Graphics Reference
In-Depth Information
r
()
¥
s
()
Æ
r
+
s
()
Ÿ
: W
MM
W
W
M
0
0
0
defined by
(
)( )
=
()
Ÿ
()
wh
Ÿ
pp
w
h
.
Again, a 0-form on
M
is just a real-valued function on
M
. Furthermore, if
f:
MR
Æ
,
then Df can really be considered to be a 1-form. The reason is that the tangent bundle
for
R
is trivial and there is a canonical identification of all the tangent spaces of
R
with
R
. We capture this idea more precisely with the following definition:
Definition.
The
differential
of f, denoted by df, is the 1-form on
M
defined by
( ()
=
()
df
pv
v
f
for
p
Œ
M
and
v
ΠT
p
(
M
).
Using coordinate neighborhoods we now relate an arbitrary k-form to those
defined on Euclidean space. Let (
U
,j), j :
U
Æ
R
n
be a coordinate neighborhood for
M
. and let
()
=
(
()
()
()
)
j
p
uu
p
,
p
,...,
n
p
,
1
2
where u
i
:
U
Æ
R
. Exercise 8.12.2 asks you to show that the differentials du
i
are the
dual basis for the tangent vectors ∂/∂u
i
. It therefore follows from the properties of the
algebra of exterior forms listed in Section 4.9 that every differential k-form w on
U
can be written in the form
Â
w
=
w
du
Ÿ
...
Ÿ
du
,
(8.31)
ii
i
i
1
...
k
1
k
1
£< < £
i
...
i
n
1
k
for functions w
i
1
...
i
k
:
U
Æ
R
.
Definition.
A differential form w on
M
is called
continuous
,
differentiable
,
C
•
,
etc., if the functions w
i
1
...
i
k
in expression (8.31) are continuous, differentiable, C
•
,
etc., respectively, with respect to all coordinate neighborhoods (
U
,j). The vector
subspace of W
k
0
(
M
) that consists of C
•
differential k-forms on
M
will be denoted by
W
k
(
M
).
It is easy to show that the definitions are well defined and do not depend on any
particular coordinate neighborhood. We shall always assume that we have C
•
mani-
folds, C
•
maps, and C
•
differential forms. Note that, although we used different def-
initions for the tangent space, this definition of W
k
(
R
n
), where
R
n
is thought of as a
manifold, and the one in Section 4.9 agree under the natural correspondence between
the tangent space definitions.
