Graphics Reference
In-Depth Information
()
[
(
]
)
=
[
]
Df
pU a
,,
j
V b
, , ,
y
(8.15b)
where (
U
,j) and (
V
,y) are coordinate neighborhoods of
p
and f(
p
), respectively and
(
)
-
1
(
()
)( )
b
=
Dyjj
oo
p
a
.
(8.15c)
The map Df(
p
) is called the
derivative
of f at
p
.
One can show that Df(
p
) is a well-defined linear transformation.
In classical terminology, tangent vectors are called contravariant tensors. This was
motivated by the definition that we just gave. The terms “covariant” and “contravari-
ant” have to do with how quantities transform under a change of coordinates. See
[Spiv70a] for a discussion of why this classical terminology is unfortunate and why
tangent vectors should really be called covariant tensors. The fact is that the termi-
nology has been around for so long, so that no one has dared to change it.
The linear functional approach to tangent vectors at p:
First, define
F(
p
) = {f :
U
Æ
R
|
U
is an open neighborhood of
p
in
M
and f is differentiable}.
A
tangent vector
of
M
k
at
p
is a map
Definition.
()
Æ
XF
:
pR
satisfying
(1) X(af + bg) = aX(f) + bX(g) for all f, g ΠF(
p
) and a, b Œ
R
.
(2) X(fg) = X(f)g(
p
) + f(
p
)X(g) for all f, g ΠF(
p
).
(The domain of af + bg and fg is the intersection of the domain of f and the domain of
g.) The
tangent space
of
M
k
at
p
, T
p
(
M
k
), is the set of all tangent vectors to
M
k
at
p
.
Note.
A map X satisfying properties (1) and (2) is usually called a
derivation
because
it acts like a derivative.
This definition is motivated by the following observations. Let us return to Section
8.3 and assume that manifolds are subsets of Euclidean space. If g : [a,b] Æ
M
k
is a
curve with g(c) =
p
, then g induces a map
()
Æ
g
*
:F
pR
defined by
d
dt
()
=
(
)( )
g
f
f
o
g
c
.
*
The map g
*
is just the directional derivative of f at
p
in the direction g¢(c). One can
show that g
*
depends only on the derivative of g(t) at c. In other words, one can think
