Graphics Reference
In-Depth Information
This exercise describes an alternate definition of the gradient of a function f :
M
n
Æ
R
.
Let
p
Œ
M
and let (
U
,j) be a coordinate neighborhood of
p
. If j(
q
) = (u
1
(
q
),u
2
(
q
),
...,u
n
(
q
)),
q
Œ
U
, then define
8.10.2.
n
∂
∂
f
uu
i
∂
Â
—
()
=
f
p
.
∂
i
i
=
1
Show that this definition of the gradient of f agrees with the definition in Section 8.10.
Show further that this reduces to the standard definition of the gradient of f when
M
=
R
n
.
Prove that the function h in the proof of Theorem 8.10.11 is a
diffeomorphism
.
8.10.3.
8.10.4.
Prove that the differentiable manifold
M
¥
M
is orientable for any differentiable
manifold
M
.
Prove that the total space of the tangent bundle of a differentiable manifold
M
n
8.10.5.
is
always an orientable manifold, even if
M
n
is not.
Section 8.11
Use transversality to prove that no differentiable retraction r :
M
n
Æ∂
M
n
exists. (
Hint:
8.11.1.
pick a regular value
p
ζ
M
and analyze
N
n-1
= r
-1
(
p
).)
Define f :
S
1
Æ
S
1
by f(
z
) =
z
n
,
z
Œ
C
. Prove that deg (f,1) = n.
8.11.2.
Section 8.12
Let f :
M
n
Æ
R
. If we use the equivalence class of vectors approach to defining
tangent vectors show that the corresponding definition for the differential of f, df,
would be
8.12.1.
(
)
()
-
1
df
()
[
pU a
(
,,
j
]
)
=
D f
o
j
(
j
pa
)
,
for
p
Œ
M
, (
U
,j) a coordinate neighborhood for
p
, and
a
Œ
R
n
. In particular, show that
df is a well-defined element of the dual space of T
p
(
M
).
Let (
U
,j) be a coordinate neighborhood for a manifold
M
n
. Let
8.12.2.
(
)
()
=
()
()
()
j
p
uu
p
,
p
,...,
n
p
,
1
2
where u
i
:
U
Æ
R
. Show that du
1
(
p
), du
2
(
p
) ,..., du
n
(
p
) is the dual basis in T
p
(
M
) for
the basis ∂/∂u
1
, ∂/∂u
2
,..., ∂/∂u
n
of tangent vectors.
8.12.3.
Prove Green's Theorem, Theorem 8.12.16.
8.12.4.
Prove the Divergence Theorem, Theorem 8.12.17.
Section 8.14
Prove that the Stiefel manifold V
n
(
R
n+k
) is a C
•
manifold. What is its dimension?
8.14.1.
