, F : Rn is C k , k 1, p0 ,
q0 = F (p0 ). Suppose that DF (p0 ) is invertible. Then there is a neighborhood
U of p0 and a neighborhood V of q0 such that F : U V is a bij