LECTURE NOTES: THEOREM OF TURNING ANGLE OF A
CURVE ON A SURFACE
WEIYONG HE
1. Angle between two vector fields
Let x : U R3 be a smooth regular surface patch with a xed orientation (given
normal vector n). And U is an open set in R2 which is homeomorphic t
DIFFERENTIAL GEOMETRY HOMEWORK 1
BRANDON HEPOLA
Problem 1.2:
(a) Which of the following are regular curves.
Before we start, note that each of the component functions in each case are smooth,
so we need only check that the velocity vectors of each vanish
REVIEW OF GEOMETRY OF CURVES AND SURFACES
WEIYONG HE
1. Curves: local theory
(1) A parametrized curve : I Rn is regular if (t) = 0 for all t I (we
only consider n = 2, 3).
(2) Review of basic notions from multivariable calculus:
(a) Dierentiability of vec
DIFFERENTIAL GEOMETRY HOMEWORK 2
BRANDON HEPOLA
Problem 3.2: Show that
(s) =
(1 + s)3/2 (1 s)3/2 s
,
,
3
3
2
is a unit speed curve and compute its Frenet apparatus.
We have
(s) =
(1 + s)1/2 (1 s)1/2 1
,
,
2
2
2
and
1+s 1s 1
+
+ =1
4
4
2
so that is unit s