Graphics Reference
In-Depth Information
Definition.
The differential form w is called
continuous
,
differentiable
,
C
•
, etc., if the
functions w
i
i
...i
k
are continuous, differentiable, C
•
, etc., respectively.
We shall always assume that differentiable forms are C
•
in order to avoid
problems with functions not being differentiable enough.
Notation.
The vector space of C
•
differential k-forms on
R
n
will be denoted by
W
k
(
R
n
). (We do not use the notation L
k
(
R
n
) because that already refers to the alter-
nating multilinear maps of the
vector space R
n
, which is something quite different.)
The next theorem expands on the result expressed by equation (4.29) by describ-
ing the expansion in more detail for the case of differentials of functions.
If f :
R
n
4.9.2. Theorem.
Æ
R
is a differentiable function, then
df
=
D f dx
+
D f dx
+
...
+
D f dx
,
1
1
2
2
n
n
that is, using the classical notation,
∂
∂
f
x
∂
∂
f
x
∂
∂
f
x
df
=
dx
+
dx
+
...
+
dx
.
(4.31a)
1
2
n
1
2
n
In particular, the differential df can be expressed in terms of the directional deriva-
tive by the formula
( ( )
=
(
.
df
pv
D f
p
(4.31b)
p
v
Proof.
Simply note that
n
n
Â
Â
( ( )
=
( ()
=
()
=
( ( )
()
df
pv
Df
pv
v D f
p
dx
pv
D f
p
p
ii
i
p
i
i
=
1
i
=
1
and
n
Â
1
()
=—
()
∑=
()
vDf
p
f
p
v
D f
p
.
ii
v
i
=
Next, consider a differentiable map f :
R
n
Æ
R
m
. The map f induces a map f
*
on
tangent spaces and its dual map
)
(
)
Æ
(
)
m
n
fT
*:
R
*
T
R
*
(
f
p
p
leads to the induced map
(
)
(
)
)
Æ
(
(
)
)
k
m
k
n
(4.32)
f
*:
L
T
R
L
T
R
.
(
f
p
p
