Graphics Reference
In-Depth Information
Figure 8.14.
Normal lines to a sphere pass
through its center.
L
S
p
P
0
z
y
x
Df
p
(v)
f
M
N
V
q = f(p)
p
g
f
g
°
Figure 8.15.
The derivative of a
differentiable map
between manifolds.
0
)
¢
()
p
()
=
(
Df
f
o
g
0.
(8.7)
See Figure 8.15.
Definition.
The map Df
p
is called the
derivative of f at the point
p
.
8.4.9. Theorem.
(1) The map Df
p
is a well-defined linear map.
(2) If
M
n
=
R
n
and
N
k
=
R
k
, then Df
p
= Df(
p
).
Proof.
There are two parts to the proof of (1). To show that the map is well defined
one must show that its definition does not depend on the curve g. In general, there
will be lots of curves g(t) satisfying g(0) =
p
and g¢(0) =
v
. To show the linearity prop-
erty, observe that
