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
nobo
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 spac
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 ca
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
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 o
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 i
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 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
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
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
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 parametrize
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