Graphics Reference
In-Depth Information
()
=
(
( ()
)
(
f
v
Df
p v
.
p
*
f
p
)
is called the
induced map
on tangent spaces.
A
vector field
on
R
n
is a map F defined on
R
n
that sends
p
Œ
R
n
Definition.
to an
element of T
p
(
R
n
). If we express F in the form
n
Â
()
=
( ( )
(4.27)
F
p
F
i
p e
,
i
p
i
=
1
then the real-valued functions F
i
(
p
) are called the
component functions
of F.
Definition.
A
differential k-form
on
R
n
, or simply
k-form
or
differential form
, is a map
w defined on
R
n
that sends
p
Œ
R
n
to an element w(
p
) ŒL
k
(T
p
(
R
n
)).
The set of k-forms on
R
n
actually forms a vector space if we define the addition
and scalar multiplication in a pointwise fashion. Note that differential 0-forms are
just functions
R
n
Æ
R
. Also, given a differentiable function f :
R
n
Æ
R
, Df(
p
) is linear
transformation from
R
n
to
R
. As such it can be considered an element of L
1
(
R
n
), in
fact, an element of L
1
(T
p
(
R
n
)).
Definition.
The
differential
of a function f :
R
n
Æ
R
, denoted by df, is the differential
1-form in L
1
(
R
n
) defined by
( ( )
=
( (
.
df
pv
Df
pv
(4.28)
p
As a special case, consider the projection functions p
i
:
R
n
Æ
R
defined by p
i
(x
1
,x
2
,
...,x
n
) = x
i
.
Notation.
In order to arrive at the classical notation for differential forms we shall
abuse the notation and write dx
i
instead of dp
i
.
Now, if
v
= (v
1
,v
2
,...,v
n
), then
( ( )
=
( ()
=
dx
pv
Dx
pv
v
.
i
p
i
i
Therefore, the linear maps dx
i
(
p
) are just the dual basis of the standard basis (
e
1
)
p
,
(
e
2
)
p
,..., (
e
n
)
p
of T
p
(
R
n
), so that every differential k-form w can be expressed in the
form
Â
w
=
w
dx
Ÿ
...
Ÿ
dx
,
(4.29)
i
...
i
i
i
1
k
1
k
1
£<
i
...
< £
i
n
1
k
for functions w
i
i
...i
k
:
R
n
Æ
R
. In particular, every n-form w has the form
(4.30)
w=
fdx
Ÿ
dx
Ÿ
...
Ÿ
dx
n
,
1
2
for some function f :
R
n
Æ
R
.
