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Lecture XXII
Surfaces
Recall that an elementary region
ˆ
D
contained in
E
2
is called
convex
if for
D
, the line conecting them is contained in
ˆ
any points in
ˆ
D
.
Also recall that a
R
on
D
is called
injective
if distinct points in
ˆ
R
.
map
ˆ
D
give diﬀerent values of
Using these notions we will give a deFnition for surfaces.
Defnition 1 (Parametric expression For a surFace)
Let
O
be a fxed point
in
E
3
and let
ˆ
D
be a convex elementary region in
E
2
, where
E
2
is the Cartesian
R
(
u, v
)
be an injective continuous Function on
ˆ
plane oF coordinates
u, v
. Let
D
with values in
E
3
.
The
surface
S
is the set oF all points that have
R
(
u, v
)
as
position vector For some
(
u, v
)
∈
ˆ
D
.
±or example, in the
uv
plane we consider the rectangle 0
≤
u
≤
π
, 0
≤
v
≤
2
π
,
ˆ
ˆ
and the function
R
(
u, v
) = sin
u
cos
v
ˆ
i
+sin
u
sin
vj
+cos
uk
. Then the surface
S
that has parametric representation given by
R
is the sphere of radius 1 centered
D
to be the disk of radius 1 centered at
O
, and
at
O
. Also, if we take
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 Two '04
 Duorg
 Math, Cone

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