Lecture 22
In
R
3
we
had the
operators
grad, div, and
curl. What
are the analogues
in
R
n
?
Answering this
question is
the
goal of today’s
lecture.
4.9
Tangent Spaces
and
k
forms
Let
p
∈
R
n
.
Definition 4.36.
The
tangent space of
p
in
R
n
is
T
p
R
n
=
{
(
p, v
) :
v
∈
R
n
}
.
(4.144)
R
n
We
identify the
tangent
space with
via
the identification
�
⏐
⏐
T
p
R
n
=
R
n
(4.145)
∼
(
p, v
)
→
v.
(4.146)
according to the
following diagram:
⏐
⏐
�
Via
this identification,
the
vector
space structure on
R
n
gives
a
vector
space structure
on
T
p
R
n
.
Let
U
be
an open set
in
R
n
,
and
let
f
:
U
→
R
m
be a
C
1
map. Also, let
p
∈
U
and
define
q
=
f
(
p
).
We
define
a
linear
map
df
p
:
T
p
R
n
T
q
R
m
(4.147)
→
T
p
R
n
df
p
T
q
R
m
−−−→
=
∼
=
∼
(4.148)
R
n
Df
(
p
)
−−−→
R
m
.
So,
df
p
(
p, v
) = (
q, Df
(
p
)
v
)
.
(4.149)
Definition 4.37.
The
cotangent
space of
R
n
at
p
is
the space
T
p
∗
R
n
≡
(
T
p
R
n
)
∗
,
(4.150)
which is
the
dual
of
the
tangent
space of
R
n
at
p
.
Definition 4.38.
Let
U
be
an open
subset
of
R
n
. A
k
form
on
U
is
a
function
ω
which
assigns
to every point
p
∈
U
an
element
ω
p
of Λ
k
(
T
∗
R
n
) (the
k
th
exterior power
p
of
T
∗
R
n
).
p
1
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�
�
�
�
Let
us
look at
a simple
example. Let
f
∈ C
∞
(
U
),
p
∈
U
, and
c
=
f
(
p
). The map
df
p
:
T
p
R
n
T
c
R
=
R
(4.151)
→
is a
linear
map of
T
p
R
n
into
R
.
That is,
df
p
∈
T
∗
R
n
. So,
df
is
the oneform
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 Fall '04
 unknown
 Vector Space, Tangent space, Tp Rn, Cotangent space

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