AA/EE/ME 570: Manifolds and Geometry for Systems and Control
Homework #1
Due: Thursday January 14, 11:30am
The purpose of these problems is to recall some properties of functions and surfaces from calculus.
All problems have equal value.
1.
Enneper’s surface.
Define a surface in
R
3
by,
x
(
u, v
) =
u

u
3
3
+
uv
2
, v

v
3
3
+
vu
2
, u
2

v
2
.
(a) Using any program of your choice, make a plot of the surface. You should get the result in Fig.
1.
Figure 1: Enneper’s surface.
(b) Find the inverse mapping (
u, v
) in terms of (
x
1
, x
2
, x
3
) and the conditions under which it exists.
When such inverses exist and are continuous, a mapping is termed
onetoone
.
(c) Show that for
u
2
+
v
2
<
3, Enneper’s surface has no selfintersections (meaning that given
(
u
1
, v
1
)
6
= (
u
2
, v
2
) one must have
x
1
6
=
x
2
). Hint: use polar coordinates
u
=
r
cos(
θ
),
v
=
r
sin(
θ
),
and show that the equality (
x
1
)
2
+ (
x
2
)
2
+
4
3
(
x
3
)
2
=
1
9
r
2
(3 +
r
2
)
2
holds, where
x
1
,
x
2
,
x
3
are the
coordinate functions of Enneper’s surface in
R
3
. Then show that the equality implies that points
in the (
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 KristiA.Morgansen
 u2, Inverse function, Surface, Injective function

Click to edit the document details