1.
x dy - y dx
Let ω = -------------- , (x , y) 屬於 R^2 - {0} .
x^2 + y^2
Show that ω is a closed 1-form , but not exact on R^2 - {0} .
2. Let S = {x屬於R^n | ||x|| ≦ 1} and f : S→R be a nonnegative continuous
function.
(a) Prove that f has the absolute maximum value on S.
(b) Let M be the absolute maximum value of f on S .
k 1/k
Show that lim (∫(f(x)) dx_1 dx_2 ... dx_n) = M .
k→∞ S
3. Prove or disprove (by a counter example) the following statements:
If f : R→R is a continuous function , then f is an open mapping .
4. Let f : [0,1]→R be defined by
0 if x is irrational
f(x) =
1 p
--- if x = ---
q q
where p , q ≧ 0 with no common factor .
Is f integrable on [0,1] ？
5. Let C([0,1]) be the set of real continuous functions on [0,1] .
Show that the complement of the following set
1
A = { f屬於 C([0,1]) | 0 < ∫ f(x) dx < 3 } is closed .
0
6. Let (M,d) be a metric space . If A is compact in M and B is closed in M ,
and A∩B = ψ . Show that there is an δ > 0 such that d(x,y) > δ for
all x 屬於 A and y 屬於 B .
7. Let k(x,y) be a continuous real-valued function on the square
S = [0,1] ╳ [0,1] . Assume that |k(x,y)| < 1 for each (x,y) 屬於 S .
Let A : [0,1]→R be continuous . Prove that there is a unique continuous
real-valued function f(x) on [0,1] such that
1
f(x) = A(x) + ∫ k(x,y)f(y) dy by contraction mapping theorem .
0
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