Lecture Notes for 120 - UCLA Department of Mathematics

5.1. CURVES ON SURFACES 104
We also restrict the domain to be (µ, ) 2 R =(0, 2⇡) ⇥
⇡
2 , ⇡ 2
and consider a
unit speed curve
q (s) =q (µ (s) ,
(s))
where (µ (s) , (s)) 2 R for s 2 [0,L]. The area under consideration will then be
the area bounded by this curve inside this region. We shall further assume that
the curve runs counter clockwise in this region so that S points inwards. Thus the
rotation of the curve is 2⇡ if we think of it as a planar curve on the rectangle.
We know that
2 3
sin µ
E µ = 4 cos µ 5 = 1 @q
cos @µ = 1 @q
r @µ
0
E
2
= @q
@ = 4
sin cos µ
sin sin µ
cos
3
5
form an orthonormal basis for the tangent space at every point so the unit tangent
can be written as
and thus
T = dq
ds = cos (✓ (s)) E µ +sin(✓ (s)) E
S =
sin ✓E µ + cos ✓E
The geodesic curvature can then be computed as
apple g = S · dT
✓
ds
d✓
= S ·
ds S + cos ✓ dE µ
ds +sin✓ dE ◆
ds
= d✓ sin 2 ✓E µ · dE ds
ds + cos2 ✓E · dE µ
ds
= d✓
ds + E · dE µ
ds
2
= d✓
ds + 4
= d✓ dµ
+sin
ds ds
= d✓
ds
dr
d
sin cos µ
sin sin µ
cos
dµ
ds
3
2
5 · 4
We can use Green’s theorem to conclude that
ˆ
ˆ ˆ dr dµ
d ds ds = d2 r
d 2 dµd
cos µ
sin µ
0
3
5 dµ
ds