課程名稱︰高等微積分
課程性質︰系定必修
課程教師︰王振男 教授
開課系所︰數學系
考試時間︰94/05/22 14:00-17:00
試題： Advanced Calculus Second Midterm (A)
1.(10%) Find conditions on a point (x_0,y_0,u_0,v_0) such that there exist
real-valued functions u(x,y) and v(x,y) that are continously differentiable
near (x_0,y_0) and satisfy the simultaneous equations
x‧u^2 + y‧v^2 + xy = 9
x‧v^2 + x‧u^2 - xy = 7.
Prove that the solutions satisfy u^2 + v^2 = 16/(x+y).
2
2.(15%) Let V be open in Ｒ , (a,b)屬於V, and f: V →Ｒ have second-order
partial derivatives on V with f_x(a,b) = f_y(a,b) = 0. If the second-order
partial derivatives of f are continuous at (a,b) and exactly two of the
three numbers f_xx(a,b), f_xy(a,b) are zero, prove that (a,b) is a saddle
point if f_xy(a,b)≠0.
n
3. Suppose that E_1, E_2 are Jordan regions in Ｒ .
(a) (5%) Prove that if E_1包含等於E_2, then Vol(E_1)≦Vol(E_2).
(b) (5%) Prove that E_1∩E_2 and E_1\E_2 are Jordan regions.
(c) (5%) Prove that if E_1, E_2 are nonoverlapping, then Vol(E_1∪E_2) =
Vol(E_1) + Vol(E_2).
(d) (5%) Prove that Vol(E_1∪E_2) = Vol(E_1) + Vol(E_2) - Vol(E_1∩E_2).
n
4. Let E be a Jordan region in Ｒ . o
(a) (5%) Prove that Vol(E) > 0 if and only if E ≠φ.
(b) (5%) Let f:→ [a,b] → Ｒ be integrable on [a,b]. Prove that the graph
2
of y = f(x), x屬於[a,b], is a Jordan region in Ｒ .
n
5.(a) (10%) Suppose that E is a Jordan region in Ｒ and that f_k: E →Ｒ
are integrable on E for k屬於Ｎ. If f_k → f uniformly on E as k→∞,
prove that f is integrable on E and
lim ∫ f_k(x) dx = ∫ f(x) dx.
k→∞ E E
(b) (5%) Prove that
lim ∫∫ coa(x/k)‧e^(y/k) dA
k→∞ E
2
exists, and find its value for any Jordan region E in Ｒ .
n 1
6. Let f:Ｒ →Ｒ be of C . Answer the following questions and prove or
disprove your answers.
n n
(i) Can the set {x屬於Ｒ : f(x)≠0, ▽f(x)≠0} be open in Ｒ ? (5%)
n n
(ii) Can the set {x屬於Ｒ : f(x)＝0, ▽f(x)≠0} have measure zero in Ｒ ?
(5%)
n 2 2
7.(5%) Let f: Ｒ →Ｒ be of C . Suppose D f(x) is positive definite for
n n n
x屬於Ｒ . Can ▽f:Ｒ →Ｒ be an open mapping, ie., ▽f(U) is open in Ｒ
n
for U open in Ｒ ?
╭ 0 ╮n ╭ ╮n
8. Let A_0 = ╰ a_ij ╯i,j=1 and A = ╰ a_ij ╯i,j=1 be n×n real symmetric
matrices. Assume that μ_0 is the smallest eigenvalue of A_0 and μ(ε)
is the smallest eigenvalue of A_0 + εA, where ε> 0.
(i) (10%) Show that if μ_0 is simple then μ(ε) is also simple provided
ε is sufficiently small.
(ii) (5%) Prove that |μ(ε)-μ_0|/ε is bounded as ε→ 0.
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.250.148