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be two parameterizations of an n-dimensional manifold
M
n
in
R
k
. Assume that F=
Y
s, where s :
U
Æ
V
, s(x
1
,x
2
,..., x
n
) = (u
1
,u
2
,..., u
n
), is a change of coordinate trans-
formation. Let Q
I
(Y) and Q
I
(F) denote the first fundamental forms of Y and F, respec-
tively. Using the notation in equations (9.35-37), let l=s
m. Then the previous
equations and a generous use of the chain rule shows that
n
∂
∂
FF
∂
∂
Â
() ()
=
Q
F
g
∑
m m
¢¢
1
ij
xx
i
j
ij
,
=
1
n
n
n
È
Í
Ê
Á
∂
∂
Y
∂
∂
s
ˆ
˜
∑
Ê
Á
∂
∂
Y
∂
∂
s
ˆ
˜
˘
˙
Â
Â
Â
s
i
t
j
=
mm
¢¢
i
j
s
x
s
x
s
t
ij
,
=
1
s
=
1
t
=
1
n
∂
∂
YY
∂
∂
Â
=
∑
ll
¢¢
st
x
x
s
t
st
,
=
1
() ()
=
Q
1
Y
l
.
This is what we wanted to show.
Volume also was independent of parameterizations. The metric coefficients on the
other hand are not but the next theorem shows that they transform in a well-defined
way.
9.8.12. Theorem.
In the case of surfaces the metric coefficients transform as
follows (using the notation above and writing (u,v) =s(x,y)):
()
=
()
+
2
()
()
2
EEuFuv Gv
FEuuFuv
F
F
y
2
y
+
y
x
x
x
x
()
=
()
()
(
)
+
()
y
+
y
+
v uGv v
y
xy
xy
xy
xy
()
=
()
+
2
()
()
2
GEuFuv
F
y
2
y
+
Gv
y
y
y
y
y
Proof.
One again simply uses the appropriate chain rule. See [Lips69].
9.9
The Geometry of Surfaces
Our overview of the geometry of surfaces in
R
3
will start with a sketch of the histori-
cal development of a few of the really fundamental invariants that are central to any
understanding of surfaces before we develop the results rigorously using modern
terminology. It should not be surprising that the initial attempt to study the geome-
try of surfaces was by means of curves. The first such result was due to Euler and
dates from 1760.
Let
S
be a surface in
R
3
and
p
any point of
S
. Let
n
p
be a unit vector normal to
S
at
p
. For each unit vector
u
in the tangent plane to
S
at
p
, let
X
u
be the plane through
p
generated by the vectors
u
and
n
p
. Consider the curve that is the intersection of
X
u
with
S
. Such a curve is called a
normal section
of
S
at
p
. Let k
u
denote the signed cur-
vature of this curve, where the plane
X
u
is given the orientation induced by the ordered
basis (
u
,
n
p
). See Figure 9.18.
