math 4210: homework problemshomepages.rpi.edu/~kovacg/classes/analysis2/421hw.pdf · 2017. 2....
TRANSCRIPT
Math 4210: Homework Problems
Gregor Kovacic
1. Derive the formula
1
1 + x= 1− x+ x2 − x3 + · · ·+ (−1)n−1xn−1 +
(−1)nxn
1 + x,
for x 6= −1 and deduce that
log(1 + x) = x− x2
2+x3
3− x4
4+ · · ·+ (−1)n−1x
n
n+ · · ·
on −1 < x ≤ 1. What is log 2?
HINT: First show that
log(1 + x) =
∫ x
0
dt
1 + t
for x > −1. Use this to obtain
log(1 + x) = x− x2
2+x3
3− x4
4+ · · ·+ (−1)n−1x
n
n+
∫ x
0
(−1)ntn dt
1 + t.
Estimate the remainder directly in a way similar to deriving Lagrange’s form of the remainderin the general Taylor series derivation. It is straight forward on 0 ≤ x ≤ 1. For −1 < x ≤t ≤ 0, use
1
1 + t≤ 1
1− |x|.
2. Show that for any real α and 0 < |x| < 1,
(1 + x)α = 1 +∞∑n=1
α(α− 1)(α− 2) · · · (α− n+ 1)
n!xn,
by completing the following outline:
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First, show that the n-th derivative of (1+x)α at x = 0 is equal to α(α−1)(α−2) · · · (α−n+1).Use Cauchy’s form of the remainder to obtain
Rn =(1− θ)n
n!α(α− 1)(α− 2) · · · (α− n)xn+1(1 + θx)α−n−1
with some 0 ≤ θ ≤ 1. Since |x| < 1, show that
0 ≤ (1− θ)(1 + θx)
≤ 1,
and deduce that
|Rn| ≤ (1 + θx)α−1|αx|∣∣∣(1− α
1
)x∣∣∣ ∣∣∣(1− α
2
)x∣∣∣ · · · ∣∣∣(1− α
n
)x∣∣∣ .
There exists a number q such that |x| < q < 1. Convince yourself that∣∣∣(1− α
m
)x∣∣∣ < q
for all sufficiently large m, say m > N . Deduce that for n > N ,
|Rn| ≤ (1 + θx)α−1|α|(1 + |α|)Nqn−N .
Show that the factor (1 + θx)α−1 is bounded by 2α−1 when α ≥ 1 and by (1 − q)α−1 whenα < 1, and thus conclude the proof.
3. Let f(x) have a continuous derivative in the interval [a, b], and let f ′′(x) ≥ 0 for everyx ∈ [a, b]. Then if ξ is any point in the interval [a, b], show that the curve nowhere fallsbelow its tangent at the point x = ξ, y = f(ξ). Draw a picture.
HINT: Use a three-term Taylor expansion.
4. Use Taylor’s formula to show that if f ′(x0) = 0, the sign of f ′′(x0) determines whether x0
is a maximum or a minimum. What happens if f ′′(x0) = 0?
5. Suppose a ∈ R, f is a twice-differentiable function on (a,∞), and M0, M1, M2 arethe least upper bounds of |f(x)|, |f ′(x)|, and |f ′′(x)|, respectively, on (a,∞). Prove thatM2
1 ≤ 4M0M2.
HINT: If h > 0, use Taylor’s theorem to show that
f ′(x) =f(x+ 2h)− f(x)
2h− hf ′′(ξ)
2
for some ξ ∈ (x, x+ 2h). Hence, show that
|f ′(x)| ≤ hM2 +M0
h.
To show that M21 = 4M0M2 can actually happen, take a = −1, define
f(x) =
2x2 − 1, −1 < x < 0,
x2 − 1
x2 + 1, 0 ≤ x <∞,
and show that M0 = 1, M1 = 4, M2 = 4.
6. Alternative derivation of Taylor’s formula: Let f(x) be (n+ 1)-times continuouslydifferentiable on an interval containing the points a and b. Consider a as the independentvariable and keep b fixed. Differentiate the expression
f(b) = f(a) + (b− a)f ′(a) + · · ·+ (b− a)n
n!f (n)(a) +Rn(a)
on a sufficiently many times to show that
0 =(b− a)n
n!f (n+1)(a) +R′n(a).
Deduce that
Rn(a) =
∫ b
a
(b− t)n
n!f (n+1)(t) dt.
7. Prove that the function
f(x) =
e−1/x2 , x 6= 0
0, x = 0,
is infinitely many times differentiable everywhere, yet it cannot be expanded in a Taylorseries about x = 0. Nevertheless, write down a suitable series expansion for f(x) valid forall x 6= 0.
HINT: For the first part, compute that f (n)(0) = 0 = limx→0 f(n)(x) for every n.
8. Asymptotic Property of the Taylor Expansion: For simplicity, consider a functionf(x), which is (n + 1)-times continuously differentiable on the symmetric interval [−a, a].Show that Taylor’s formula is an asymptotic formula in the following sense: If
f(x) = f(0) + xf ′(0) +x2
2f ′′(0) + · · ·+ xn
n!f (n)(0) +Rn(x) ≡ fn(x) +Rn(x),
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then
limx→0
f(x)− fn(x)
xn= lim
x→0
Rn(x)
xn= 0,
regardless of whether Rn(x) → 0 as n → ∞ or not. Also, interpret the result of problem 7in view of this fact.
9. Consider the series
S =∞∑n=1
1
n(n+ 1).
Does this series converge? If yes, why, and what is its sum?
10. (i) Let a1 ≥ a2 ≥ a3 ≥ · · · ≥ 0. Show that the series S =∞∑n=1
an converges if and only if
the series Σ =∞∑m=1
2ma2m does.
HINT: If Sn and Σm are the respective partial sums, show that for n < 2m, Sn ≤ Σm, andthat for n > 2m, 2Sn ≥ Σm.
(ii) Use part (i) to conclude that∞∑n=1
1
nαconverges for α > 1 and diverges for α ≤ 1.
11. Investigate for convergence or divergence of the series∑an with the general term
(i) an =√n+ 1−
√n,
(ii) an =
√n+ 1−
√n
n.
HINT: Use 10 (ii) for (ii).
12. Let all an ≥ 0. Show that the convergence of the series∑an implies the convergence of
the series∞∑n=1
√ann
.
HINT: Use the Cauchy-Schwartz inequality for sums.
13. If the series∑an converges and the sequence bn is monotonic and bounded, show that∑
anbn converges.
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HINT: Show that there is no loss of generality in assuming that bn is increasing. Letb = limn→∞ bn. (Show that b exists!) Use Abel’s test proven in class to show that the series∑an(b− bn) converges, and thus conclude the validity of the claim you had to prove.
14. Show that the series∞∑n=1
sinπ
ndiverges, but the series
∞∑n=1
sinπ
n2converges.
HINT: First, from the graph of sinx, find the estimate 2x/π ≤ sinx ≤ x on 0 ≤ x ≤ π.
15. (i) For what values of α does the series∞∑n=1
(−1)n
nαconverge?
(ii) For what values of α does it converge absolutely?
HINT: Use the alternating series and problem 10 or the integral test.
16. Find the sums of the following rearrangements of the series
1− 1
2+
1
3− 1
4+
1
5− 1
6+ · · ·
for log 2:
(i) 1− 1
2− 1
4+
1
3− 1
6− 1
8+
1
5− 1
10− 1
12+−− · · · ,
HINT: Insert pairs of parentheses according to some appropriate simple pattern, and evaluatethe sum in each pair of the parentheses explicitly.
(ii) 1 +1
3+
1
5− 1
2− 1
4− 1
6+ + + · · · .
HINT: Look carefully at blocks of length 6.
17. Show that the series∞∑n=1
cosnx
nconverges for all x which are not integer multiples of 2π.
HINT: Restrict your analysis to x ∈ [0, π]. (Why can you do it?) Multiply the sum
σn(x) =1
2+ cosx+ cos 2x+ · · ·+ cosnx (1)
by sin 12x and use appropriate trigonometric identities to show that
σn(x) =sin(n+ 1
2
)x
2 sin 12x
. (2)
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For x ∈ [0, π], show that sin(x/2) ≥ x/π. Deduce that |σn(x)| < π/2x for x 6= 0, then useAbel’s test.
18. Show that if n is an arbitrary integer greater than 1,
∞∑m=1
am(n)
m= log n,
where am(n) is defined as
am(n) =
1, if n is not a factor of m,
−(n− 1), if n is a factor of m.
HINT: If γ is the Euler-Mascheroni constant, then
γ = limM→∞
(M∑m=1
1
m− logM
)= lim
M→∞
(nM∑m=1
1
m− log nM
).
19. Show that the series
1− 1
2α+
1
3− 1
4α+
1
5− 1
6α+
1
7−+ · · ·
only converges for α = 1.
HINT: For α > 1, show that it is the sum of a convergent and a divergent series. For0 < α < 1, write the series in the form
1− 1
2+
(1
2− 1
2α
)+
1
3− 1
4+
(1
4− 1
4α
)+
1
5− 1
6+
(1
6− 1
6α
)+
1
7− · · ·
and show that the series(1
2− 1
2α
)+
(1
4− 1
4α
)+
(1
6− 1
6α
)+ · · ·
diverges. What happens for α ≤ 0.
20. Show that the series∞∑n=1
(1− 1√
n
)nconverges.
6
HINT: Write
(1− 1√
n
)nin terms of an exponential and use the series for log(1− x) with
small x to show that (1− 1√
n
)n≤ e−
√n.
Then use the integral test.
21. By comparison with∞∑n=1
1
nα, prove Raabe’s test:
The series∑|an| converges or diverges according as
n
(|an||an+1|
− 1
)is greater than 1 + ε or less than 1 − ε for every sufficiently large n and for some ε > 0independent of n.
HINT: First show that the binomial series for (1 + x)α converges absolutely for |x| < 1.Conclude that for |x| ≤ q < 1, the estimate∣∣∣∣∣
∞∑k=2
(αk
)xk
∣∣∣∣∣ ≤ C|x|2
holds for some constant C depending only on q and α.
Infer that, for sufficiently large n,
1 +1 + ε
n≥(
1 +1
n
)1+ε/2
=
(n+ 1
n
)1+ε/2
and
1 +1− εn≤(
1 +1
n
)1−ε/2
=
(n+ 1
n
)1−ε/2
.
You will need these inequalities at some appropriate points in your proof of the test.
22. Show that∞∑n=1
n!
(α + 1)(α + 2) · · · (α + n)converges if α > 1 and diverges if α ≤ 1.
23. (i) If fn and gn converge uniformly on a set E and fn and gn are sequences ofbounded functions, then show that fngn converges uniformly on E.
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(ii) Construct sequences fn and gn which converge uniformly on some set E, but suchthat fngn converges only pointwise on E.
24. Consider
f(x) =∞∑n=1
1
1 + n2x.
For what values of x does the series converge absolutely? On what intervals does it convergeuniformly? On what intervals does it fail to converge uniformly? Is f continuous whereverthe series converges? Is f bounded?
25. Let
fn(x) =
0, x <1
n+ 1,
sin2 π
x,
1
n+ 1≤ x ≤ 1
n,
0,1
n< x.
Show that fn converges to a continuous function, but not uniformly. Use the series∑fn
to show that absolute convergence, even for all x, does not imply uniform convergence.
26. Show that the series∞∑n=1
(−1)nx2 + n
n2
converges uniformly in every bounded interval, but does not converge absolutely for anyvalue of x.
27. Letfn(x) =
x
1 + nx2, n = 1, 2, · · · .
Show that fn converges uniformly to a function f , and that the equation
f ′(x) = limn→∞
f ′n(x)
holds for x 6= 0 and does not hold for x = 0.
28. Let
H(x) =
0, x ≤ 0,
1, x > 0,
8
let xn be a sequence of distinct points in (a, b), and let∑cn converge absolutely. Show
that the series
f(x) =∞∑n=1
cnH(x− xn), a ≤ x ≤ b,
converges uniformly, and that f is continuous at every x 6= xn.
29. Let fn be a sequence of continuous functions which converges uniformly to a function fon a set D. Show that
limn→∞
fn(xn) = f(x)
for every sequence of points xn ∈ D such that xn → x, and x ∈ D. By finding a counterexample, show that the converse is not true if D is not compact.
30. Let fn be Riemann integrable on [a, b] for n = 1, 2, 3, . . . , and let fn → f uniformly on[a, b]. Then f is Riemann integrable on [a, b], and∫ b
a
f(x) dx = limn→∞
∫ b
a
fn(x) dx.
HINT: Letεn = sup
a≤x≤b|fn(x)− f(x)| .
Let U(f, a, b) and L(f, a, b) be the upper and lower Riemann integrals, respectively, definedas
U(f, a, b) = supK−1∑k=0
Mk (xk+1 − xk) , L(f, a, b) = infK−1∑k=0
mk (xk+1 − xk)
where the supremum and infimum are taken over all possible partitions a = x0 < x1 < · · · <xK−1 < xK = b of the interval [a, b], and
Mk = supxk≤x≤xk+1
f(x), mk = infxk≤x≤xk+1
f(x).
Show that fn − εn < f < fn + εn implies both
0 ≤ U(f, a, b)− L(f, a, b) ≤ 2εn(b− a)
and ∣∣∣∣∫ b
a
f(x) dx−∫ b
a
fn(x) dx
∣∣∣∣ ≤ εn(b− a).
31. Suppose fn(x) and gn(x) are defined on an interval I and
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(a)∑fn(x) has uniformly bounded partial sums;
(b) gn(x)→ 0 uniformly on I;
(c) g1(x) ≥ g2(x) ≥ g3(x) ≥ · · · at every x ∈ I.
Show that∑fn(x)gn(x) converges uniformly on I.
32. On what intervals of x does the series∞∑n=1
cosnx
nconverge uniformly?
HINT: Use the solutions of problems 17 and 31.
33. From the appropriate geometric and binomial (see problem 2) series, derive the seriesexpansions in powers of x of the functions arctanx and arcsin x. What are their respectiveradii of convergence? Show that the series for arctanx also converges at the endpoints ofthe convergence interval.
34. Show thatlog(1 + x)
1 + x=∞∑n=1
(−1)n+1
(n∑k=1
1
k
)xn.
35. Letx
ex − 1=∞∑n=0
Bn
n!xn.
Using complex variables, one can show that the radius of convergence of this series is 2π.
(i) Multiply the above series by ex − 1 to show that the Bernoulli numbers Bn satisfy theequation (
n+ 1
1
)Bn +
(n+ 1
2
)Bn−1 +
(n+ 1
3
)Bn−2 + · · ·+
(n+ 1
n+ 1
)B0 = 0
for n > 0, where B0 = 1. Conclude that these numbers are rational.
(ii) Show thatx
ex − 1+x
2=x
2coth
x
2
and thus that, for n > 0, B2n+1 = 0 and
x cothx =∞∑n=0
22nB2n
(2n)!x2n.
10
What is the radius of convergence of this series?
(iii) Replace x by ix, to find
x cotx =∞∑n=0
(−1)n22nB2n
(2n)!x2n.
(iv) Derive the formula 2 cot 2x = cotx− tanx to conclude that
tanx =∞∑n=1
(−1)n−1 22n (22n − 1)B2n
(2n)!x2n−1.
Where does this series converge?
36. (i) Integrate by parts to obtain∫ π2
0
sinm x dx =m− 1
m
∫ π2
0
sinm−2 x dx
for all integer m > 1. Deduce that∫ π2
0
sin2n x dx =2n− 1
2n
2n− 3
2n− 2· · · 1
2
π
2.
(ii) Let |x| < 1, and
K(x) =
∫ π2
0
dt√1− x2 sin2 t
.
Find the power series expansion for K(x) in powers of x. Where does this series converge?
HINT: Use problem 2 or 33.
(iii) The Bessel function of order zero is given by the formula
J0(x) =2
π
∫ π2
0
cos(x sin t) dt.
By expanding the integrand in a power series and carrying out the integration term-by-term(justify it!), show that J0(x) has a power series expansion
J0(x) =∞∑n=0
(−1)n
(n!)2
(x2
)2n
.
Where does this series converge?
11
37. Use the result of problem 2 to show that
√1− x = 1− 1
2x− 1
2 · 4x2 − 1 · 3
2 · 4 · 6x3 − 1 · 3 · 5
2 · 4 · 6 · 8x4 − 1 · 3 · 5 · 7
2 · 4 · 6 · 8 · 10x5 − · · · ,
where the series converges for −1 < x < 1 and all the coefficients an, n ≥ 1, are negative.
(i) Show that this power series still converges to√
1− x at x = 1 by completing the followingoutline:
Denote g(x) =√
1− x, and let Sn(x) be its n-th partial sum.
(a) For finite n and 0 ≤ x < 1, show that
Sn(x) =n∑k=0
akxk ≥ g(x) > 0.
(b) Conclude that Sn(1) ≥ 0 and that Sn(1)→ S ≥ 0.
(c) Show that, for fixed n, Sn(1) < Sn(x).
(d) Given ε > 0, choose x so close to 1 that g(x) < ε/2 and n so large that Rn(x) =Sn(x)− g(x) < ε/2. Conclude that 0 ≤ Sn(1) < Sn(x) < ε.
(ii) Show that for every ε > 0, there exists an n such that∣∣√1− x− Sn(x)
∣∣ < ε uniformlyon 0 ≤ x ≤ 1.
HINT: 0 ≤ Rn(x) ≤ Rn(1)→ 0.
(iii) Replace x by 1− x2 in (ii), and show that there exist polynomials Pn(x) that convergeto |x| uniformly on −1 ≤ x ≤ 1.
(iv) Replace x by 1− (x−a)2/A2 in (ii) to generalize the result of (iii) to the function |x−a|on the interval [a− A, a+ A].
38. (i) Show that if f(x) is continuous on [a, b], then given ε > 0, there exists a piecewiselinear function φ(x) such that |f(x)− φ(x)| < ε on [a, b].
(ii) Show that every polygonal function φ(x) can be represented as
φ(x) = c0 +n∑i=1
ci (x− xi + |x− xi|) = a+ bx+n∑i=1
ci |x− xi| .
HINT: First investigate the behavior of function x+ |x|.
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(iii) Prove the Weierstrass approximation theorem: For every continuous function f(x)on [a, b] and every ε > 0, there exists a polynomial P (x) such that |f(x)− P (x)| < ε for allx ∈ [a, b].
39. Prove the following existence and uniqueness theorem for second-order linear differentialequations with power-series coefficients: Let
p(x) =∞∑n=0
pnxn, q(x) =
∞∑n=0
qnxn,
where both series converge at least for |x| < R. Then the initial-value problem
y′′ + p(x)y′ + q(x)y = 0, y(0) = y0, y′(0) = y1
has a unique power-series solution converging at least for |x| < R.
HINT: (i) Assume y(x) =∞∑n=0
ynxn. By differentiation and multiplication of power series,
show that the coefficients yn, n ≥ 2, can be computed recursively, provided y0 and y1 aregiven.
(ii) From the recursion formulas for the coefficients yn, show that if you can find an auxiliarydifferential equation y′′+P (x)y′+Q(x)y = 0 such that the coefficients of −P (x) and −Q(x)are all positive and larger than the corresponding |pn| and |qn|, then the coefficients of itssolution series will also be positive and larger than the corresponding |yn|. The series fory(x) will therefore converge at least as far as this auxiliary solution.
(iii) Using the ratio test, show that an appropriate auxiliary equation is
y′′ −M(
1− x
ρ
)−1
y′ −N(
1− x
ρ
)−2
y = 0
for some appropriate constants M and N . Here, ρ is is any number 0 < ρ < R.
(iv) Show that the general solution of the auxiliary equation is
Y (x) = Y0
(1− x
ρ
)α0
+ Y1
(1− x
ρ
)α1
,
where α0 and α1 are the two roots of the quadratic equation
α(α− 1)− αM −N = 0.
Show that its power series expansion about x = 0 has radius of convergence ρ. Since ρ < Ris arbitrary, the radius of convergence of the series for y(x) is R.
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40. Let M be a metric space. Show that both M and the null set are open. Show that anarbitrary union of open sets and the intersection of a finite number of open sets are open.
41. Let M be a metric space and N ⊂ M . Show that A ⊂ N is open in N if and only ifA = N ∩B, where B is some (not necessarily unique) open set in M .
42. Show that if f : M → N is continuous, then every preimage of an open set is open, andevery preimage of a closed set is closed.
43. (i) Prove that the interval I = [0, 1] is compact by carrying out the following argument:Assume that it is not compact. Then there exists a cover Gα of I which does not containa finite subcover. One of the intervals
[0, 1
2
]and
[12, 1]
cannot be covered by any finitesub-collection of Gα. Repeat the argument, until you arrive at a single point not coveredby any finite sub-collection of Gα. Argue that this is a contradiction.
(ii) Generalize this result to any rectangle x | ak ≤ xk ≤ bk, k = 1, . . . , n ⊂ Rn.
44. Show that every closed subset of a compact set is compact. Then use the result ofproblem 43 (ii) to show that any closed and bounded subset of Rn is compact.
45. Show that if A is closed and B is compact, then A ∩B is compact.
46. Let M be a metric space and let Kα be a collection of its compact subsets. Show thatif the intersection of every finite sub-collection of Kα is nonempty, then ∩αKα is nonempty.
HINT: The complement of each Kα is an open set. If no point of some Kβ belongs to everyother Kα, their complements form an open cover of Kβ.
47. The closure A of a set A is the union of A and all its limit points. Show that A is closed,that A = A precisely when A is closed, and that A is the smallest closed set containing A,that is, if A ⊂ B and B is closed then A ⊂ B.
48. A point x is an interior point of the set A if some open ball Br(x) ⊂ A. The set Aof all the interior points if A is called the interior of A. Show that A is open, that A = Aprecisely when A is open, and that A is the largest open set contained in A. (Formulate thelast statement precisely!)
49. Show that if M is compact and f : M → N is continuous and one-to-one, then its inversef−1 is continuous.
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50. If A is connected and f is continuous, show that f(A) is connected.
51. (i) Show that any open connected subset of the real line is an open interval.
(ii) Show that every open subset of the real line is a union of (at most) countably manydisjoint open intervals.
52. Show that a continuous real function f on [a, b] achieves its maximum, minimum, andevery point in-between.
53. Show that the closed unit ball in C[0, 1] is not compact.
HINT: Look at all the powers xn.
54. Let
fn(x) =x2
x2 + (1− nx)2, 0 ≤ x ≤ 1, n = 1, 2, 3, . . . .
Show that fn is uniformly bounded on [0, 1], limn→∞ fn(x) = 0 for all x ∈ [0, 1], but
fn
(1
n
)= 1, n = 1, 2, 3, . . . ,
so that no subsequence of fn can converge uniformly on [0, 1]. Show that f ′n is un-bounded, and so fn cannot be equicontinuous.
HINT: For the last statement, compute f ′n
(1
n− 1
n2
).
55. Suppose f is a real continuous function on R, fn(x) = f(nx) for n = 1, 2, . . ., and fnis equicontinuous on [0, 1]. What conclusion can you draw about f?
56. Let fn be a uniformly bounded sequence of continuous function on [a, b]. Show thatthe sequence Fn of functions given by
Fn(x) =
∫ x
a
fn(t) dt, a ≤ x ≤ b,
has a subsequence which converges uniformly on [a, b].
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57. Use the Arzela-Ascoli Theorem to prove Peano’s existence theorem: Let the functionf(t, x) be continuous and bounded on the strip defined by 0 ≤ t ≤ 1, −∞ < x < ∞. Thenthere exists at least one continuously differentiable solution of the initial-value problem
x = f(t, x), x(0) = x0
on the interval 0 ≤ t ≤ 1.
HINT: Fix n. For i = 0, . . . , n put ti = i/n. Let φn be a continuous function on 0 ≤ t ≤ 1such that φn(0) = x0,
φn(t) = f(ti, φn(ti)) if ti < t < ti+1,
and put∆n(t) = φn(t)− f(t, φn(t)),
except at the points ti, where ∆n(t) = 0. Then
φn(t) = x0 +
∫ t
0
[f(τ, φn(τ)) + ∆n(τ)] dτ.
Choose M so that f < M . Verify the following assertions:
(i) |φn| ≤ M , |∆n| ≤ 2M , ∆n Riemann integrable, and |φn| ≤ |x0| + M = M1, say, on0 ≤ t ≤ 1, for all n.
(ii) φn is equicontinuous on 0 ≤ t ≤ 1, since |φn| ≤M .
(iii) Some φnk converges to some φ, uniformly on 0 ≤ t ≤ 1.
(iv) Since f is uniformly continuous on the rectangle 0 ≤ t ≤ 1, |x| ≤M1,
f(t, φnk(t))→ f(t, φ(t))
uniformly on 0 ≤ t ≤ 1.
(v) ∆n(t)→ 0 uniformly on 0 ≤ t ≤ 1 since
∆n(t) = f(ti, φn(ti))− f(t, φn(t))
for ti < t < ti+1.
(vi) Hence
φ(t) = x0 +
∫ t
0
f(τ, φ(τ)) dτ.
This φ is the solution of the given problem.
16
58. Consider the space `2 of real sequences an such that∑a2n < ∞. If a = an and
b = bn define their sum to be a + b = an + bn, and if α ∈ R define αa = αan.
(i) Show that `2 is a vector space.
(ii) If a,b ∈ `2, let 〈a,b〉 =∑anbn. Show that |〈a,b〉| <∞ and that 〈a,b〉 defines an inner
product on `2.
(iii) Show that the induced norm and metric are
‖a‖ =√〈a, a〉 =
√√√√ ∞∑n=1
a2n and d(a,b) = ‖a− b‖ =
√√√√ ∞∑n=1
(an − bn)2,
respectively.
(iv) Show that `2 is complete: if an is a Cauchy sequence (of sequences) in `2, that is, if‖an − am‖ → 0 as n,m→∞, then there exists a sequence a ∈ `2 such that ‖an − a‖ → 0.
HINT: Define the limit component-wise. Use ‖am‖ ≤ ‖an − am‖+ ‖an‖ at some opportunemoment.
(v) Show that the closed unit ball a | ‖a‖ ≤ 1 is not compact.
HINT: Consider the sequences en = 0, . . . , 0, 1, 0, . . ., n = 1, 2, . . ., in which 1 is in the n-thspot.
(vi) Show that the Hilbert cube, a | 0 ≤ an ≤ 1/n, is compact.
HINT: Proceed component-wise and mimic the proof of the Arzela-Ascoli theorem.
(vii) Show that the Hilbert cube has no interior points. In other words, it is not a neighbor-hood of any of its points.
(viii) Show that sequences with rational terms are dense in `2, so that `2 is separable.
(ix) Show that the vectors en, defined in (v), form a complete orthonormal set. In otherwords,
〈en, em〉 =
0, m 6= n
1, m = n,
and any sequence a ∈ `2 can be expressed as
a =∞∑n=1
〈a, en〉en.
17
Here, the sum of the series is to be interpreted as the limit in the `2-norm of its partial sums.
The space `2 is a prototypical Hilbert space.
59. Suppose f is a real function on (−∞,∞). Call x a fixed point of f if f(x) = x.
(i) If f is differentiable and f ′(t) 6= 1 for all real t, show that f has at most one fixed point.
HINT: Mean-value theorem.
(ii) Show that the function f defined by
f(t) = t+1
1 + et
has no fixed point although 0 < f ′(t) < 1 for all real t.
(iii) However, if there is a constant A < 1 such that |f ′(t)| ≤ A for all real t, prove that afixed point x of f exists, and that x = limn→∞ xn, where x1 is an arbitrary number and
xn+1 = f(xn)
for n = 1, 2, 3, . . . .
A proof without using the contraction mapping theorem will bring you extra points.
(vi) Show that the process described in (iii) can be visualized by the zig-zag path
(x1, x2)→ (x2, x2)→ (x2, x3)→ (x3, x3)→ (x3, x4)→ · · · .
60. Use the contraction principle to prove Picard’s existence theorem: Let the functionf(t, x) be continuous in t for 0 ≤ t ≤ 1 and let it satisfy the Lipschitz continuity condition|f(t, x) − f(t, y)| < L|x − y| for −∞ < x, y < ∞. Then there exists a unique continuouslydifferentiable solution of the initial-value problem
x = f(t, x), x(0) = x0, (3)
on some interval 0 ≤ t ≤ T with T ≤ 1.
HINT: (i) Show that every continuous solution of the the integral equation
x(t) = x0 +
∫ t
0
f(τ, x(τ)) dτ. (4)
must be continuously differentiable, and so (3) and (4) are equivalent.
18
(ii) Set up the Picard iteration procedure: Let x0(t) ≡ x0, and let
xn+1(t) = x0 +
∫ t
0
f (τ, xn(τ)) dτ ≡ x0 + Axn(t).
Using the Lipschitz continuity condition, derive the estimate
|Ax(t)− Ay(t)| ≤ Lt sup0<τ<t
|x(τ)− y(τ)| ≤ LT sup0<τ<T
|x(τ)− y(τ)|
for 0 ≤ t ≤ T ≤ 1.
(iii) Let C[0, T ] denote the space of continuous functions on the interval [0, T ] with thedistance induced by the norm ‖f‖T = sup0<τ<T |f(t)|. Show that for a sufficiently small T ,the mapping A maps C[0, T ] into itself, and is a contraction.
(iv) Deduce that the integral equation (4) has a unique continuous solution on [0, T ].
61. (i) Let f(x) be piecewise smooth and periodic with period 2π. Let its Fourier expansionbe
f(x) =a0
2+∞∑n=1
(an cosnx+ bn sinnx) . (5)
Show that if f(x) is even, bn = 0, and (5) becomes a Fourier cosine series with
an =2
π
∫ π
0
f(x) cosnx dx. (6)
Likewise, if f(x) is odd, an = 0, and (5) becomes a Fourier sine series with
bn =2
π
∫ π
0
f(x) sinnx dx. (7)
(ii) Let f(x) be a piecewise smooth real function on the interval [0, π]. Show that for0 < x < π,
f(x) =a0
2+∞∑n=1
an cosnx =∞∑n=1
bn sinnx,
with an and bn given by fomulas (6) and (7) respectively. Do these two series convergeoutside of the interval [0, π], and if yes, to what functions?
62. Let f(x) be periodic with period 2L. Show that its Fourier series is
a0
2+∞∑n=1
(an cos
nπx
L+ bn sin
nπx
L
),
19
with
an =1
L
∫ L
−Lf(x) cos
nπx
Ldx, bn =
1
L
∫ L
−Lf(x) sin
nπx
Ldx.
63. (i) Show the complex counterpart of the orthogonality relation for trigonometric func-tions: ∫ π
−πeinxe−imx dx =
0, m 6= n,
2π, m = n.
(ii) Show that the complex form of the Fourier series for a piecewise-smooth, 2π-periodic,real function f(x) is
f(x) =∞∑
n=−∞
cneinx, cn =
1
2π
∫ π
−πf(x)e−inx dx, c−n = cn,
where the overbar denotes the complex conjugate.
(iii) If the real form of the Fourier series for f(x) is (5), what is the connection between thetwo sets of coefficients an, bn and cn?
64. Differentiation of Fourier Series: Let f(x) be 2π-periodic. Let it also have continu-ous derivatives up to order k and a piecewise continuous derivative of order k + 1.
(i) Show that there exists a constant B, depending only on f and k, such that the Fouriercoefficients of f satisfy
|an|, |bn| <B
nk+1.
HINT: If cn is the n-the complex Fourier coefficient of f , then integrate by parts to show
2πcn =
(−in
)k+1 ∫ π
−πf (k+1)(x)e−inx dx.
(ii) Use the (i) to conclude that for k > 2 the Fourier series for f(x) can be differentiatedk − 1 times and yields the Fourier series for the differentiated function.
65. Let x not be an integer, and let f(t) = cosxt for −π < t < π. Extend f(t) periodicallyin t outside the interval −π < t < π, and then expand it in a Fourier series.
(i) Show that this series is
cosxt =2x sinπx
π
(1
2x2− cos t
x2 − 12+
cos 2t
x2 − 22− cos 3t
x2 − 32+ · · ·
).
20
Convince yourself that this series represents a continuous function near t = ±π. Settingt = π thus conclude that
cot πx =2x
π
(1
2x2+
1
x2 − 12+
1
x2 − 22+
1
x2 − 32+ · · ·
).
This is the so-called partial fraction decomposition of the cotangent.
(ii) Let 0 ≤ x ≤ q < 1 for some q. Write the above formula as
cotπx− 1
πx= −2x
π
(1
12 − x2+
1
22 − x2+
1
32 − x2+ · · ·
).
Using the Weierstrass M-test, convince yourself that the series on the right-hand side con-verges uniformly. Integrate term-by-term between 0 and x to conclude that
logsin πx
πx=∞∑n=1
log
(1− x2
n2
).
Interpret the series on the right-hand side as the limit of its partial sums, and invoke thecontinuity of the exponential function to show that
sin πx
πx= lim
N→∞
N∏n=1
(1− x2
n2
)=∞∏n=1
(1− x2
n2
).
Therefore,
sin πx = πx∞∏n=1
(1− x2
n2
).
This is the infinite product expansion of the sine function. It can be shown that it is notonly valid for 0 ≤ x < 1, by for all complex x.
66. Use Fourier analysis to derive the formal solution of the heat equation
∂u
∂t= α2∂
2u
∂x2, 0 < x < π, t > 0, (8)
with the boundary conditions
u(0, t) = 0, u(π, t) = 0, t > 0, (9)
and the initial conditionu(x, 0) = f(x), 0 < x < π. (10)
21
HINT: (i) Assume solutions of the form u(x, t) = X(x)T (t), substitute into (8), and divideby X(x)T (t). Argue that a function of x can only be equal to a function of t for all x and tif both functions are equal to the same constant. In this way, derive the two equations
X ′′(x) + λX(x) = 0, T (t) + α2λT (t) = 0, λ = const.
(ii) Show that the boundary conditions (9) translate into the boundary conditions X(0) =X(π) = 0.
(iii) Show that all possible nonzero solutions X(x) are proportional to Xn(x) = sinnx,n = 1, 2, 3, . . ., with the corresponding λn = n2.
(iv) Show that the corresponding nonzero solutions T (t) are proportional to Tn(t) = e−n2α2t.
(v) Conclude that un(x, t) = e−n2α2t sinnx satisfy both the heat equation (8) and the bound-
ary conditions (9), and, since both are homogeneous, so does any sum∑
n cne−n2α2t sinnx.
(vi) At t = 0, the initial condition (10) becomes f(x) =∑
n cn sinnx. Compute cn to findthe solution u(x, t) of the original problem.
(vii) Let f(x) be piecewise smooth on 0 < x < π. By comparing it to the series∑e−n
2α2ε andits derivatives, show that the series solution u(x, t) you just obtained converges absolutelyand uniformly for 0 < x < π and t ≥ ε, for any ε > 0. Show that it can be differentiated anarbitrary number of times, and is therefore a true solution of the heat equation (and satisfiesthe boundary conditions).
(viii) If f(x) is continuous and piecewise smooth on 0 < x < π, and also f(0) = f(π) = 0,show that u(x, t→ 0)→ f(x). Therefore, u(x, t) also truly satisfies the initial condition.
HINT: Derive the estimate |f(x) − u(x, t)| ≤∑∞
n=1(1 − e−n2α2t)|cn| ≤∑∞
n=1 |cn|, then usethe result of problem 64 (i) to show that the last series converges. Show that this impliesuniform convergence of the first series, so that limt→0
∑∞n=1 =
∑∞n=1 limt→0 in that series.
67. (i) Derive the formal solution of the heat equation
∂u
∂t= α2∂
2u
∂x2, 0 < x < π, t > 0,
with the boundary conditions
∂u
∂x(0, t) = 0,
∂u
∂x(π, t) = 0, t > 0,
and the initial conditionu(x, 0) = f(x), 0 < x < π.
22
(ii) If f(x) is piecewise smooth on 0 < x < π, show that the formal solution u(x, t) is a truesolution of the heat equation. Moreover, if f(x) is also continuous, show that it also trulysatisfies the initial condition.
HINT: Proceed along the lines of 66 (vii) and (viii).
68. (i) Derive the formal solution of the wave equation
∂2u
∂x2=
1
c2
∂2u
∂t2, 0 < x < π, t > 0,
with the boundary conditions
u(0, t) = 0, u(π, t) = 0, t > 0,
and the initial condition
u(x, 0) = f(x),∂u
∂t(x, 0) = g(x), 0 < x < π.
(ii) Try to obtain a rigorous justification for the formal Fourier series solution u(x, t) as inparts (vii) and (viii) of problem 66. Just how smooth must f(x) be for this justification tosucceed?
69. (i) Derive the formal solution of Laplace’s equation on a rectangle
∂2u
∂x2+∂2u
∂y2= 0, 0 < x < a, 0 < y < b,
with the boundary values
u(x, 0) = 0, u(x, b) = 0, 0 < x < a,
u(0, y) = 0, u(a, y) = f(y), 0 < y < b.
HINT: The solution is
u(x, y) =∞∑n=1
cn sinhnπx
bsin
nπy
b,
where
cn =2
b
(sinh
nπa
b
)−1∫ b
0
f(y) sinnπy
bdy.
(ii) Obtain a rigorous justification for the formal Fourier series solution u(x, t) as in parts(vii) and (viii) of problem 66. State precisely the smoothness assumptions on the functionf(y).
23
(iii) Write down the solution of Laplace’s equation
∂2u
∂x2+∂2u
∂y2= 0, 0 < x < a, 0 < y < b,
with the boundary values
u(x, 0) = f1(x), u(x, b) = f2(x), 0 < x < a,
u(0, y) = f3(y), u(a, y) = f4(y), 0 < y < b.
70. Derive the formal solution of Laplace’s equation on a circle
∂2u
∂x2+∂2u
∂y2= 0, 0 ≤ x2 + y2 = r2 < a2,
with the boundary valuesu(r = a, θ) = f(θ).
HINT: (i) In polar cordinates, x = r cos θ, y = r sin θ, Laplace’s equation becomes
∂2u
∂r2+
1
r
∂u
∂r+
1
r2
∂2u
∂θ2= 0.
(ii) Show that the boundary conditions in polar coordinates become
u(a, θ) = f(a), u(0, θ) is bounded, u(r, θ + 2π) = u(r, θ).
(iii) After separating variables u(r, θ) = R(r)Θ(θ), show that the resulting equations become
r2R′′(r) + rR′(r)− λR(r) = 0, Θ′′(θ) + λΘ(θ) = 0,
with the conditionsR(0) is bounded, Θ(θ + 2π) = Θ(θ).
(iv) Show that the eventual solution has the form
u(r, θ) =c0
2+∞∑n=1
rn (cn cosnθ + dn sinnθ) .
What are the expressions for the coefficients cn and dn?
24
71. Derive the formal solution of Laplace’s equation on an annulus
∂2u
∂x2+∂2u
∂y2= 0, a ≤ r < b,
with the boundary values
u(a, θ) = f(θ), u(b, θ) = g(θ).
HINT: The solution has the form
u(r, θ) =c0 + e0 log r
2+∞∑n=1
rn (cn cosnθ + dn sinnθ)
+∞∑n=1
1
rn(en cosnθ + fn sinnθ) .
72. (i) Draw the graph of the function t(u) = u− ε sinu for 0 ≤ ε ≤ 1, and convince yourselffrom the graph, as well as analytically, that the following statements are true:
(a) t(u) is odd, monotonically increasing, and continuously differentiable for all real u,
(b) t(u+ 2π) = t(u) + 2π,
(ii) Consider the equationt = u− ε sinu (11)
(a) Given ε ≤ 1, show that there is a unique function u(t) that solves equation (11), which isodd and monotonically increasing for all u. If ε < 1 show that this function is continuouslydifferentiable for all t, and if ε = 1, it is continuously differentiable at t 6= 2nπ with integern. (What happens at t = 2nπ with integer n?)
(b) Use part (b) of (i) to show that u(t) can be written as u(t) = t+ f(t), where f(t+ 2π) =f(t) for all real t.
HINT: To show (a), use monotonicity and continuous differentiablity of t(u).
(iii) If 0 ≤ ε < 1, show that the function f(t) can be expanded in a Fourier sine series,
f(t) =∞∑n=1
cn(ε) sinnt,
where
cn(ε) =2
π
∫ π
0
f(t) sinnt dt. (12)
25
Calculate cn(ε) in the following way: Write f(t) = u(t) − t, and split the integrand in (12)into a sum of two terms, the first of which is
dn(ε) =2
π
∫ π
0
u(t) sinnt dt.
You will be able to calculate the second term easily, but for the first term, integrate by parts,use periodicity of u(t), and make a substitution t = t(u) to get
dn(ε) =2(−1)n+1
n+
2
nπ
∫ π
0
cosnt u′(t) dt
=2(−1)n+1
n+
2
nπ
∫ π
0
cosnt(u) du
=2(−1)n+1
n+
2
nπ
∫ π
0
cosn(u− ε sinu) du.
(Justify all your steps.) This integral involves a standard special function, called the Besselfunction Jn(x), given by the formula
Jn(x) =2
π
∫ π
0
cos(nu− x sinu) du.
Thus, express cn(ε) in terms of algebraic expressions and a Bessel function, and u(t) as thesum of t and a Fourier series in t with coefficients involving Bessel functions.
73. Integration of Fourier Series: Show that if f(x) is a piecewise continuous functionin −π ≤ x ≤ π having the formal Fourier expansion
a0
2+∞∑n=1
(an cosnx+ bn sinnx) ,
then for any two points x1 and x2,∫ x2
x1
f(x) dx =
∫ x2
x1
a0
2dx+
∞∑n=1
∫ x2
x1
(an cosnx+ bn sinnx) dx,
that is, the formal Fourier series can be integrated termwise. Moreover, the series on theright converges uniformly in x2 for fixed x1.
HINT: The function
F (x) =
∫ x
−π
[f(t)− a0
2
]dt
is continuous and piecewise smooth. Compute its Fourier coefficients and compare themwith those of the termwise-integrated series.
26
74. Show that ∫ ∞0
sinx
xdx =
π
2
by completing the following outline:
(i) For any nonnegative integer n, let
an =
∫ (n+1)π
nπ
sinx
xdx.
Show that a2m > 0 and a2m+1 < 0 for every nonnegative integer m. By introducing thesubstitution x = t + π in the integral for an+1, show that |an+1| < |an|, therefore |an| isa monotonically decreasing sequence. Introducing the substitution x = t + nπ, show thatan → 0 as n → ∞, and therefore that the series
∑∞n=0 an converges. Now let nπ ≤ A <
(n+ 1)π. Then ∫ A
0
sinx
xdx =
∫ nπ
0
sinx
xdx+
∫ A
nπ
sinx
xdx.
Introduce the substitution x = t+nπ to show that the last integral tends to zero as A→∞,and deduce that the integral ∫ ∞
0
sinx
xdx
converges.
(ii) Integrate by parts to show that
limλ→∞
∫ a
0
f(x) sinλx dx = 0
for any continuously differentiable function f(x) on [0, a]. (We showed this in class forpiecewise continuous functions!) Thus
limλ→∞
∫ π
0
sinλx
(1
x− 1
2 sin x2
)dx = 0.
(iii) Integrate formulas (1) and (2) to show that∫ π
0
sin(n+ 1
2
)x
2 sin x2
dx =π
2.
(iv) Show that ∫ a
0
sinλx
xdx =
∫ λa
0
sinx
xdx.
27
Let a = π and λ = n+ 12
to deduce that∫ ∞0
sinx
xdx =
π
2.
75. (i) Show that, for 0 < x < π,
π − x2
=∞∑n=1
sinnx
n≡ φ(x).
What is φ(x) for values of x outside this interval?
(ii) Show that φ(x) has a jump discontinuity at x = 0. What is its size? What is φ(0)?
(iii) Integrate formulas (1) and (2) to show that
Sn(x) =n∑k=1
sin kx
k= −x
2+
∫ x
0
sin(n+ 1
2
)t
2 sin 12t
dt.
(iv) Show that φ(x)− Sn(x) = σn(x) + ρn(x), with
σn(x) =π
2−∫ x
0
sin(n+ 1
2
)t
tdt, ρn(x) =
∫ x
0
2 sin 12t− t
2 sin 12t
sin(n+ 1
2
)t dt.
(v) Show that ρn(x)→ 0 uniformly on 0 < x < π as n→∞.
(vi) Show that
σn(x) =π
2−∫ (n+ 1
2)x
0
sin t
tdt.
Since, by problem 74, ∫ ∞0
sin t
tdt =
π
2,
σn(x)→ 0 as n→∞ for fixed x.
(vii) Show that σn(x) has extrema at the points xk = 2kπ/(2n + 1), for k = 1, 2, 3, . . .,minima at x1, x3, x5, . . . and maxima at x2, x4, . . . .
(viii) Show that the values σn(x2k+1) at the minima form an increasing sequence, and that
28
thus the biggest oscillation of σn(x) is at x1. Show that
σn(x1) =π
2−∫ π
0
sin t
tdt
= π
(1
2− 1 +
π2
2 · 3 · 3− π4
2 · 3 · 4 · 5 · 5
+π6
2 · 3 · 4 · 5 · 6 · 7 · 7
)≈ −0.090 · · · π.
This overshoot is called the Gibbs phenomenon.
76. Suppose f is a continuous function on R, f(x+ 2π) = f(x), and α/π is irrational. Showthat
limN→∞
1
N
N∑n=1
f(x+ nα) =1
2π
∫ π
−πf(t) dt.
HINT: Do it first for f(x) = eikx, k integer. Then use the Weierstrass approximation theoremfor trigonometric functions.
77. (i) Show that
〈f, g〉 =1
b− a
∫ b
a
f(x)g(x) dx
is an inner product on the space of piecewise continuous, complex-valued functions. Whatis the corresponding induced norm ‖f‖2?
(ii) Let f(x) be a piecewise continuous function and let αn be its complex Fourier coefficients.For any set of complex numbers βn, n = −N, . . . , N , show that
1
2π
∫ π
−π
∣∣∣∣∣f(x)−N∑
n=−N
βneinx
∣∣∣∣∣2
dx = ‖f‖22 −
N∑n=−N
|αn|2 +N∑
n=−N
|αn − βn|2.
Conclude that its N -th order Fourier partial sum minimizes the distance in the ‖ · ‖2 normbetween f and N -th order trigonometric polynomials.
(iii) Prove Parseval’s equality: For any continuous, 2π-periodic real function f(x), if an, bnare its Fourier coefficients, show that
1
2a2
0 +∞∑n=1
(a2n + b2
n
)= ‖f‖2
2.
29
In the process, also prove that if Sn(x) is the n-th Fourier partial sum of the function f(x),
limn→∞
‖f − Sn‖22 → 0.
In other words, the Fourier series converges to f in the ‖ · ‖2 norm.
HINT: By the Weierstrass approximation theorem for trigonometric polynomials (How didwe prove this theorem in class?), there exist trigonometric polynomials Tn(x) such thatf(x)− Tn(x)→ 0 uniformly in x. Use this fact and (ii).
(iv) Show that Parseval’s equality and the convergence of the Fourier series in the ‖·‖2 normremain valid if f has a finite number of jump discontinuities.
HINT: Put a sufficiently steep straight line through each discontinuity to approximate fwith a continuous function in the ‖ · ‖2 norm.
78. (i) Let f(x) be piecewise continuous and 2π-periodic, and let an, bn be its Fouriercoefficients. Use Parseval’s equality to show that the mapping
f →a0√
2, a1, b1, a2, b2, . . .
defines an isometry (distance-preserving mapping) from the space of piecewise continuousfunctions equipped with the norm ‖.‖2 into the Hilbert space `2, discussed in problem 58.
(ii) Show that this mapping is 1-1. Speculate whether it is onto or not.
79. If f is continuous on [0, 1] and if∫ 1
0
f(x)xn dx = 0, n = 0, 1, . . . ,
show that f(x) = 0 on [0, 1].
HINT: The integral of the product of f with any polynomial is zero. Use the Weierstrasstheorem to show that ∫ 1
0
f 2(x) dx = 0.
80. Following the outline below, provide another proof of the Weierstrass approximationtheorem: For every continuous function f(x) on [a, b] and every ε > 0, there exists apolynomial P (x) such that |f(x)− P (x)| < ε for all x ∈ [a, b].
30
(i) Show that, with no loss of generality, [a, b] = [0, 1], and f(0) = f(1) = 0.
Define f(x) = 0 for x outside [0, 1].
(ii) Let g(x) = (1− x2)n − 1 + nx2. Show that g(0) = 0 and g′(x) > 0 on (0, 1) to concludethat g(x) ≥ 0 on [0, 1]. Conclude that∫ 1
−1
(1− x2)ndx ≥ 2
∫ 1/√n
0
(1− nx2) dx ≥ 1√n.
(iii) Let
Qn(x) = cn(1− x2)n, cn =
(∫ 1
−1
(1− x2)ndx
)−1
, n = 1, 2, . . . .
Given δ > 0, show that Qn(x)→ 0 uniformly on δ ≤ |x| ≤ 1.
HINT: Qn(x) ≤√n(1− δ2)n there.
(iv) Let
Pn(x) =
∫ 1
−1
f(x+ t)Qn(t) dt, 0 ≤ x ≤ 1.
Change the integration variable t→ t− x to show that Pn(x) is a polynomial.
HINT: f(x) = 0 for x outside [0, 1].
(v) Given ε > 0, estimate
|Pn(x)− f(x)| =∣∣∣∣∫ 1
−1
[f(x+ t)− f(x)]Qn(t) dt
∣∣∣∣to show it is < ε.
HINT: Use∫ 1
−1Qn(t) dt = 1, then consider the integral on three intervals [−1,−δ], [−δ, δ],
and [1, δ]. Use (iii) for the outer two intervals, and uniform continuity of f on the middleinterval.
REMARK: Reading pages 296 through 307 of the Strichartz book will be very illuminating.
81. Following the outline below, provide yet another proof of the Weierstrass approxima-tion theorem: For every continuous function f(x) on [0, 1] and every ε > 0, there exists apolynomial Bn(x) such that |f(x)−Bn(x)| < ε for all x ∈ [0, 1]. In fact, one can take
Bn(x) =n∑k=0
(nk
)xk(1− x)n−k f
(k
n
)(13)
31
for some large enough n.
(i) Show thatn∑k=0
(nk
)xk(1− x)n−k = 1. (14)
Differentiate (14) on x and multiply by x(1 − x); differentiate again and use (14); finallymultiply by x(1− x)/n2. You should obtain
n∑k=0
(nk
)xk(1− x)n−k
(x− k
n
)2
=x(1− x)
n. (15)
(ii) Use (14) to show
f(x)−Bn(x) =n∑k=0
(nk
)xk(1− x)n−k
[f(x)− f
(k
n
)]. (16)
Given ε > 0, argue that there exists a δ > 0 such that∣∣∣∣f(x)− f(k
n
)∣∣∣∣ < ε
2
if ∣∣∣∣x− k
n
∣∣∣∣ < δ.
Taking appropriate absolute values, estimate the size of (16) by two sums,∑
1 and∑
2. Thesum
∑1 runs over all the terms for which∣∣∣∣x− k
n
∣∣∣∣ < δ.
Show that∑
1 < ε/2.
Now show that∑
2 can be made less than ε/2 as follows: Let K = max |f(x)| on [0, 1]. Then∑2≤ 2K
∑(nk
)xk(1− x)n−k ≡ 2K
∑3,
where∑
3 is taken over all k such that ∣∣∣∣x− k
n
∣∣∣∣ ≥ δ.
Use (15) to show that ∑3≤ x(1− x)
δ2n,
32
and conclude that for n large enough∑
3 < ε/4K. This should let you finish the proof.
82. Let K be the unit circle in the complex plane (i.e., the set of all z with |z| = 1) and letA be the algebra of all functions of the form
f(eiθ) =N∑n=0
cneinθ, θ ∈ R.
Show that A vanishes nowhere and separates points, yet there are continuous functions onK which are not in the uniform closure of A. What are those functions?
HINT: What is∫ 2π
0f(eiθ)eiθ dθ?
83. (i) If f(0, 0) = 0 and
f(x, y) =xy
x2 + y2, (x, y) 6= (0, 0)
show that D1f(x, y) and D2f(x, y) exist at every point of R2, although f is not continuousat (0, 0).
(ii) If f is a real-valued function defined in an open set E ⊂ R2, and if the partial derivativesD1f and D2f are bounded in E, then f is continuous.
HINT: f(x+ h, y + k)− f(x, y) = f(x+ h, y + k)− f(x+ h, y) + f(x+ h, y)− f(x, y).
84. If f and g are differentiable real functions in Rn, show that
∇(fg) = f∇g + g∇f,
and that
∇(
1
f
)= − 1
f 2∇f
whenever f 6= 0.
85. Suppose f is a differentiable mapping of R1 into Rn such that ‖f(t)‖ = 1 for every t.Show that f ′(t) · f(t) = 0 for every t. Interpret this result geometrically for n = 2, 3.
86. Define f(0, 0) = 0 and
f(x, y) =x3
x2 + y2, (x, y) 6= (0, 0).
33
(i) Show that D1f and D2f are bounded functions in R2. (Hence f is continuous by problem83 (ii).)
(ii) Let u be any unit vector in R2. Show that the directional derivative Duf(0, 0) exists,and that its absolute value is at most 1.
(iii) Let γ be a differentiable mapping of R1 into R2 (in other words, γ is a differentiablecurve in R2), with γ(0) = (0, 0) and ‖γ′(0)‖ > 0. Put g(t) = f(γ(t)) and show that g isdifferentiable for every t ∈ R1. If γ is continuously differentiable, show that so is g.
HINT: Dividing the tops and bottoms of fractions by some appropriate power of t will helpin the limit as t→ 0.
(iv) Despite of this, show that Duf(0, 0) 6= D1f(0, 0)u1 + D2f(0, 0)u2, so that f is notdifferentiable at (0, 0).
87. Define f(0, 0) = 0 and
f(x, y) =xy (x2 − y2)
x2 + y2, (x, y) 6= (0, 0).
Show that
(i) f , D1f , and D2f are continuous in R2.
(ii) D12f and D21f exist at every point in R2, and are continuous except at (0, 0).
(iii) D12f(0, 0) = 1 and D21f(0, 0) = −1.
88. (i) Let f and g be twice continuously differentiable. Show that the function u(x, t) =f(x− ct) + g(x+ ct) solves the wave equation
∂2u
∂x2=
1
c2
∂2u
∂t2.
(ii) Let φ be twice and ψ be once continuously differentiable. Show that the solution of thewave equation satisfying the initial conditions
u(x, 0) = φ(x),∂u
∂t(x, 0) = ψ(x),
is
u(x, t) =1
2
[φ(x− ct) + φ(x+ ct)
]+
1
2c
∫ x+ct
x−ctψ(s) ds.
34
89. A smooth function f : Rn → R is called homogeneous of degree h if f(tx) = thf(x) forevery t ∈ R and x ∈ Rn. Show that f is homogeneous of degree h if and only if it satisfiesthe differential equation x · ∇f(x) = hf(x).
HINT: To show the only if part, derive a differential equation for the function g(t) = f(tx)−thf(x).
90. Let f = (x, y, z) : A → R3, with A ⊂ R2, be continuously differentiable, i.e., a smoothparametrization of a surface. Let Ij, j = 1, 2, be two intervals, and let γj : Ij → A be twosmooth curves in A.
(i) At any point in A where the curves γ1 and γ2 cross, i.e., γ1(t) = γ2(s), show that theangle θ between the two image curves f (γ1(t)) and f (γ2(s)) in R3 is given by the formula
cos θ =γ′1(t) ·M(u, v)γ′2(s)√
γ′1(t) ·M(u, v)γ′1(t)√γ′2(s) ·M(u, v)γ′2(s)
,
where (u, v) are the coordinates in A, and M(u, v) is the matrix
M(u, v) =
(E FF G
),
with
E =
(∂x
∂u
)2
+
(∂y
∂u
)2
+
(∂z
∂u
)2
= Duf ·Duf ,
F =∂x
∂u
∂x
∂v+∂y
∂u
∂y
∂v+∂z
∂u
∂z
∂v= Duf ·Dvf ,
G =
(∂x
∂v
)2
+
(∂y
∂v
)2
+
(∂z
∂v
)2
= Dvf ·Dvf .
(ii) A continuously differentiable planar map
x = φ(u, v), y = ψ(u, v)
is called conformal if it maps two intersecting curves into two others enclosing the same angleas the original ones. Show that the necessary and sufficient condition that a planar map isconformal is is that the Cauchy-Riemann equations
∂φ
∂u− ∂ψ
∂v= 0,
∂φ
∂v+∂ψ
∂u= 0
or∂φ
∂u+∂ψ
∂v= 0,
∂φ
∂v− ∂ψ
∂u= 0
35
hold.
Use the result of problem 91 below to show that in the first case the direction of the anglesis preserved, while in the second case it is reversed.
HINT: Adapt part (i) to planar maps. If (u, v)→ (x, y) is conformal, it must map orthogonalcurves into orthogonal curves. Choose a pair of straight lines parallel to the (u, v) coordinateaxes and the same pair rotated by π/4 to show that F = E −G = 0, and infer the Cauchy-Riemann equations. The converse is straight forward.
91. Let f : A → R2, with A ⊂ R2, be given in components as x = φ(u, v) and y = ψ(u, v).Show that f preserves or reverses orientation, depending on whether the Jacobian
det f ′(u, v) =∂(φ, ψ)
∂(u, v)
is positive or negative, by carrying out the following outline:
(i) Let γ(t) = (u(t), v(t)) be a curve in A. Argue that its slope is m(t) = v′(t)/u′(t).
(ii) Show that the slope of the curve f (γ(t)) is given by
µ(t) =c+ dm
a+ bm,
where the quantities a, b, c and d are the partial derivatives of the function f , that is,
f ′(u, v) =
(a bc d
).
(iii) Compute dµ/dm to show that µ increases or decreases with m depending on whetherdet f ′ is positive or negative. Argue that this implies the counterclockwise or clockwiserotation of the curve f (γ(t)) if the curve γ(t) is rotated counterclockwise, which is thepreservation or reversal of orientation.
92. (i) Let φij(t), i, j = 1, . . . , n, be continuously differentiable, and let W (t) be the deter-minant
W (t) =
∣∣∣∣∣∣∣φ11(t) . . . φ1n(t)
.... . .
...φn1(t) . . . φnn(t)
∣∣∣∣∣∣∣ .
36
Show that
W ′(t) =n∑i=1
∣∣∣∣∣∣∣∣∣∣∣
φ11(t) . . . φ1n(t)...
. . ....
φ′i1(t) . . . φ′in(t)...
. . ....
φn1(t) . . . φnn(t)
∣∣∣∣∣∣∣∣∣∣∣.
HINT: First use known facts from linear algebra to show that
∂W
∂φij= (−1)i+j+1Wij,
where Wij is the cofactor of the element φij, i.e., the determinant obtained by erasing thei-th row and j-th column from the determinant W .
(ii) Let A(t) be and n× n matrix with continuous entries aij(t). Let the n× n matrix Φ(t)with entries φij(t) be a solution of the matrix differential equation Φ′(t) = A(t)Φ(t), i.e., eachcolumn of Φ(t) solves the linear system x′ = A(t)x. Use (i) to show that W (t) = det Φ(t)satisfies the equation
W (t) = W (t0) exp
(∫ t
t0
traceA(s) ds
), traceA =
n∑i=1
aii.
HINT: Derive a differential equation for W (t).
93. (i) If f is a differentiable mapping of a connected open set E ⊂ Rn into Rm, and iff ′(x) = 0 for every x ∈ E, show that f is constant in E.
HINT: Apply the mean-value theorem to the appropriate directional derivative to show thatif f(x0) = c at some point x = x0, then f(x) = c in some open ball Br(x0), so that the set ofpoints on which f(x) = c is open. On the other hand, show that the set of points on whichf(x) 6= c must also be open, and thus empty. (Why?)
(ii) A subset of Rn is called convex if, together with every pair of its points, it also containsthe straight line connecting them.
If f is a real function defined in a convex open set E ⊂ Rn, such that D1f(x) = 0 for everyx ∈ E, show that f(x) depends only on x2, . . . , xn.
Show that the convexity of E can be replaced by a weaker condition, but that some conditionis required. For example, if n = 2 and E is shaped like a horseshoe, the statement may befalse.
37
HINT: The condition is that points in E with the same coordinate x1 be connected by astraight line contained in E.
94. Define f(0, 0) = 0 and
f(x, y) = x2 + y2 − 2x2y − 4x6y2
(x4 + y2)2 , (x, y) 6= (0, 0).
(i) Show, for all (x, y) ∈ R2, that
4x4y2 ≤(x4 + y2
)2.
Conclude that f is continuous.
(ii) For 0 ≤ θ ≤ 2π, −∞ < t <∞, define
gθ(t) = f(t cos θ, t sin θ).
Show that gθ(0) = 0, g′θ(0) = 0, g′′θ (0) = 2. Each gθ has therefore a strict local minimumat t = 0. (In other words, the restriction of f to each line through (0, 0) has a strict localminimum at (0, 0).)
(iii) Show that (0, 0) is nevertheless not a local minimum for f , since f(x, x2) = −x4.
95. (i) Let f(r) = 1/r, with r = (x, y, z) and r = ‖r‖ =√x2 + y2 + z2. Find the four-term
(including the remainder) Taylor expansion of the function f(r − h), with h = (h1, h2, h2)for small values of ‖h‖. Show that the remainder can be bounded by C‖h‖3/r4 for someappropriate constant C. Conclude that, up to a scale difference, the small ‖h‖ expansiongives the same result as the large ‖r‖ expansion.
HINT: For sufficiently small ‖h‖/r, we have ‖r− θh‖ ≥ r/2 for any 0 ≤ θ ≤ 1.
(ii) Let ej, j = 1, . . . , N , be electrostatic charges forming a neutral charge cloud,∑N
j=1 ej = 0.Let the charge ej be fixed at the position rj = (xj, yj, zj). Using the result of (i), show thatthe electrostatic potential U(r) produced by these charges is given by the formula
U(r) ≡N∑j=1
ej‖r− rj‖
=p · rr3
+r ·Qr
2r5+O
(R3
r4
),
where R = maxj=1,...,N
rj,
p =N∑j=1
ejrj,
38
and
Q =N∑j=1
ej(3rj ⊗ rj − r2
j
).
Here r⊗ r is the “tensor product,” i.e., the matrix product rrT , with the r being a columnand rTa row vector.
REMARK: The vector p is known as the dipole moment of the charge cloud, and Q as thequadrupole matrix.
96. Fix two real numbers, 0 < a < b. Define a mapping f = (f1, f2, f3) of R2 into R3 by
f1(s, t) = (b+ a cos s) cos t
f2(s, t) = (b+ a cos s) sin t
f3(s, t) = a sin s.
(i) Show that the range K of this mapping is a torus in R3.
(ii) Show that there are exactly four points on this torus for which ∇f1 vanishes. Find thesepoints, and show that one corresponds to a local maximum of f1, one to a local minimum,and two to saddles.
(iii) Determine the set of points on the torus for which ∇f3 vanishes. Which of these pointscorrespond to maxima, minima, or saddles?
97. Show that the continuity of f ′ at the point a is needed in the inverse function theorem,even in the case n = 1: If
f(t) = t+ 2t2 sin
(1
t
)for t 6= 0 and f(0) = 0, then f ′(0) = 1, f ′ is bounded in (−1, 1), but f is not one-to-one inany neighborhood of 0.
98. Let f = (f1, f2) be the mapping R2 into R2 given by
f1(x, y) = ex cos y, f2(x, y) = ex sin y.
(i) What is the range of f?
(ii) Show that the Jacobian of f is nonzero at any point of R2. Thus every point of R2 hasa neighborhood in which f is one-to-one. Nevertheless, f is not one-to-one on R2.
39
(iii) Put a = (0, π/3), b = f(a), and let g be the continuous inverse of f , defined in aneighborhood of b, such that g(b) = a. Find an explicit formula for g, compute f ′(a) andg′(b), and verify that they are each other’s inverses.
(iv) What are the images under f of lines parallel to the coordinate axes?
99. Show that the system of equations
3x+ y − z + u2 = 0
x− y + 2z + u = 0
2x+ 2y − 3z + 2u = 0
can be solved for x, y, u, in terms of z; for x, z, u, in terms of y; for y, z, u, in terms of x;but not for x, y, z, in terms of u.
100. Define f in R2 byf(x, y) = 2x3 − 3x2 + 2y3 + 3y2.
(i) Find the four points at which the gradient of f is zero. Show that f has exactly one localmaximum and one local minimum in R2.
(ii) Let S be the set of all (x, y) ∈ R2 at which f(x, y) = 0. Show that S is the union of astraight line and an ellipse. Find those points of S that have no neighborhoods in which theequation f(x, y) = 0 can be solved for y in terms of x or x in terms of y.
HINT: Rewritef(x, y) = (x+ y)(2x2 − 2xy + 2y2 − 3x+ 3y).
Diagonalize the quadratic form in the second factor and complete a square to find the ellipse.
101. Define f in R3 byf(x, y1, y2) = x2y1 + ex + y2.
Show that f(0, 1,−1) = 0, D1f(0, 1,−1) 6= 0, and that there exists therefore a differentiablefunction g in some neighborhood of (1,−1) in R2 such that g(1,−1) = 0 and
f(g(y1, y2), y1, y2) = 0.
Find D1g(1,−1) and D2g(1,−1).
102. By following the outline below, prove the one-dimensional version of the implicitfunction theorem: Let F : A → R, with A ⊂ R2, be continuously differentiable. Letthe equation F (x0, y0) = 0 hold at some point (x0, y0), and let DyF (x0, y0) 6= 0. Then
40
there exists an open interval x1 < x < x2 containing x0 on which the equation F (x, y) = 0defines a unique function y = f(x) with f(x0) = y0 and F (x, f(x)) = 0. Moreover, f(x) iscontinuously differentiable on x1 < x < x2, with
f ′(x) = −DxF (x, f(x))
DyF (x, f(x)).
(i) Assume DyF (x0, y0) > 0 with no loss of generality. (Why?) Using the continuousdifferentiability of F , conclude that, for all x and y in some rectangle x1 < x < x2, y1 < y <y2 around (x0, y0), the function F (x, y) increases monotonically in y along every line x =constant. Use F (x0, y0) = 0 to show that F (x, y1) < 0 and F (x, y2) > 0 on x1 < x < x2. Inferthat for every x in x1 < x < x2, there is a unique value y = f(x) such that F (x, f(x)) = 0.
(ii) Let x and x + h be two points in x1 < x < x2, let y = f(x) and y + k = f(x + h). Usethe two term Taylor formula to show that
k
h= −DxF (x+ θh, y + θk)
DyF (x+ θh, y + θk). (17)
Bound the right-hand side of this equation and conclude that |k| < C|h| for some constantC, and therefore f(x) is continuous.
(iii) Use (17) again to show that
limh→0
f(x+ h)− f(x)
h= −DxF (x, f(x))
DyF (x, f(x)).
Conclude that f ′(x) exists and is continuous.
103. (i) Let r = (x, y, z), and let g : A→ R3, with A ⊂ R2 open, be a smooth parametrizationof a surface S, i.e., rank (g′) = 2. Show that the tangent plane Tg(u,v)S of the surface Sat the point g(u, v) is spanned by the vectors gu and gv. (Here the subscripts denote thepartial derivatives on the variable in the subscript.) Find a normal to S at g(u, v), and showthat the equation of the tangent plane is
gu × gv · [r− g(u, v)] = 0.
(ii) Repeat the discussion of (i) for the explicit parametrization of the surface S given byg(x, y) = (x, y, f(x, y)). Find the explicit expressions all the vectors involved in terms of fxand fy.
(iii) Find the equation of the tangent plane and the normal at any point of the surface Sdescribed implicitly by F (x, y, z) = 0.
41
(iv) Following (i), compute two basis vectors of the tangent plane, the equation of the tangentplane, and a normal at any point of the torus given in problem 96.
104. Let f : Mm → R be a C1 function, where Mm ⊂ Rn is a C1 surface. Show that everypoint in Mm lies in a neighborhood U ⊂ Rn such that there exists a C1 function F : U → Rthat extends f , i.e., F (y) = f(y) for y in Mm ∩ U .
HINT: Use the local explicit (graph) representation of the surface Mm.
105. Let M2 be any compact two-dimensional surface in R3. Show that for any two dimen-sional vector subspace V in R3, there exists a point x on M2 whose tangent space equals V .
HINT: If u is a vector perpendicular to V , what happens at points on M2 where x ·u achievesa maximum or a minimum?
106. A matrix M ∈ Rn×n is orthogonal if MTM = I, where T denotes the transpose and I then×n identity matrix. (Therefore also MMT = I.) Show that the orthogonal n×n matricesform a C1 surface of dimension n(n− 1)/2 in Rn×n. How many connected components doesit consists of?
HINT: detMT = detM .
107. A linear map T : Rn → Rn is self adjoint if 〈Tx,y〉 = 〈x, Ty〉 for every x,y ∈ Rn.
(i) Let T be a self-adjoint linear map, with matrix A = (aij), which is symmetric, so thataij = aji. If f(x) = 〈Tx,x〉 =
∑aijxixj, show that Dkf(x) = 2
∑nj=1 akjxj. Then, by
considering the maximum of 〈Tx,x〉 on the unit sphere Sn−1 = x ∈ Rn : ‖x‖ = 1 ⊂ Rn
show that there is x ∈ Sn−1 and λ ∈ R with Tx = λx.
(ii) If V = y ∈ Rn : 〈x,y〉 = 0, show that T (V ) ⊂ V and T : V → V is self-adjoint.
(iii) Show that T has a basis of eigenvectors.
HINT: In any orthonormal basis on V , the matrix of T : V → V is symmetric.
108. (i) Find the maximum of the function f(x, y, z) = x2y2z2 on the sphere x2+y2+z2 = c2.Conclude the inequality (
x2y2z2) 1
3 ≤ x2 + y2 + z2
3,
which states that the geometric mean of three nonnegative numbers x2, y2, z2 is never greaterthan their arithmetic mean.
42
(ii) Prove the same result in Rn.
109. (i) Let two positive numbers α and β be such that
1
α+
1
β= 1.
Find the minimum of the expression
f(u, v) =uα
α+vβ
β
subject to the condition uv = 1. Conclude that
uv ≤ uα
α+vβ
β. (18)
HINT: If uv 6= 1, 0, consider ut1α and vt
1β , where t = 1/uv.
(ii) Let x1, . . . , xn and y1, . . . , yn be nonnegative numbers and let at least one xj and at leastone yk be nonzero. Prove Holder’s inequality
n∑i=1
xiyi ≤
(n∑i=1
xαi
) 1α(
n∑i=1
yβi
) 1β
.
HINT: Letu =
xj(n∑i=1
xαi
) 1α
, v =yj(
n∑i=1
yβi
) 1β
, j = 1, . . . , n,
in (18) and sum over j.
110. (i) Show that the point on the closed surface φ(x, y, z) = 0 that is the closest to (orfarthest from) the given point (ξ, η, ζ) lies on the straight line
(x− ξ)φx
=(y − η)
φy=
(x− ζ)
φz,
normal to the surface.
(ii) Extend the result of (i) to m-dimensional surfaces in Rn: Let Mm be a C1 surface in Rn
and let y be a in Rn not on Mn. If x is a point on Mm that minimizes or maximizes the
43
distance to y, prove that the line joining x and y is perpendicular to the surface Mm, i.e.,its tangent space at x.
HINT: It is easier to consider the square of the distance.
111. Let f, g : A→ R, where A is a rectangle in Rn, be integrable.
(i) For any partition and P and subrectangle S, show that
mS(f) +mS(g) ≤ mS(f + g) and MS(f + g) ≤MS(f) +MS(g),
and therefore
L(f, P ) + L(g, P ) ≤ L(f + g, P ) and U(f + g, P ) ≤ U(f, P ) + U(g, P ).
(ii) Show that f + g is integrable and∫Af + g =
∫Af +
∫Ag.
(iii) For any constant c, show that∫Acf = c
∫Af .
(iv) If f ≤ g, show that∫Af ≤
∫Ag.
(v) Show that |f | is integrable and∣∣∫Af∣∣ ≤ ∫
A|f |.
112. Let f : [0, 1]× [0, 1]→ R be given by
f(x, y) =
1 x irrational,
1 x rational, y irrational,
1− 1/q x = p/q in lowest terms, y rational.
(i) Show that f is integrable and∫
[0,1]×[0,1]f = 1.
HINT: Show that f is only discontinuous when x is rational.
(ii) Show that∫ 1
0f(x, y) dy = 1 if x is irrational and does not exist if x is rational. In general,
a common “cure” for h(x) not being defined at a few isolated points is to assign an arbitrarynumber to be h(x) at all those points, and then proceed with the integration on x. Show
that this is not necessarily possible here: h is not integrable if h(x) =∫ 1
0f(x, y) dy is set
arbitrarily to any number other than 1 when the integral does not exist. However, computethe lower and upper y-integrals of f(x, y) for any x, show that they are integrable, and thatthey integrate to 1.
44
HINT: If x is rational, f(x, y) is discontinuous at every y. Also, show that the lower y-integralof f(x, y) is
L
∫ 1
0
f(x, y) dy =
1 x irrational,
1− 1/q x = p/q in lowest terms,
which is only discontinuous at rationals.
113. Let f : C → R, where C ⊂ Rn is connected and Jordan measurable, be integrable. Ifm = infC f and M = supC f , show that
∫Cf = µv(C), where m ≤ µ ≤ M and v(C) is the
volume of the set C. If C is compact and f continuous, show that µ = f(ξ) for some ξ ∈ C.
114. Let f : [a, b]→ R be integrable and non-negative and let
Af = (x, y) : a ≤ x ≤ b and 0 ≤ y ≤ f(x).
Show that Af is Jordan measurable and has area∫ baf(x) dx.
HINT: Let P = a = x0 < x1 < · · · < xn−1 < xn = b be a partition of [a, b] such thatU(f, P ) − L(f, P ) < ε. (Why does such a partition exist?) What is the total area ofall the rectangles of the form [xi, xi+1]×
[m[xi,xi+1](f),M[xi,xi+1](f)
]?
115. Use Fubini’s theorem to derive an expression for the volume of a subset of R3 obtainedby revolving a Jordan-measurable set in the yz-plane about the z-axis.
116. Show Cavalieri’s Principle: Let A and B be Jordan measurable subsets of R3. LetAc = (x, y) : (x, y, c) ∈ A and define Bc similarly. Suppose each Ac and Bc are Jordan-measurable and have the same area. Then A and B have the same volume.
117. Let A = [a1, b1] × · · · × [an, bn] and let f : A → R be continuous. Define IA(f) to bethe n-fold integral
IA(f) =
∫ bn
an
· · ·∫ b1
a1
f(x1, . . . , xn) dx1 . . . dxn,
carried out in the order so that x1 is integrated first, then x2, and so on, until xn. Usethe Stone-Weierstrass theorem to show that this order is immaterial, and can thus be inter-changed arbitrarily.
HINT: This is clear for functions of the form h(x1, . . . , xn) = h1(x1) · · ·hn(xn) and theirsums. Show that the latter form an algebra A on the space of continuous real functions onA that separates points.
45
118. Let g1, g2 : R2 → R be continuously differentiable and suppose that D1g2 = D2g1. Let
f(x, y) =
∫ x
0
g1(t, 0) dt+
∫ y
0
g2(x, t) dt.
Show that D1f(x, y) = g1(x, y).
119. Let f be a continuously differentiable function that vanishes outside of a boundedinterval, and let g be continuous. Let their convolution f ∗ g be defined as
(f ∗ g)(x) =
∫ ∞−∞
f(x− y)g(y) dy.
Show that f ∗ g is continuously differentiable and that (f ∗ g)′ = f ′ ∗ g. Show the analogousresult if f is n-times continuously differentiable.
120. Consider again the Weierstrass approximation theorem as proved in Problem 80. Letf(x) be the function on [0, 1] that is to be uniformly approximated by polynomials. Supposethat f is k-times continuously differentiable.
(i) Show that we can assume f and all its derivatives up to f (k) to vanish at 0 and 1 withno loss of generality.
(ii) Use problem 119 to show that you can approximate f and all its derivatives up to f (k)
by polynomials P through P (k).
121. (i) Suppose that f : (0, 1)→ R is a non-negative continuous function. Show that∫
(0,1)f
exists if and only if limε→0
∫ 1−εε
f exists.
HINT: Recall from class that if Φ is a partition of unity on (0, 1), then every finite sum∑ϕ
vanishes outside some [a, b] ⊂ (0, 1), and that for every [c, d] ⊂ (0, 1), there is a finite sum∑ϕ such that
∑ϕ = 1 on [c, d].
(ii) Let An = [1− 1/2n, 1− 1/2n+1]. Suppose that f : (0, 1) → R satisfies∫Anf = (−1)n/n
and f(x) = 0 for x /∈ any An. Show that∫
(0,1)f does not exist, but limε→0
∫ 1−εε
f = log 2.
HINT:∫
(0,1)f =
∑∫(0,1)
ϕf regardless of the order in the sum.
122. For (x, y) ∈ R2, define
F(x, y) = (ex cos y − 1, ex sin y).
46
Show that F = G2 G1, where
G1(x, y) = (ex cos y − 1, y)
G2(u, v) = (u, (1 + u) tan v)
in some neighborhood of (0, 0). Compute the Jacobians of G1, G2, and F at (0, 0).
DefineH2(x, y) = (x, ex sin y)
and findH1(u, v) = (h(u, v), v)
so that F = H1 H2 in some neighborhood of (0, 0).
123. (i) Derive the formula ∫ ∞0
dy
x2 + y2=π
2
1
x,
and by repeated differentiations show that∫ ∞0
dy
(x2 + y2)n=π
2
1 · 3 · · · (2n− 3)
2 · 4 · · · (2n− 2)
1
x2n−1.
(ii) Use (i) to show that ∫ ∞0
dy
(1 + y2/n)n=π
2
1 · 3 · · · (2n− 3)
2 · 4 · · · (2n− 2)
√n.
(iii) Write the integral ∫ ∞0
[e−y
2 − 1
(1 + y2/n)n
]dy
as∫ T
0+∫∞T
. Use (1 + y2/n)n > y2 and the growth/decay properties of the exponential toshow that the second integral is smaller in magnitude than ε/2 for large enough T . Use theproperty of the alternating series that, in absolute value, the remainder is smaller than thefirst omitted term, to show that
x− n log(
1 +x
n
)→ 0
as n→∞ uniformly in 0 ≤ x ≤ X. Conclude that
e−y2 − 1
(1 + y2/n)n→ 0
47
as n → ∞ uniformly in 0 ≤ y ≤ T so that the magnitude of∫ T
0can also be brought below
ε/2. Thus conclude that
limn→∞
1 · 3 · · · (2n− 3)
2 · 4 · · · (2n− 2)
√n =
1√π.
124. (i) Let Ik be the set of all u = (u1, . . . , uk) ∈ Rk with 0 ≤ ui ≤ 1 for all i; let Qk be theset of all x = (x1, . . . , xk) ∈ Rk with xi ≥ 0 and
∑ni=1 xi ≤ 1. (Ik is the unit cube and Qk is
the standard simplex in Rk.) Define x = T (u) by
x1 = u1
x2 = (1− u1)u2
...
xk = (1− u1) · · · (1− uk−1)uk.
Show thatk∑i=1
xi = 1−k∏i=1
(1− ui).
Show that T maps Ik onto Qk, that T is 1-1 in the interior of Ik, and that its inverse S isdefined in the interior of Qk by u1 = x1 and
ui =xi
1− x1 − · · · − xi−1
for k = 2, . . . , k. Show that
JT ≡∂(x1, . . . , xk)
∂(u1, . . . , uk)= (1− u1)k−1(1− u2)k−2 . . . (1− uk−1),
and
JS ≡∂(u1, . . . , uk)
∂(x1, . . . , xk)=
1
(1− x1)(1− x1 − x2) · · · (1− x1 − · · · − xk−1).
HINT: To compute the Jacobians JT and Js, note that the matrices T ′(u) and S ′(x) aretriangular, so that their determinants are the products of their diagonal elements.
(ii) Let r1, . . . , rk be nonnegative integers, and prove that∫Qkxr11 . . . xrkk dx =
r1! · · · rk!(k + r1 + · · · rk)!
.
HINT: Use B(x, y) = Γ(x)Γ(y)/Γ(x+ y).
48
125. Derive the formula for the volume of the n-dimensional ball in the following way.
(i) Consider the hyperspherical coordinates
x1 = r cosφ1
x2 = r sinφ1 cosφ2
...
xk = rk−1∏j=1
sinφj cosφk
...
xn−1 = r
n−2∏j=1
sinφj cosφn−1
xn = rn−1∏j=1
sinφk
Argue that, since the transformation is linear in r,
∂(x1, . . . , xn)
∂(r, φ1, . . . , φn−1)= rn−1Ωn(φ1, . . . , φn−1)
for some function Ωn(φ1, . . . , φn−1). (No need to compute this Ωn.)
(ii) Show that for any function f(r) of r only,∫BR(0)
f(r) dx1 · · · dxn = ωn
∫ R
0
f(r)rn−1 dr (19)
where
ωn =
∫ π
0
dφ1 . . .
∫ π
0
dφn−2
∫ 2π
0
Ωn(φ1, . . . , φn−1) dφn−1.
Here BR(0) is the ball of radius R around the origin.
(iii) Put f(r) = e−r2, and let R→∞ in (19) to show that(∫ ∞
−∞e−x
2
dx
)n= ωn
∫ ∞0
e−r2
rn−1 dr.
Conclude that
ωn =2√πn
Γ(n
2
) .49
(iv) Use (19) to show that the volume of the ball BR(0) equals
vn(R) =Rn√πn
Γ
(n+ 2
2
) .Express this volume explicitly for odd and even n.
126. For any nonnegative integer index n the Bessel function Jn(x) may be defined by
Jn(x) =xn
1 · 3 · 5 · · · (2n− 1)π
∫ 1
−1
(1− t2)n−1/2 cosxt dt.
Show that
J ′′n +1
xJ ′n +
(1− n2
x2
)Jn = 0, n ≥ 0.
HINT: Use integration by parts to get the same trigonometric function in all parts of theintegrand.
127. Using Fourier transforms show that
2
π
∫ ∞0
sin τ cos τx
τdτ =
1 for |x| < 1
1
2for x = ±1
0 for |x| > 1.
128. Find the Fourier transform of Jn(x)/xn, with Jn defined as in problem 126.
HINT: Do not perform any integrals.
129. Assume that f : R→ R is as smooth as you need.
(i) Show that if ∫ ∞−∞|x|n|f(x)| dx <∞
and
F (k) =1√2π
∫ ∞−∞
f(x)e−ikx dx
50
is the Fourier transform of f(x), then
F (n)(k) =1√2π
∫ ∞−∞
(−ix)nf(x)e−ikx dx.
(ii) If ∫ ∞−∞|f (j)(x)| dx <∞, j = 1, . . . n,
then1√2π
∫ ∞−∞
f (n)(x)e−ikx dx = (ik)nF (k).
130. Find the solution to the heat equation
ut = α2uxx, −∞ < x <∞, t > 0,
with the initial condition
u(x, 0) = f(x),
∫ ∞−∞|f(x)| dx <∞,
with f(x) also being continuous and piecewise smooth, by completing the following outline:
(i) Let
U(t, k) =1√2π
∫ ∞−∞
u(x, t)e−ikx dx
be the Fourier transform of u(x, t). Show that it satisfies the initial-value problem
Ut = −α2k2U, U(0, k) = F (k), F (k) =1√2π
∫ ∞−∞
f(x)e−ikx dx,
and solve this problem for every k.
(ii) Deduce from (i) and the Fourier integral theorem that
u(x, t) =1
2π
∫ ∞−∞
dk
∫ ∞−∞
f(ξ)e−α2k2t+ik(x−ξ) dξ.
Show that the repeated integral exists as an integral over R2 for all t ≥ t0 > 0 and reversethe order of integration.
(iii) Use (ii) and the formula√2
π
∫ ∞0
e−y2/2 cosλy dy = e−λ
2/2,
51
proven in class, to show that
u(x, t) =1
2α√πt
∫ ∞−∞
f(ξ) exp
(−(x− ξ)2
4α2t
)dξ (20)
for all x and all t > 0. Show uniform convergence of the integral and its partial derivativesfor all t ≥ t0 > 0 and thus verify directly that (20) indeed satisfies the heat equation. Infact, show that u ∈ C∞(R, t ≥ t0 > 0).
(iv) Show that
limt→0+
1
2α√πt
exp
(− y2
4α2t
)= 0, y 6= 0,
1
2α√πt
∫ ∞−∞
exp
(− y2
4α2t
)dy = 1, t > 0,
and
limt→0+
1
2α√πt
∫ ρ
−ρexp
(− y2
4α2t
)dy = 1
for any ρ > 0.
(v) Consider the integral
1
2α√πt
∫ ∞−∞
[f(ξ)− f(x)
]exp
(−(x− ξ)2
4α2t
)dξ.
Break the integral up into∫ x−ρ−∞ +
∫ x+ρ
x−ρ +∫∞x+ρ
and carefully estimate each of these terms
using (iv). Thus, deduce that, with u(x, t) as in (20),
limt→0+
u(x, t) = f(x).
In other words, u(x, t) indeed satisfies the initial value problem.
131. Consider the curve γ : [a, b]→ Rn. For each partition P = a = t0 < t1 < · · · < tN = bof [a, b] define
Λ(γ, P ) =N∑i=1
‖γ(ti)− γ(ti−1)‖.
(i) What does Λ(γ, P ) represent geometrically?
(ii) Define the length of γ asΛ(γ) = sup
PΛ(γ, P ).
52
and call γ rectifiable if Λ(γ) <∞.
By carrying out the outline below, prove the following Theorem: If γ ∈ C1[a, b], then γ isrectifiable, and
Λ(γ) =
∫ b
a
‖γ′(t)‖ dt.
(a) Show that
‖γ(ti)− γ(ti−1)‖ ≤∫ ti
ti−1
‖γ′(t)‖ dt,
and conclude that
Λ(γ) ≤∫ b
a
‖γ′(t)‖ dt.
(b) To show the opposite inequality, first choose ε > 0. Prove and use the uniform continuityof γ′ on [a, b] to show that for every sufficiently fine partition P ,
‖γ′(ti)− γ′(t)‖ < ε and ‖γ′(t)‖ ≤ ‖γ′(ti)‖+ ε
whenever ti−1 ≤ t ≤ ti. Thus, writing in some appropriate place γ′(ti) = γ′(t)+γ′(ti)−γ′(t),derive the estimate ∫ ti
ti−1
‖γ′(t)‖ dt ≤ ‖γ(ti)− γ(ti−1)‖+ 2ε(ti − ti−1).
Conclude that ∫ b
a
‖γ′(t)‖ dt ≤ Λ(γ) + 2ε(b− a),
and thus the statement of the theorem.
(iii) For a ≤ t ≤ b, define the arclength, s(t), of γ as
s(t) =
∫ t
a
‖γ′(τ)‖ dτ.
Compute s′(t)and deduce that s(t) is monotonically increasing. Conclude that the curve γcan be re-parametrized in terms of the arclength by σ(s) = γ(t(s)), where s runs throughthe interval [0,Λ(γ)]. Show that ‖σ′(s)‖ = 1, and so the integral for Λ(σ) becomes trivial.
132. (i) Consider the curve γ : [a, b]→ R3 and the line integral
Iγ =
∫γ
f dx+ g dy + h dz =
∫ b
a
[f(γ(t))γ′1(t) + g(γ(t))γ′2(t) + h(γ(t))γ′3(t)] dt.
53
Show that Iγ can be written as
Iγ =
∫γ
F · t ds,
where F = (f, g, h), t is the unit tangent to the curve γ(t), and ds is the differential of thearclength.
(ii) What must F = (f, g, h) be so that you can write
Λ(γ) =
∫γ
f dx+ g dy + h dz ?
(iii) Generalize the result of (i) and (ii) to curves in Rn.
133. (i) What are the arclength and the length of one turn of the helix
γ(t) = (a cos t, a sin t, bt)?
(ii) Compute the arclength of the curve of intersection between the sphere x2 + y2 + z2 = 4and the cylinder (x− 1)2 + y2 = 1. What is its total length?
134. Let Σ be a smooth surface in R3, expressed in terms of a single coordinate patch withparameters u and v in the parameter domain ∆. Show that the surface integral
IΣ =
∫∫Σ
f dy ∧ dz + g dz ∧ dx+ h dx ∧ dy
can be written in the form
IΣ =
∫∫Σ
F · n dA =
∫∫∆
F(r(u, v)
)· n(r(u, v)
)√EG− F 2 du dv.
Here F = (f, g, h), r(u, v) =(x(u, v), y(u, v), z(u, v)
)is the parametrization of the surface
Σ,
n =ru × rv‖ru × rv‖
is the unit normal to the surface Σ,
E = ru · ru, F = ru · rv, G = rv · rv,
anddA =
√EG− F 2 du dv
54
is the surface area element. (The subscripts denote partial derivatives.) Observe that
EG− F 2 = ‖ru × rv‖2 .
Choose F = n; what is the result?
135. Compute the volume and the area of the torus parametrized by
x(r, s, t) = (b+ r cos s) cos t
y(r, s, t) = (b+ r cos s) sin t
z(r, s, t) = r sin s,
with 0 ≤ t, s ≤ 2π and 0 ≤ r ≤ a, with 0 < a < b constants.
136. Let E be an open rectangle in R3, with edges parallel to the coordinate axes. Let(a, b, c) ∈ E and fi ∈ C1(E) for i = 1, 2, 3. Consider
ω = f1 dy ∧ dz + f2 dz ∧ dx+ f2 dx ∧ dy,
and assume that dω = 0 in E. Define
λ = g1 dx+ g2 dy,
where
g1(x, y, z) =
∫ z
c
f2(x, y, s) ds−∫ y
b
f3(x, t, c) dt
g2(x, y, z) = −∫ z
c
f1(x, y, s) ds
for (x, y, z) ∈ E. Prove that dλ = ω in E.
137. Vector analysis: Let F = (F1, F2, F3) be a smooth mapping of a star-shaped open setE ⊂ R3 into R3, which we will now call a vector field in E. With every such vector field F,we associate a 1-form
λF = F1 dx+ F2 dy + F3 dz,
and a 2-formωF = F1 dy ∧ dz + F2 dz ∧ dx+ F3 dx ∧ dy.
For any smooth function u : E → R, define its gradient as the vector
∇u = (D1u,D2u,D3u).
55
For a smooth vector field F in E define its curl
∇× F = (D2F3 −D3F2, D3F1 −D1F3, D1F2 −D2F1),
and its divergence∇ · F = D1F1 +D2F2 +D3F3.
Use Poincare’s lemma to show:
(i) F = ∇u for some smooth function u if and only if ∇× F = 0 in E.
(ii) F = ∇×G for some smooth vector field G if and only if ∇ · F = 0.
138. Let E = R2 − 0, the plane with the origin removed.
(i) Show that the 1-form
η =x dy − y dxx2 + y2
is closed in E.
(ii) Fix r > 0 and consider the circle
γ(t) = (r cos t, r sin t), 0 ≤ t ≤ 2π.
Since γ(0) = γ(2π), we have ∂γ = 0. Show directly that∫γ
η = 2π.
Use Stokes’ theorem to show that:
(a) η is not exact in E,
(b) γ is not the boundary of any 2-chain in E.
(iii) Let Γ be a smooth curve in R2 with parameter interval [0, 2π] such that no straight linesegment [γ(t),Γ(t)], with 0 ≤ t ≤ 2π, contains the origin. Prove that∫
Γ
η = 2π.
HINT: For 0 ≤ t ≤ 2π, 0 ≤ u ≤ 1, define
Φ(t, u) = (1− u)Γ(t) + uγ(t).
56
Then Φ is a 2-surface in R2−0 whose parameter domain is the indicated rectangle. Showthat ∂Φ = Γ− γ and use Stokes’ theorem to deduce that∫
Γ
η =
∫γ
η
because dη = 0.
(iv) Take the ellipse Γ(t) = (a cos t, b sin t), where a > 0 and b > 0 are fixed. Use part (iii)to show that ∫ 2π
0
ab
a2 cos2 t+ b2 sin2 tdt = 2π.
(v) Show that
η = d(
arctany
x
)in any convex open set in which x 6= 0, and that
η = d
(− arctan
x
y
)in any convex open set in which y 6= 0.
Explain why this justifies the notation η = dθ despite the fact that η is not exact in R2−0.
(vi) Show that (iii) can be derived from (v).
139. (i) Let ω =∑ai(x) dxi be a 1-form in a convex open set E ⊂ Rn. (See Problem 93 (ii)
for the definition of a convex set. Note that a convex set is always star-shaped.) Assumedω = 0 and prove that ω is exact in E by completing the following outline:
Fix p ∈ E, and let [p,x] be the straight line segment between the points p and x. Define
f(x) =
∫[p,x]
ω, x ∈ E.
Apply Stokes’ theorem to the appropriately oriented triangles [p,x,y] in E, with the verticesp, x, and y. Deduce that
f(y)− f(x) =n∑i=1
(yi − xi)∫ 1
0
ai
((1− t)x + ty
)dt
for x,y ∈ E. Hence Dif(x) = ai(x).
57
(ii) Assume that ω is a smooth 1-form in an open set E such that∫γ
ω = 0
for every smooth closed curve γ in E. Prove that ω is exact in E by imitating part of theargument sketched in (i).
(iii) Assume ω is a smooth 1-form in R3−0 and dω = 0. Prove that ω is exact in R3−0.
HINT: Every closed smooth curve in R3 − 0 is the boundary of a 2-surface in R3 − 0.Apply Stokes’ theorem and (ii).
140. State conditions under which the formula∫Φ
f dω =
∫∂Φ
fω −∫
Φ
df ∧ ω
is valid and show that it generalizes the formula for integration by parts.
HINT: d(fω) = df ∧ ω + f dω.
141. Using the notation of the problems 132, 134, and 137, as well as dV = dx dy dz,formulate precisely and prove the two classical formulas in R3:∫
Σ
(∇× F) · n dA =
∫∂Σ
F · t ds, (Stokes),
and ∫Ω
∇ · F dV =
∫∂Ω
F · n dA, (Gauss).
HINT: Simply translate them in the language of differential forms and use the results shownin class.
142. Let E ⊂ R3 be open, and g, h : E → R smooth. Consider the vector field
F = g∇h.
(i) Show that∇ · F = g∇2h+∇g · ∇h,
where
∇2h = ∇ · (∇h) =3∑i=1
∂2h
∂x2i
58
is the Laplacian of h.
(ii) If Ω is a 3-dimensional manifold-with-boundary in E with positively oriented boundary∂Ω, show that ∫
Ω
(g∇2h+∇g · ∇h
)dV =
∫∂Ω
g∂h
∂ndA,
where we have written ∂h/∂n in place of ∇h · n. (Thus ∂h/∂n is the directional derivativeof h in the direction of the outward normal to ∂Ω, the so-called normal derivative of h.)Interchange g and h and subtract the resulting formula from the first one, to obtain∫
Ω
(g∇2h− h∇2g
)dV =
∫∂Ω
(g∂h
∂n− h∂g
∂n
)dA.
These formulas are called Green’s identities.
(iii) Assume h is harmonic in E; this means that ∇2h = 0. Take g = 1 and conclude that∫∂Ω
∂h
∂ndA = 0.
Take g = h and conclude that h = 0 in Ω if h = 0 on ∂Ω.
(iv) Show that, with appropriate changes, Green’s identities are also valid in R2.
143. Let Σ be a “tube” in R3, that is, a surface parametrized by a function r(t, z) =(x(t, z), y(t, z), z
)defined on the rectangle 0 ≤ t ≤ 1, a ≤ z ≤ b, such that r(0, z) = r(1, z)
for every z ∈ [a, b]. In other words, each z-slice through the surface Σ is a closed curve. UseStokes’ theorem to show that ∫
Σ
dx ∧ dy = A(b)− A(a),
where A(z) is the area enclosed by the curve(x(t, z), y(t, z)
)in the xy-plane.
144. The physical principles of electricity and magnetism can be stated in the following way:
(i) Faraday’s law: The total electromotive force induced in a closed loop ∂Σ equals minusthe time rate of change of the magnetic flux through this loop. In the appropriate units, thislaw reads ∮
∂Σ
E · t ds = −1
c
d
dt
∫∫Σ
B · n dA.
(ii) Ampere’s law: The total magnetic force induced in a loop ∂Σ equals the total of theenclosed currents and the time rate of change of the electric displacement flux through the
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loop: ∮∂Σ
H · t ds =4π
c
∫∫Σ
J · n dA+1
c
d
dt
∫∫Σ
D · n dA.
(iii) Coulomb’s law: The electric displacement flux through any closed surface ∂Ω equals theenclosed charge: ∫∫
∂Ω
D · n dA =
∫∫∫Ω
ρ dv.
(iv) Absence of magnetic monopoles: There is no flux of the magnetic induction throughany closed surface: ∫∫
∂Ω
B · n dA = 0.
Show that these laws result in Maxwell’s equations
∇× E +1
c
∂B
∂t= 0, ∇ ·B = 0,
∇×H =4π
cJ +
1
c
∂D
∂t, ∇ ·D = ρ.
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