MATH 321: Real Variables II
Lecture #32
University of British Columbia
Lecture #32:
Instructor:
Scribe:
March 31, 2008
Dr. Joel Feldman
Peter Wong
Theorem. (Uniform Convergence of Fourier Series)
If f : R → C is 2π-periodic and is piecewise C 1 , then the Fourier series for f converges uniformly to f on any
closed interval that does not contain a point of discontinuity of f .
Proof. Step 1: If, in addition, f is continuous, then the Fourier series of f converges uniformly to f .
Proof. Done.
Step 2: Define
y
y = F (x)
1
2
b
−3π
b
−π
−2π
b
π
2π
3π
x
The Fourier series converges uniformly on any closed interval that does not contain a point in 2πZ =
{ 2πk | k ∈ Z }
Proof. Done.
Step 3: The general case:
−
Let the points of discontinuity of f on [−π, π) be p1 , . . . , pm . Let the jump heights be δj = f (p+
j ) − f (pj ).
Pm
Pm
Write f (x) = g(x) + j=1 δj F (x − pj ) with g(x) = f (x) − j=1 δj F (x − pj ). The Fourier series for
δj F (x − pj ) converges uniformly on any closed interval not containg a point of pj + 2πZ. It suffices to show
that g(x) is continuous. g(x) is obviously continuous except possibly at the pk ’s, which are in [−π, π). To
check continuity at pk ’s
−
+
−
g(p+
k ) − g(pk ) = f (pk ) − f (pk ) −
= δk −
n
X
j=1
=0
δj
(
n
X
δj [F (( pk − pj )+ ) − F ((pk − pj )− )]
| {z }
j=1
∈(−2π,2π]
0, j 6= k,
1, j = k
Manifolds (See web notes and references therein)
(1) Definition
(2) Examples
(3) Integration on Manifolds & Stoke’s theorem
2
MATH 321: Lecture #32
Rough Definition
• an n-dimensional manifold is something that looks locally like Rn
• it is a union of subsets with each subset having coordinates that run over any open subset of Rn
Definition. (An n-dimensional manifold)
Let M be a metric space.
(a) A chart on M is a pair { U, ϕ } with U ⊂ M open and ϕ : U → Rn a homeomorphism between U and ϕU.
Think of ϕ as assigning coordinates in Rn to each m ∈ U.
(b) Two charts { U, ϕ } and { V, ψ } are compatible if
ψ ◦ ϕ−1 : ϕ(U ∩ V) ⊂ Rn → ψ(U ∩ V) ⊂ Rn
ϕ ◦ ψ −1 : ψ(U ∩ V) ⊂ Rn → ϕ(U ∩ V) ⊂ Rn
are C ∞ (i.e., all partial derivatives are defined and continuous.)
MATH 321: Real Variables II
Lecture #33
University of British Columbia
Lecture #33:
Instructor:
Scribe:
April 2, 2008
Dr. Joel Feldman
Peter Wong
Definition. (Manifold of dimension n) Let M be a metric space.
(a) A chart is a pair { U, ϕ } with
• U an open subset of M
• ϕ a homeomorphism from U to an open subset of Rn
(b) { U, ϕ } and { V, ψ } are compatible if ψ ◦ ϕ−1 and ϕ ◦ ψ −1 are C ∞
(c) An atlas A for the M is a set A = { { Ui , ϕi } | i ∈ I }, with I being a completely arbitrary index set, of
charts obeying
(a) { Ui } covers M (i.e.,
S
i∈I
Ui = M)
(b) Every pair of charts in A is compatible.
A maximal atlas is an atlas A with the property that { U, ϕ } being any chart that is compatible with every
chart in A implies { U, ϕ } ∈ A.
(d) A manifold of dimension n is a metric space M together with a maximal atlas A.
Lemma. Let { U, ϕ } and { V, ψ } be
• U, V ∈ M with U ∪ V =
6 ∅
• ϕ : U → Rn be a homeomorphism from U to ϕ(U) ⊂ Rn
open
• ψ : V → Rm be a homeomorphism from V to ϕ(V) ⊂ Rm
open
• ψ ◦ ϕ−1 and ϕ ◦ ψ −1 are C ∞
Proof. Homework problem.
Lemma. If A is any atlas, then there exists a unique maximal atlas containing A
Proof. Homework problem. (Hint: Try to guess what the maximal atlas is.)
Example 1. Let M be an open subset of Rn (e.g., M = Rn ) and A = { { U = M, ϕ = identity map } }. Then
(U, ϕ) is an identity map.
Example 2. Let M = S 1 = { (x, y) ∈ Rn | x2 + y 2 = 1 } and
U1 = S 1 \ { (−1, 0) } , ϕ1 (x, y) = arctan xy ∈ (−π, π)
(−1, 0)
bc
= a unique angle with (x, y) = (cos ϕ1 , sin ϕ1 )
ϕ
1
U2 = S \ { (1, 0) } ,
ϕ2 (x, y) = arctan xy ∈ (0, 2π)
= a unique angle with (x, y) = (cos ϕ2 , sin ϕ2 )
2
MATH 321: Lecture #33
Then { { U1 , ϕ1 } , { U2 , ϕ2 } } is an atlas for S 1 verification of compatibility:
ϕ1 (U1 ∩ U2 ) = (−π, 0) ∪ (0, π)
ϕ2 (U1 ∩ U2 ) = (0, π) ∪ (π, 2π)
 
θ

, if − π < θ < 0,
ϕ2
(cos θ, sin θ)
−1
ϕ2 ◦ ϕ1 (θ) =

(cos θ, sin θ)

ϕ2
, if 0 < θ < π,
θ
(
θ + 2π, if − π < θ < 0,
=
U1 ∩ U2 = S 1 \ { (−1, 0), (1, 0) }
θ,
if 0 < θ < π,
bc
b
bc
b
bc
b
bc
bc
b
bc
∈ C∞
Similarly for ϕ1 ◦ ϕ2−1 .
Example. (Any n-dimensional surface in Rn+m ) Handwavy definition of an n-dimensional surface in Rn+m
• a subset of Rn+m such that locally
– m coordinates of points on the surface are determined by the other n coordinates in a C ∞ way.
b
x=−
p
1 − y2
b
√
y = − 1 − x2
MATH 321: Real Variables II
Lecture #34
University of British Columbia
Lecture #34:
Instructor:
Scribe:
April 4, 2008
Dr. Joel Feldman
Peter Wong
Example. (n-dimensional surface in Rm+n )
Rough definition: A subset of Rn+m is an n-dimensional surface if locally, m coordinates of points on the surface
are determined by the n other coordinates (i.e. are functions of the other n coordinates) in a C ∞ way.
Example.
S 1 = { (x, y) ∈ R2 | x2 + y 2 = 1 }
p
For y > 0, y = 1 − x2
p
For x < 0, x = − 1 − y 2
Formal definition: M ⊂ Rm+n is an n-dimensional surface if, for each ~z ∈ M, there are
• a neighborhood U~z of ~z in Rm+n
• n natural numbers 1 ≤ j1 < j2 < · · · < jn ≤ m + n
(indices of the independent coordinates)
Rm
M
U~z
• m C ∞ functions fk (xj1 , . . . , xjn ) with
k ∈ { 1, . . . , m + n } \ { 1 ≤ j1 < · · · < jn }
such that the point ~x ∈ U~z is in M if and
only if xk = fk (xj1 , . . . , xjn ) for all
k ∈ { 1, . . . , m + n } \ { 1 ≤ j1 < · · · < jn }
b
b
~
x ~z
b
Rn
(xj1 , . . . , xjn )
This is a manifold with the atlas
A = { { M ∩ U~z , ϕ~z } | ~z ∈ M }
where ϕ~z (x1 , . . . , xn+m ) = (xj1 , . . . , xjn ).
Equivalent definition: M ⊂ Rm+n is an n-dimensional surface if, for each ~z ∈ M, there are
• a neighborhood U~z of ~z in Rm+n
• m C ∞ functions
Rm
U~z
gk : U~z → R, k = 1, . . . , m
The two definitions are equivalent because of the . . .
0
b
b
~ k (~z) | 1 ≤ k ≤ m } are
such that { ∇g
linearly independent such that the
point ~x ∈ U~z is in M if and only if
gk (~x) = 0 for all 1 ≤ k ≤ m.
M
=
x)
g(~
z
~
x ~
b
(xj1 , . . . , xjn )
Rn
2
MATH 321: Lecture #34
~y ∈ Rm
Implicit Function Theorem
U
Let
g(~x) = 0
V
(~
x, ~
y)
• m, n ∈ N
• U ⊂R
m+n
b
b
(~
x0 , ~
y0 )
be open
• ~g : U → Rm be C ∞
b
b
W
~x ~x0
obey
~x ∈ Rn
• ~g (~x0 , ~y0 ) = 0 for some ~x0 ∈ Rn , ~y0 ∈ Rm with (~x0 , ~y0 ) ∈ U
∂gi
• det
(~x0 , ~y0 )
6= 0
1≤i≤m
∂yi
1≤j≤m
Then there exist open sets V ⊂ Rm+n and W ⊂ Rn with ~x0 ∈ W and (~x0 , ~y0 ) ∈ V such that for each ~x ∈ W , there
is a unique (~x, ~y) ∈ V with ~g(~x, ~y) = ~0. Call ~y ∈ f~(~x). Furthermore, f~ : W → Rm is C ∞ and f~(~x0 ) = ~y0 and
~g (~x, f~(~x)) = ~0 for all ~x ∈ W .
Example. (Simple – S 1 )
g(x, y) = x2 + y 2 − 1,
(x0 , y0 ) = (0, 1)
U = R2 , V = { (x, y) ∈ R2 | y > 0 }, W = (−1, 1), f (x) =
Examples. (SO(3), O(3))
p
1 − x2
O(3) = { 3 × 3 real matrices R | R† R = I }
SO(3) = { 3 × 3 real matrices R | R† R = I, det R = 1 }
Think of the matrix

a1
R =  a2
a3
b1
b2
b3

c1
c2 
c3
as a vector in R9 . Write R† R = I as a linear system of equation.