Graphics Reference
In-Depth Information
generalizations to n-dimensional submanifolds
M
n
of
R
k
are clear. Up to now we have
used the definition of tangent vectors and tangent planes as presented in Section 8.4.
In this section it will be convenient to use a variant of those definitions, one that is
implicit in equation (8.30) in Section 8.10. Tangent vectors will now be associated to
points. The formal way to do this was to define the tangent space T
p
(
M
n
) at
p
Œ
M
n
by
(
)
=
(
{
}
n
)
(
)
n
T
p
Mp v
,
v
is a tangent vector in the sense of Section 8.4 to
Mp
at
k
Õ¥
MR
.
There is one caveat, however. If we were to always use this pair (
p
,
v
) notation for a
tangent vector, even if we were to abbreviate it somewhat to
v
p
as we shall do at times,
the readability of many of expressions and equations in the rest of this section would
suffer greatly unfortunately. Things would look much more complicated than they
really are. Therefore, we shall be sloppy and use “
v
” to denote either the pair (
p
,
v
) or
v
itself whenever it is convenient. Sometimes an expression may contain two instances
of “
v
” where each has a different meaning. The reader will know which is being meant
at any given time because there will only be one obvious correct meaning. To help out
making the distinction, we shall, when possible, refer to the “tangent vector
v
” or the
“tangent vector
v
at
p
” when we mean (
p
,
v
) and simply the “vector
v
” if we do mean
just
v
. By the way, the only reason that we run into this notational problem is so that
the reader can see the close connection between abstract concepts, such as tangent
vectors, with “ordinary” functions and derivatives for Euclidean space.
Specializing to the case
M
=
R
k
, let
X
be a vector field of
R
k
defined over some
open subset
A
. We can write
X
in the form
()
=
(
()
)
Xp
p
,
g
p
,
for some function g :
A
Æ
R
k
. Given a curve g : [a,b] Æ
A
, define
d
dt
gt
Ê
Ë
ˆ
¯
(
()
)
=
()
(
()
)
X
g
g
¢
t
g
t
,
g
(9.91)
This definition of course only makes sense when g is one-to-one but everything in this
section involves local properties and so assuming that parametric curves are locally
one-to-one is not a problem.
Definition.
The vector field
X
g
is called the
derivative of
X
along the curve
g.
Now let
X
be a vector field of
R
3
defined in a neighborhood of
p
and let
v
ΠT
p
(
R
3
).
Definition.
The
covariant derivative
, of
X
at
p
with respect to
v
, denoted by —
v
X
(
p
),
is the tangent vector of
R
3
at
p
defined by
()
=¢
()
—
v
Xp
X p
,
g
where g is the curve g(t) =
p
+ t
v
.
