課程名稱︰分析導論優二
課程性質︰數學系大二必修
課程教師︰王振男
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2015/03/31
考試時限（分鐘）：180
試題 :
n n 1
1. (20%) Let S be an open set in |R . Assume that F: S → |R is a C mapping.
Let x ∈ S and the Jacobian matrix DF(x ) is invertible. Show that there
0 0
exists an open set O satisfying x ∈ O ⊂ S such that
0
∥F(u) - F(v)∥ ≧ c∥u - v∥ ∀u, v ∈ O (1)
with some constant c > 0. Hint: A matrix A is invertible iff there exists a
n
positive constant k > 0 such that ∥Ah∥≧ k∥h∥ for all h ∈ |R . Show that
if ∥B - A∥≦ k/2, then ∥Bh∥≧ (k/2)∥h∥. For F, use the mean value
theorem.
n n
2. (20%) A mapping F satisfies (1) is said to be stable in O. Let F: |R → |R
1 n -1
be a C mapping and be stable in |R . Show that F exists globally, i.e., F
n n
is a bijective map on |R . Hint: Show that F(|R ) is both open and closed in
n n
|R . Note that |R is connected.
2
3. (20%) This is a global implicit function theorem. We consider F: |R → |R a
1
C function. Assume that there exists some constant c > 0 such that F (x, y)
y
2 1
≧ c for all x, y ∈ |R . Show that there exists a unique C function g: |R
→ |R such that F(x, g(x)) = 0 for all x ∈ |R. Hint: Use the injectivity of
a strictly monotone function and the usual implicit function theorem.
n ×n 1 m
4. (20%) Let A = A(x) ∈ |R , where A is a C function of x ∈ |R . Assume
m
that for some x ∈ |R , λ(x ) is a simple eigenvalue of A(x ). Show that
0 0 0
1
there exists a C simple eigenvalue λ(x) of A(x) for x near x . Give an
0
∞
example showing that even for a C matrix-valued function A(x), a non-simple
1
eigenvalue may not be C .
2 2 2
5. (20%) Find the maximum of (x x ...x ) under the restriction x + x + ... +
1 2 n 1 2
2
x = 1. Use the result to derive the following inequality, valid for positive
n
real numbers a , ..., a :
1 n
a + ... + a
1/n 1 n
(a ...a ) ≦ ───────.
1 n n