Lecture Notes for 120 - UCLA Department of Mathematics

6.4. THE GAUSS AND CODAZZI EQUATIONS 143
This means that the unit tangent can be calculated without reference
to the parametrization of the curve.
(b) Show that if we use this formula for the velocity, then the geodesic
curvature can be computed as
apple g =
@
@u
⇣ ⌘ ⇣
˙q · @q @
@v @v
p
det [I]
⌘
˙q · @q
@u
(c) Generalize this to the situation where a curve satisfies a differential
relation
P ˙u + Q ˙v =0
where P = P (u, v) and Q = Q (u, v).
(2) Define the Hessian of a function on a surface by
Hessf (X, Y )=I(D X rf,Y )
Show that the entries in the matrix [Hessf] defined by
Hessf (X, Y )= ⇥ X u X ⇤ apple Y v u
[Hessf]
are given as
@ 2 f
@w 1 @w 2
+ ⇥ @f
@u
@f
@v
⇤ apple u
w1w 2
v
w 1w 2
Further relate these entries to the dot products
@rf
@w 1
·
@q
@w 2
Y v
6.4. The Gauss and Codazzi Equations
Recall the Gauss formulas and Weingarten equations in combined form:
@ ⇥ @q
@w
@u 2
@q
@v
N ⇤ = ⇥ @q
@u
Taking one more derivative on both sides yields
@ 2 ⇥ @q @q
@w 1 @w
@u @v
N ⇤ ✓ @ ⇥
=
@q
2 @w
@u 1
Now using that
@ 2 ⇥ @q
@w 1 @w
@u 2
+ ⇥ @q
@u
= ⇥ @q
@u
+ ⇥ @q
@u
@q
@v
N ⇤ [D w2 ]
@q
@v
@q
@v
N ⇤◆ [D w2 ]
N ⇤ ✓ ◆
@
[D w2 ]
@w 1
@q
@v
N ⇤ [D w1 ][D w2 ]
@q
@v
@q
@v
N ⇤ @ 2 ⇥
=
@q
@w 2 @w
@u 1
N ⇤ ✓ @
@w 1
[D w2 ]
@q
@v
N ⇤
◆