FINAL FOR ADVANCED CALCULUS
Date: Monday, Jun 17, 2002 Instructor: Shu-Yen Pan
No credit will be given for an answer without reasoning.
1. [10%]
(i) Give an example of a set A ⊂ Rn such that A is of measure zero but not of volume zero.
(ii) Give an example of a function f on [−π, π] such that the Fourier series of f converges uniformly.
(iii) Give an example of a function f on [−π, π] such that the Fourier series of f converges to f at every point x in [−π, π] but not uniformly on [−π, π].
2. [10%] Integrate the function f (x, y, z) = x2+ y2+ z2over the set B = { (x, y, z) | x2+ y2+ z2≤ 1 }.
3. [10%] Suppose that A ⊂ Rn and A has zero volume. Suppose f : A → R is a bounded function. Prove that f is integrable andR
Af = 0.
4. [10%] Let fn(x) =Pn
k=1 1
2ksin kx for x ∈ R.
(i) Show that limn→∞fn(x) exists for all x.
(ii) Show that the sequence converges uniformly on the whole real line.
(iii) Show that R2π
0 (limn→∞fn(t)) dt = 0
5. [10%] Let ϕn(x) = √1πsin nx. Check that { ϕn(x) | n ∈ N } is an orthonormal family of functions on [0, 2π].
6. [10%] Compute the Fourier series a20 +P∞
n=1(ancos nx + bnsin nx) of the function f (x) = x on the interval [−π, π].
7. [10%] Let V be a Hilbert space and let ϕ0, ϕ1, ϕ2, . . . be a complete orthonormal set. Let c0, c1, c2, . . . be complex numbers and suppose thatP∞
k=0|ck|2< ∞. Let fn=Pn
k=0ckϕk. (i) Show that kfn− fmk2=Pn
k=m+1|ck|2. (ii) Show that fn is a Cauchy sequence in V.
(iii) Show that P∞
k=0ckϕk is the Fourier series of some f . 8. [10%] Let f : R2→ R be defined by
f (x, y) =
xy(x2−y2)
x2+y2 , if (x, y) 6= (0, 0);
0, if (x, y) = (0, 0).
Show that ∂x∂y∂2f and ∂y∂x∂2f exist at (0, 0) but are not equal.
9. [10%] Let f (x) = cos xx . Show thatR∞
1 f (x) dx converges butR∞
1 |f (x)| dx does not converge.
10. [10%] Let f : R → R be of class C1 and u = f (x), v = −y + xf (x). If f0(x0) 6= 0, show that this transformation is invertible near (x0, y0) and the inverse has the form x = f−1(u), y = −v + uf−1(u).
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