THE GAUSS LEMMA
DANIEL S. FREED
I proved this in class, but perhaps a bit quickly, so I write out a proof here.
Let S E3 be a surface and p S . Fix a unit tangent vector 0 Tp S and also an orientation
of Tp S . The orientation gives a well-dened map : R T
The d operator
M427L-AP
October 22, 2001
This note discusses the algorithm for computing with d, not the theory behind it. After giving
the rules and some examples, Ill discuss the interpretation and give some problems.
The operator d acts on real-valued
DIFFERENTIAL OF A SMOOTH FUNCTION
DANIEL S. FREED
Here I give a quick informal summary of some basics about derivatives of functions of several
variables. This applies to functions of any number of variables, and with some enhancements to
functions of an
CURVES AND SURFACES
HOMEWORK 9
OTHER PROBLEM 2
We want to compute
d
dt
1 , 2
2
df(t,p) (Tp S )
.
t=0
By denition, this is equal to
d
Df(t,p) (1 ), Df(t,p) (2 )
dt
Applying the Leibnitz rule, this equals
D
d
f(t,p)
dt
t=0
(1 ), Df(0,p) (2 ) + Df(0,p) (1 ),
CURVES AND SURFACES
HOMEWORK 12
OTHER PROBLEM 1
(a) You can get a geodesic that intersects itself on a cone, as you saw in problem 9.4.1.
(b) I took being tangent to itself to mean that the geodesic hits the same point twice with the
same tangent directio
CURVES AND SURFACES
HOMEWORK 11
OTHER PROBLEM 2
Youre supposed to give an example of a complete surface and an incomplete surface. The plane E2
is complete: the geodesics are line segments, rays, and lines. Each of these can of course be extended
to a ful
HOMEWORK 8 (OR 7)
OTHER PROBLEM 2
Let : (a, b) R2 be a plane curve (although nothing would really change if we used space curves
instead). We want to dene the rst fundamental form E (t)dt2 in a way such that if you apply this
form to a vector v in the tan