Lecture Notes for 120 - UCLA Department of Mathematics

4.3. THE ABSTRACT FRAMEWORK 85
Note that r ˙✓ 2 corresponds to the centrifugal force that you feel when forced to move
in a circle.
The term ¨✓ 2ṙ ˙✓
+
r
is related to Kepler’s second law under a central force field.
In that context
2ṙ ˙✓
¨✓ + =0
r
as the force and hence acceleration is radial. This in turn implies that r 2 ˙✓ is
constant as Kepler’s law states.
The general goal will be to develop a similar set of ideas for surfaces and in
addition to find other ways of calculating [ w] that depend on the geometry of the
tangent fields.
Before generalizing we make another rather startling observation. Taking one
more derivative we obtain
@ 2 ⇥ @q
⇤
@q @ ⇥
=
@q
⇤
@q
@w 2 @w
@u @v
1 @w
@u @v [ w 1
]
✓ 2
@ ⇥
=
@q
⇤ ◆ @q
[
@w
@u @v w 1
]+ ⇥ ⇤ apple @q @q @ w1
@u @v
2 @w 2
= ⇥ @q
@u
= ⇥ @q
@u
@q
@v
@q
@v
⇤
[ w 2
][ w 1
]+ ⇥ @q
@u
⇤ ✓ [ w 2
][ w 1
]+apple @ w1
@w 2
@q
@v
⇤ apple @ w1
@w 2
◆
Switching the order of the derivatives should not change the outcome,
@ 2 ⇥ @q
⇤ ⇥
@q
@w 1 @w
@u @v = @q
⇤ ✓ ◆
@q
@ w2
[
@u @v w 1
][ w 2
]+apple
2 @w 1
but it does look different when we use w 1 = u and w 2 = v. Therefore we can
conclude that
apple apple @ u
@ v
[ v][ u]+ =[ u][ v]+
@v
@u
or
apple @ v
@u
apple @ u
@v
+[ u][ v] [ v][ u] =0.
For polar coordinates this can be verified directly:
apple apple apple
apple @ r @ ✓
0 1 0 1
= 0 1 = 1
@✓ @r
r
0 2 r
0
apple apple apple 2
0 0 0 r 0
[ r][ ✓] [ ✓][ r] =
1 1
0
r r
0
apple 0 1
=
1
r 2 0
r
1
r
0
apple 0 0
0
1
r
This means that the two matrices of functions [ u] , [ v] have some nontrivial
relations between them that are not evident from the definition.
For a surface q (u, v) in R 3 we add to the tangent vectors the unit normal
N (u, v) =
@q
@u ⇥ @q
@v
@q
@u ⇥ @q
@v