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Instructor’s Solution Manual for
ADVANCED CALCULUS
Gerald B. Folland
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Contents
1
2
3
4
Setting the Stage
1.1 Euclidean Spaces and Vectors
1.2 Subsets of Euclidean Space . .
1.3 Limits and Continuity . . . . .
1.4 Sequences . . . . . . . . . . .
1.5 Completeness . . . . . . . . .
1.6 Compactness . . . . . . . . .
1.7 Connectedness . . . . . . . .
1.8 Uniform Continuity . . . . . .
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1
1
1
2
3
4
5
5
7
Differential Calculus
2.1 Differentiability in One Variable . . . . . . . . . . . . . .
2.2 Differentiability in Several Variables . . . . . . . . . . . .
2.3 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . .
2.4 The Mean Value Theorem . . . . . . . . . . . . . . . . .
2.5 Functional Relations and Implicit Functions: A First Look
2.6 Higher-Order Partial Derivatives . . . . . . . . . . . . . .
2.7 Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . .
2.8 Critical Points . . . . . . . . . . . . . . . . . . . . . . . .
2.9 Extreme Value Problems . . . . . . . . . . . . . . . . . .
2.10 Vector-Valued Functions and Their Derivatives . . . . . .
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8
8
9
10
10
10
11
12
14
15
17
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The Implicit Function Theorem and its Applications
3.1 The Implicit Function Theorem . . . . . . . . . .
3.2 Curves in the Plane . . . . . . . . . . . . . . . .
3.3 Surfaces and Curves in Space . . . . . . . . . . .
3.4 Transformations and Coordinate Systems . . . .
3.5 Functional Dependence . . . . . . . . . . . . . .
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19
19
20
21
22
24
Integral Calculus
4.1 Integration on the Line . . . . . . . . . .
4.2 Integration in Higher Dimensions . . . .
4.3 Multiple Integrals and Iterated Integrals .
4.4 Change of Variables for Multiple Integrals
4.5 Functions Defined by Integrals . . . . . .
4.6 Improper Integrals . . . . . . . . . . . .
4.7 Improper Multiple Integrals . . . . . . . .
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25
25
26
27
28
30
31
32
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iii
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download instant at http://testbankinstant.com
Contents
iv
5
6
7
8
Line and Surface Integrals; Vector Analysis
5.1 Arc Length and Line Integrals . . . . .
5.2 Green’s Theorem . . . . . . . . . . . .
5.3 Surface Area and Surface Integrals . . .
5.4 Vector Derivatives . . . . . . . . . . . .
5.5 The Divergence Theorem . . . . . . . .
5.6 Some Applications to Physics . . . . .
5.7 Stokes’s Theorem . . . . . . . . . . . .
5.8 Integrating Vector Derivatives . . . . .
Infinite Series
6.1 Definitions and Examples . . . . . . . .
6.2 Series with Nonnegative Terms . . . . .
6.3 Absolute and Conditional Convergence
6.4 More Convergence Tests . . . . . . . .
6.5 Double Series; Products of Series . . . .
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Functions Defined by Series and Integrals
7.1 Sequences and Series of Functions . . . . . . . .
7.2 Integrals and Derivatives of Sequences and Series
7.3 Power Series . . . . . . . . . . . . . . . . . . .
7.4 The Complex Exponential and Trig Functions . .
7.5 Functions Defined by Improper Integrals . . . . .
7.6 The Gamma Function . . . . . . . . . . . . . . .
7.7 Stirling’s Formula . . . . . . . . . . . . . . . . .
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Fourier Series
8.1 Periodic Functions and Fourier Series . . . . . . . .
8.2 Convergence of Fourier Series . . . . . . . . . . . .
8.3 Derivatives, Integrals, and Uniform Convergence . .
8.4 Fourier Series on Intervals . . . . . . . . . . . . . .
8.5 Applications to Differential Equations . . . . . . . .
8.6 The Infinite-Dimensional Geometry of Fourier Series
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34
34
35
36
38
38
39
40
41
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43
43
44
45
46
47
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49
49
50
51
53
54
56
57
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59
59
60
61
62
63
65
download instant at http://testbankinstant.com
NOTE: Users of Advanced Calculus should be aware of the web site
www.math.washington.edu/ folland/Homepage/index.html
where a list of corrections to the book can be found. In particular, some errors in the exercises and in
the answers in the back of the book were discovered in the course of preparing this solution manual. The
solutions given here pertain to the corrected exercises.
v
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download instant at http://testbankinstant.com
Chapter 1
Setting the Stage
1.1 Euclidean Spaces and Vectors
1.
2.
! #"$%&'
(
)
*",+-./ 12433657$/8"$ , 9:12433657$; 8 <=>?8@ ,
,0
.
BAC DEFAG"HEBACI-J K AL%"MN
O . Taking the plus sign gives (a); adding these
IPG
C"""L
QSR& T U)R V P U W<T PYX[Z]\ U X R ^Z_" U
. The Pythagorean theorem follows immediately.
Eabcd efQagh
Eab
With `
proof,
efag as in theacik
j equality holds precisely when the minimum value of ` e is 0,
that
for some
. Thus equality holds in Cauchy’s inequality precisely when and
g is, when
are linearly dependent.
el"gNO e' gh
e g
The triangle inequality
is an equality preciselyg when
, that is, when the angle from to
e
is 0, or when is a positive scalar multiple of or vice versa.
e'mnoep%gI^
ghqr es%g*
gh
e'Lt g*q eu%gh
ghMt e'vq epwgh
, so
. Likewise,
.
es"gN+
epxg
e#ykghMm eG gh
eky#gNz
epz gNz
(a) If
then
, so
; hence if also
then
or
.
ef"M{kg%"{
eNyN{kg|yN{
oef gI"{F+
oe}WgI;yN{k~z
(b)
then
and
, so by (a), either
es%Ifg%z {z and
or
; the latter possibility is excluded.
eFy#ep€z e
g
eky#g%z
e#y#g
(c) We always have
. If
then
too. If not, then
is a
e and
ekyoare
eky#proportional,
g:(
‚ z
identities with the plus and minus signs gives (b).
3.
4.
5.
6.
7.
nonzero vector perpendicular to , so
.
8. This follows from the definitions by a simple calculation.
1.2 Subsets of Euclidean Space
1. (a)–(d): See the answers in the back of the text.
ˆ
EŽb+9*[q fqL‘
ƒ ƒk‹FŒ
(e) ƒI„†…b‡
and ‰Šƒ
.
o+Žb+94‘ E“?ŽY[*v“  qL‘
(f) ƒ „†…b‡ ƒk
, and
Žb’;+LŒ •Iy +v‘ , ƒ Œ
Π.
segment ”
ˆ
+Ž,)•Iy +Ž,)•
(g) ƒ „†…b‡
and ‰Šƒ
ƒ ”
” .
1
‰_ƒ
is the union of the unit circle and the line
download instant at http://testbankinstant.com
Chapter 1. Setting the Stage
2
i
$ ŽY'
ƒI„†…b‡ , there is a ball
2. If
contained in
ƒ . is open, so every point of is an interior
point of and hence of ƒ , so in fact €ƒ „†…4‡ and is an interior
point of ƒ „†…4 ‡ . Thus
ƒ „†…b‡ is open by
Proposition 1.4a. Next, ƒ and ‰_ƒ are the complements of ƒ „†…4‡ and ƒI„†…b‡I‹ ƒ „†…4‡ , respectively, so
they are closed by Proposition 1.4b.
i
P
P
4. The complements of ƒ ‹kƒ and ƒ
by Exercise 3 and Proposition 1.4b.
P
ƒ are
ƒ P
ƒ and
5. This
1.4:
follows from the
remarks preceding Proposition
ƒ „†…b‡ , whereas ƒ ƒI„†…b‡^‹#‰Šƒ and ƒ ƒ „†…4‡^‹}‰_ƒ .
6. One example (in
7.
j
and
ˆ
j P
j
is the disjoint union of
4.
5.
6.
7.
, and
.
#%eG
then
KmnEk%eŠ^
We'vq
e'
. Thus, if
ƒ
LŽbe
then
ƒ
e' ŽŠz
.
o+ŽY[<.
o+ŽY[<.
 +
 +
for
and `
for
.
E“'Žb+9<C“ !
“" +
as
.
a( +
Ea)ŽYaY: $8 L# a&%'!
as
.
P
“HSvq E“ Q E“'ŽY&q P E“ Q + “?ŽY) +
%
(a) Since
, we have `
as
.
L“ % w % q|&E“ % W % E“'ŽY&&q|[ “I* + “?ŽY) +
(b) Since
, we have `
as
.
E“?ŽY[+ J o+ŽY[<C
“ +
as
, so take `
.
`
E“?-Ž ,&
,ŽY[
.
,B
‚ +
o+Žb+9
E“?Žb+9
` o+ŽY&and
`
are
continuous
for
since
`
is
continuous
except
at
.
Moreover,
`
+
“'ŽY
for all
, also continuous.
`
f +
}“ “t.
‚ +
The two formulas for ` agree along the curves
except at
o+Žb+9<+ and E“'Ž P “ ,< P ‚ , so+ ` is“continuous
+
the origin. It is discontinuous there since `
but `
as
.
E“^q “:
“
E“^ +- o+9 "
“ +
,BC
‚ + ,
Since
for all , we have
` as
.“_Suppose
. If is irrational,
then
.,/` ,F
‚ +
E
:
+
.,&:
+
“ `
,
,
P
, but there
are
points
arbitrarily
close
to
with
.
If
is
rational,
then
,
`
`
`
“
,
E“_ ,_
,
. In both cases ` is discontinuous at .
but there are points arbitrarily close to with `
E“^&q “I
“
,w
‚ +
,
.
‚ +
Clearly“ `
for all , , so ` isE“_continuous
at 0. If
is rational,
then, `
, but there are
:+
,
`
`
points arbitrarily
close
to
with
;
hence
is
discontinuous
at
.
If
is
irrational
and
,
q21
E“_38$1
“4
,_0 is the
distance from to the nearest
rational number with denominator
, then `
for 5
0 ;
,
q 1
in any
hence ` is continuous at . (There are only finitely many rational numbers with denominator
1. (a) `
(b) `
(c) `
3.
ƒ „†…4‡ , ‰_ƒ
+Ž'c
P •
+Ž,
P
U
) is ƒ
, for which ƒ U
”
” .
1.3 Limits and Continuity
2.
ƒ
or ƒ and
centered
, so again
ƒ P k
‹ ƒ , respectively, which are both open
8. The sets in Exercise 1a and 1f are both examples.
9. If
P
ƒ ‹Bƒ , some
3. We use Proposition
If
centered at is contained in either ƒ
P 1.4a.
P ball
. If i ƒ P ƒ , there are balls P and
ƒ
‹
ƒ
ƒ
‹
ƒ
hence
in
,
so
is
an
interior
point
of
P
, respectively; the smaller of these balls is contained in ƒ P at
and
contained
in
and
ƒ
ƒ
P
ƒ .
is an interior point of ƒ
bounded interval.)
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1.4. Sequences
ewiwj
3
€+
ŽoeY
 E'i
‘
Œ
, let
. Then Žbeis open, and hence so is .
8. Given eti and
+
E'hvoe+
We have sF e', so
there exists 0
suche that
0
. But this says that
0 , so
whenever
is continuous at . One can replace “open” by “closed” in the hypothesis by
the reasoning of the second paragraph of the proof of Theorem 1.13.
9. The fact that since is a one-to-one correspondence between the
pointslof
v P
’
the
consequences
%following
vE'hi
‘( that we shall use: (i) If ,
Œ
.
g i
P Žbg:Y
Suppose
+
and
the points of has
v .’
. (ii) If !
,
ŽbgI
P small enough so that
. Since
, and let
be
is continuous,
og:
is a neighborhood of
by Theorem 1.13
it. Hence it
Žbg: and the remarks following
v ƒ
ƒ
ƒ
and
points
not
in
,
and
therefore
contains
points
in
and
points
not in
contains
points
in
gWi Y
ƒ . It follows that
‰ ƒ .
P
‰_ƒ
gWi
P suppose ‰ Conversely,
Since
is continuous,
Hence it contains pointse#
ini
not in ƒ . It follows that
Y
e og:
+
Žbe P P ŽbeY; let
be small enough
.
ƒ ŽbeY,and
let
g so that
is
a
neighborhood
of
by
Theorem
1.13
again.
Žbe
ƒ and pointsgQnot
in
ƒ
,
and
so
contains points in ƒ and points
i v ‰_ƒ and hence
‰Šƒ .
1.4 Sequences
1. (a) Divide top and bottom by
1
1 P
“ŠR c
1 P to get
7 [1 8$S1 &q8$1 +
.
“_R + P P 1
(c) Diverges since
is ,
, and
for infinitely many
“R!%[.,/8& 1u=[
1"|
=c
,/ P
whenever
.
1 “R.*" " """ 1 1 +
.
“R ,
R E“ŠRŽYRL . ,Ž6 .
(b)
2.
3.
each.
4. If
and
, then
. By continuity of addition and multiplication (Theorem
1.10) and the sequential characterization of continuity (Theorem 1.15), the result follows.
vE'+ e
+
+
vE' 3
+ K(e'
0 .
5. If ?R e as
, for any
there
such that1
whenever
'R exists
We' 0
E'RMK
If
, there exists
such
that
.
On
0 whenever
, and hence
vE's
‚ e
€+
+
0
the
other
hand,
if
as
,
there
exists
such
that
for
every
there
is
an
+)
ke' vE'I Y ?R
.8$1
'R e
E?R!!
‚ with
0 but
. Let
be such a point for 0
. Then
but
.
SRui
SRFr
‚ e
SR
e
?R‘
6. If
ƒ , + , and
, then the sequence Œ
must
Žbe assume
e infinitely many distinct values,
and for
,e all but finitely many of them are in
; thus is 1 an accumulation
of ƒ .
oeSŽ,8$point
1[
Conversely, if is an accumulation
point
of
ƒ
,
for
each
positive
integer
the
ball
contains
e
GR
points of ƒ other than ; let
be one.
e
ewi
e|i 8
e
7. If is an accumulation point of ƒ , then
1.14 and Exercise 6. If
ƒ by Theorem
ƒ and is not
e
an accumulation point of ƒ , there is a neighborhood
of
that
contains
only
finitely
many
points of
e
e ƒ . If
is
less
than
the
minimum
distance
from
to
any
of
these
points
(which
do
not
coincide
with
since
Žbe
eQi 8
e
eQi 8
ƒ .
ƒ ),
is a neighborhood of that is disjoint from ƒ , and hence
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Chapter 1. Setting the Stage
4
1.5 Completeness
Ž?8 8M vŽ?
1. (a) ƒ
, so the inf and sup are
and .
‹
(b) The supremum is the 0th element of the sequence; the infimum is the limit of the odd-numbered
subsequence.
>?8
>?8 &Ž ”
, so the inf and sup are
and .
(c) ƒ
“_R!7 S 1>?8
(Exercise 1c in 1.4).
,%
+ , P , , Kiro+Ž,
Š“ R# + , P , ,R
+ R
If
,
let
,
considered
as
a
fraction
with
denominator
“ŠR9‘
,
,}.+
“GRl~8$1 .
Œ
Then
is
a
subsequence
of
the
given
sequence
that
converges
to
.
For
,
take
; for
,s.
“ŠR 1l Y8$1
, take
.
W“ŠR
S
^+
, then
and hence
or 1.
(a) If ,_ , Al
,^
(b) The limit is zero if
, 1 if
, and nonexistent (or infinite) if
.
“?P )
C “R “ŠR P Q“ŠR c

“_R t
We
have
.“^IfR P ,then
. “By
induction,
for all
1

“
R

“
*
R

“
R
L
Š
“
R
)

“
;
R
,
"
R
€

“
R
.“RThis
being
the
case,
.
Thus
the
sequence
‘
Œ
is increasing andSbounded
above
, hence
[
?.
by 2, so it converges to a limit . We have
“'R +
1
2. One example is
3.
4.
5.
, and hence
6.
7.
8.
or
. The latter alternative is impossible since
for all .
Š“ R
1
MR €“R P 8$“R
“ŠR €“R P W“ŠR
(a) Let $R be theh
thterm
of
the
Fibonacci
sequence,
so
.
Since
, we
8 R R;
€Y8 R
“_R P
1
1
€
P
obtain
by dividing through by
. Replacing by
we get
R
R P Y8 R P , and substituting in LR P R*P Y8 R gives $R Rc
CY8R
.
E“_<m L“<
kY8E“
#h? E“
k I<
“
is $an
increasing
function
ofRL,4and
(b) The function
`
` I! .
$R $R
R4
G
!
R
R
`
`
`
Hence,P
if "
, Land
if then
.
then ` R 1
$R 1
Since
for odd and
for even. Next,
R R RL? andR!mI
R* , itR Yfollows
8 $RK
tthat
1
`
,
which
by
the
hint
is
positive
for
odd and negative
1
for even.
P‘
‘
(c) By (b), ΠU is an
increasing sequence
and ΠU is a decreasing
P
<
U sequence, bounded above and
below, respectively, by . Their limits and both satisfy ` U
, and hence both are equal to .
?R M‘
e
C+
Žbe
GR If Œ
converges
, then IR
contains ?R for
all sufficiently
large .Conversely,
e to , and
ci
Žbe
vŽ4vŽ&Žif
P
every ball about
contains
infinitely many , we can pick
, and then for
,
1
21
U P so that SR i Žbe ; then ?R ' e .
we can pick U
GR‘
If ƒ is bounded and infinite, let Œ
be a sequence of distinct points
e of ƒ . By Theorem 1.19, this set
1.4 its limit is an accumulation point of ƒ . (At
has a convergent
subsequence,
and
by
Exercise
6
in
?R
e
most one
can be equal to ; throw it out if necessary.)
“SR
1
,
7 “ŠR-
1
"N
! ‘
,
!
Œ “SR ,H
, then
and hence
9. If there
many for which
7#hare
“R$infinitely
,^
1
7# “Šfor
R  all
1%"N
! ‘-q,H
.
If
there
are
only
finitely
many
for
which
,
then
Œ
!
|7h
“^Rq,<
,fq
|7“^R-
q ,
for sufficiently
large, and hence . Since is arbitrary, we have
,l
7&“_R
and hence
.
“^RM ‘
10. We define
subsequence
“ŠR a~
7# “RN1'Œ so that R+ P C8 Œ 't
“R (
have
,
.
1
HP P
recursively.
We
take
, and for
, we choose U 7 U P
r
L

‘
8
v
U
. Then,Q
with
of , we
R+ P
“_R(* ) as in the definition
|7“R
(
(*)
. It follows that . Similarly for
1
1
1
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