MIDTERM EXAM 2 SOLUTIONS
Directions and rules. The exam will last 75 minutes. Open book, open notes, but no
electronic devices of any kind will be allowed, with one exception: a music player that
nobody else can hear, and whose controls you do not use dur
MIDTERM 1 SOLUTIONS
For problems 1-5: Let x be a surface in R3 , so x(u, v ) is dened for (u, v ) in some open
set D.
1. Suppose (a, b) belongs to D. What would be a natural basis for the tangent space to x
at x(a, b)?
cfw_xu , xv
2. Write down a formula
MIDTERM EXAM 1
Directions and rules. The exam will last 75 minutes. Open book, open notes, but no
electronic devices of any kind will be allowed, with one exception: a music player that
nobody else can hear, and whose controls you do not use during the ex
San Jose State University
Department of Mathematics
Fall 2010
Math 213: Advanced Dierential Geometry
Instructor: Michael Beeson, ProfBeeson at gmail.com, MH 215
Time: MW 5:30 p.m.
Prerequisite: Math 113 or instructor consent
Textbook: ONeill, Barrett, El
ANOTHER PROOF THAT STATIONARY POINTS OF THE LENGTH
FUNCTIONAL ARE GEODESICS
In this note I present (to my class) a dierent proof that shortest curves are geodesics
than the one we went over in class on Oct. 4. This proof was sketched for me in an email
fr
STATIONARY POINTS OF THE LENGTH FUNCTIONAL ARE
GEODESICS
Theorem 1. Let be a C 2 curve on a C 2 surface x; suppose is a stationary point of
the length functional with respect to C 2 variations. Then is a geodesic.
Proof. Let x be a surface dened over some
HOMEWORK PROBLEMS ABOUT THE GAUSS MAP
Composing the surface with its Gauss map, we obtain a map from the parameter domain
to the sphere. This is also often referred to as the Gauss map. The Gaussian image is
the range of the Gauss map. The Gaussian area i
FINAL EXAM
Directions and rules. The exam will last 2 hours and 15 minutes. Open book, open
notes, but no electronic devices of any kind will be allowed, with one exception: a music
player that nobody else can hear, and whose controls you do not use durin
ENNEPERS SURFACE
MICHAEL BEESON
I asked my class to compute the shape operator (or second fundamental form) of Ennepers surface directly from the denitions. The denition of Ennepers surface is
3
r cos r3 cos 3
X
u = Y = r sin + r3 sin 3
3
Z
r2 cos 2
Enne
ENNEPERS SURFACE
MICHAEL BEESON
Here are the solutions of some exercises I gave my class about Ennepers surface. The
denition of Ennepers surface is
3
r cos r3 cos 3
X
u = Y = r sin + r3 sin 3
3
Z
r2 cos 2
Ennepers curve is the curve obtained by xing a v
HOMEWORK PROBLEMS ABOUT THE CATENOID
A catenoid is a surface of revolution obtained by rotating a catenary about the Z -axis:
Z
X 2 + Y 2 = a cosh
a
We must have a > 0. This surface can be parametrized by
a cosh(Z/a) cos
u = a cosh(Z/a) sin
Z
The cateno
HOMEWORK PROBLEMS ABOUT PROJECTIONS
1. Show that the function g in the Weierstrass representation of a minimal surface is
actually the stereographic projection of the unit normal. I will spell out the problem in
more detail. The denition of stereographic