課程名稱︰高等微積分二
課程性質︰數學系大二必修
課程教師︰陳俊全
開課學院：理學院
開課系所︰數學系
考試日期︰2011年1月，10:20-12:19
考試時限：170分鐘
是否需發放獎勵金：是
試題 :
Choose 5 from the following 7 problems
1. Let f(x,y) = (u(x,y),v(x,y)) be a C1 function from R^2 to R^2. Suppose that
∂u ∂v ∂u ∂v
f = (u,v) satisfies the Cauchy Riemann condition ----- = -----; ----- = -----.
∂x ∂y ∂y ∂x
(a) Show that if Df(x0,y0)≠0, then f is locally invertible in a neighborhood
of (x0,y0) and the inverse function also satiesfies the Cauchy-Riemann
condtion.
(b) Assunme further that u^2 + v = 0 on R^2. Prove that f = (u,v) is a constant
on R^2.
2
(a) Let f:[a,b]→R be a continous function. Show that the graph E = {(x,f(x))|
x∈[a,b]} of f has content zero and measure zero in R^2.
(b) Show that if f is a continuous function on R, then its graph has measure
zero in R^2.
3 Let E = {p1/q1 , p2/q2 , p3/q3 ,...|qk≠0,pk and qk are coprime} be the set
of all rational numbers in [0,1]. Suppose f :[0,1]→R satisfies f(x)= 1 if x∈E
and f(x)=0 if x 不屬於E; and g:[0,1]→R satisfies g(pk/qk)=1/qk for k=1,2,3,...
and g(x)=0 if x 不屬於E.
(a) Prove that f is not Riemann integrable.
(b) Prove that g is Riemann integrable.
(You can use Lebesgue's Theorem)
∂f
4. Suppose f:[a,b] ×[c,d]→R is continuous and ----- exists and is continuous
b ∂y
on [a,b] ×[c,d]. Let F(y) = ∫f(x,y)dx. Show that F is differentiable and
b ∂f a
F'(y) = ∫-----(x,y)dx.
a ∂y
5. Let A = [a,b] ×[c,d] and f:A→R be a bounded function. Suppose f is Riemann
integral on A and f(x,y) is Riemann integrable with respect to the variable y
for each fixed x∈[a,b]. Prove the Fubini theorem
b d
∫f = ∫(∫f(x,y)dy)dx
A a c
6. Let f,g:[a,b]→R be Riemann integrable.
(a) Prove the Cauchy-Schwarz inequality
b b b
(∫f(x)g(x)dx)^2 ≦ (∫f^2(x)dx)(∫g^2(x)dx).
a a a
(b) Prove the Minkowski inequality
b b b
(∫|f(x)+g(x)|^2 dx)^(1/2) ≦ (∫|f(x)|^2dx)^(1/2) + (∫|g(x)|^2dx)^(1/2)
a a a
Hint: consider a suitable inner product.
7. Compute the Fourier series of e^x on [-π,π] and prove the following
identity 1 ∞ 1
-----(πcothπ-1) = Σ ----------.
2 k=1 k^2 +1
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